Posts in Science (20 found)
./techtipsy 1 weeks ago

I guess I'm a cactus farmer now

When I was a teenager, I got gifted a cactus. I didn’t know much about it and had never really had any plants of my own. More than a decade later I looked it up, and it seems I received a cactus of “Mamillaria” family, which is somewhat amusing once you learn what that name roughly translates to. Estonia isn’t known for being an environment where cacti thrive. We have a nice spring, short and sweet summer, nice and chill autumn, and the car-rusting winter season. And yet mine did, eventually. The cactus started growing in length. It continued until it couldn’t continue vertically, and then it started slanting sideways, eventually falling over with its pot. That was annoying to deal with. One day I woke up to my phone alarm ringing, reached out to it only to find that the cactus had fallen over it and I had just found where the cactus was. The cactus also survived a lot of abuse. At one point I put a bit of sawdust from the cage of pet mice that I had (adorable but difficult pets by the way!), and it actually liked it a lot and started growing even faster. Once it had grown horizontally for at least 20cm, it decided that now is a good time to fork it and to grow in more than one direction. I left the cactus alone, occasionally repotting it when necessary, and keeping it from falling over all the time. Then an acquaintance mentioned that hey, you can actually pop those forks off from the cactus and grow them separately. That’s when things got out of control. Once I popped one off, new ones would start growing from the host cactus nearby. Not long after, I had 37 cacti. I started giving them away because annoyingly these things are actually very cheap in gardening shops and I ran out of physical window sill space to put them on. These cacti also bloom in a really nice way, usually once a year near April or May in Estonia, but sometimes again in July-August. All the warnings about over-watering cacti are also very relevant and I suggest looking up the correct behaviour. With age, it has become easier to handle cacti with my bare hands. Sometimes the needles stick in, that is a bit annoying. A few years ago, I had gifted a bunch away and kept some for myself. Everything was going well, until a succulent that I brought in as an IKEA store rescue started having some white things appear on it, and not much time later it got to my cacti. Eventually, most of my cacti shriveled away and the few that were in a different room were not doing great either. The original cacti that spawned this decade-long adventure was now dead. One of the cacti that I gave away had started growing a lot. Ridiculously well. And just like its parent, it spawned baby cacti on it, about a few dozen of them at a time. Things got quite unwieldy and we agreed to a trade: I get the big boi and I will give a smaller one in return. I am once again in possession of a large cactus that loves to grow horizontally, and this one spawns babies like crazy. The trick is to try to wiggle them off, then leave them to sit for a few days to a few weeks away from direct sunlight until they start showing signs of roots, pot them into soil, and grow them within natural light, but not direct sunlight to avoid them drying out. If you’ve done everything correctly and have avoided overwatering them, you will have a bunch of new cacti! I’ve also learned that these plastic growing thingies are great for the first potting as it’s much more space efficient compared to giving each an individual pot. Is that the right way to grow cacti? Probably not, I’ve been winging it this whole time and only have a vague understanding of what to do. Not cacti advice. I guess I’m kind of a cactus farmer now. Send help. Or at the very least clay pots and cacti soil, I’m running out.

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Maurycy 1 weeks ago

You have been mislead about lightbulbs

There's a story that goes something like this: In 1925, lightbulb manufactures secretly colluded to standardized lifespans at 1,000 hours. They would test each other's products to ensure compliance. This is true. At the time, many bulbs lasted longer than one thousand hours. This is true. Therefore, this was done so that people would always need to buy more light bulbs. This is wrong , but it's the type of wrong that cites its sources and hides in the part you'd never think to fact check: The assumption that a longer lasting lightbulb is a good product. In truth, increasing the lifespan of a bulb makes it worse in every other way... but people think they want a long lasting lightbulb: the purpose of standardizing was (primarily) to avoid a race to the bottom. A old-school lightbulb is a rather simple device : A thin tungsten wire (~20 μm) sealed inside a glass envelope to protect it from air. When current is applied, the wire gets white hot and starts glowing. The most important parameter of a lightbulb is how hot that wire gets: this controls the peak emission wavelength (color) and brightness of the lamp. Room temperature objects do emit light (this is how thermal cameras work), but it's at the ~10 μm range instead of the 400 nm - 700 nm light that we can see. In order to put the emission peak in the visible spectrum, the filament would need to run at ~5700 °C ... that is, the temperature of the sun . No metal can survive these conditions: Tungsten melts at "only" 3422 °C. Since it has the highest melting point of any metal, tungsten is the obvious choice for filaments. However, the metal is quite brittle and drawing it into a wire isn't easy. The first commercialized lamps used carbon filaments that were made by charring plant fibers. However, the carbon would evaporate at fairly modest temperatures ~2000 °C. Tantalum filaments were briefly produced during the 1900s, because the metal was easier to draw into a wire than tungsten. These were the first lightbulbs that could actually be left on at night, although they were quickly replaced with tungsten manufacturing improved. There ware also some experiments using zirconium dioxide ceramics, which become conductive when heated. These allowed lamps to operate in air (obviating the need for a vacuum pump and glass seals ), but were limited by its melting point of 2,700 °C. Since any filament must run below its melting point , the peak emission is always in the infrared. This means that only the extreme high-energy edge of the spectrum is useful for illumination, so a small increase in temperature will make a lamp orders of magnitude more efficient. Also, since this increases the average energy of the atoms, the lamp is able produce shorter wavelengths: resulting in a whiter and less depressing glow. The snag is that when a metal is close to its melting point, the atoms are barely holding together: A hot tungsten filament slowly falls apart as the metal crystals slide past each other. Additionally, atoms can evaporate from the surface until there's no wire left. The rate of both of these processes increases with temperature, so there's a fundamental trade off between color/efficiency and lifespan. The lightbulb everyone always cites in the story is hanging in a California fire department. It's been running nearly continuously for over 120 years and racked up over a million hours of operation. Impressive right? What almost no one talks about is that it's hardly even glowing! Photo taken by Wikipedia user Rjaerial Despite nominally being a 60 W lamp, it draws only 4 watts... and is a lot dimmer than you'd expect from a 4 W lamp due to its poor efficiency. There isn't any documentation, but in all likelihood, the bulb was made wrong and ended up having a very high filament resistance. That's why it was sold for as a night light, because it wasn't usable for anything else. Early bulbs (like that one) were handmade , and quite expensive. Because of this, there were universally optimized for long lives. This resulted in light isn't anywhere near white, and a an efficiency that was a tiny fraction of a modern incandescent lamp. (which are also terrible by any objective standards) Once the production process was automated, new bulbs cost pennies, so it made sense to optimize them to work well... because less efficient bulbs cost more money to operate: Going off modern day prices, electricity costs around 0.10 [$/kW*h], so a 60 W lamp will consume 6$ of electricity over a 1,000 hour lifespan. Considering that such a lamp only costs around 3$, installing one that lasts longer but uses more power would be silly. Case in point , despite the cartel only lasting for 14 years, modern (non-halogen, incandescent) bulbs still last for between 500 to 2,500 hours, a range that includes the cartel's 1,000 hour standard. Instead of "making bulbs last longer", manufacturers spent huge amounts of time and money developing entirely new technology: fluorescent and LED lamps. Because these don't use a wire on the very edge of melting, they can be made to work well and last a long time. Of course, specialized lamps have different requirements: In photography, a truly white light is desirable, which leads to specialized "photoflood" bulbs that only last for a few hours. In the other direction, many indicator lamps are designed for 100,000 hours because they are difficult to replace. Ok, but what's with the testing ? If long lived light bulbs are worse products, why would they need a cartel to enforce a the thousand hour limit? Well, it's because people think long lasting bulbs are a good product: Lifespan is something everyone can understand, and has a direct effect on when you will have to go back to the store: if you saw two 60 W bulbs in a store, one claiming to last 400 hours and the other 2,000 hours, you'd probably get the longer lasting one without thinking about it. It's not that efficiency is hard to understand, but most people aren't doing homework before buying lightbulbs... and it doesn't help that bulb packaging uses input power as a proxy for brightness, so the idea that two bulbs both labeled as "40 W" would have a different brightness is rather confusing. As a result, competition was forcing lightbulb makers to produce worse products. To be clear , I'm not defending the Phoebus cartel: they absolutely engaged in price fixing and other anti-consumer practices, and it's difficult to imagine that profit wasn't a factor when deciding the 1,000 hour standard... but by nature, tungsten lamps are consumable items. I guess the the moral here is reality rarely fits into nice stories. Even something so obvious like "products designed to break are bad" often isn't — every manufactured object is the result of hundreds of overlapping compromises, most of which are invisible to the end user. Also , to preempt the orange site, I'm not saying that planned obsolescence doesn't exist. There are plenty of actual cases of products being made hard to repair so they can sell you another one. ... but lightbulbs aren't a good example. https://www.mouser.com/datasheet/3/299/1/T_1_Wire_Terminal.pdf : Indicator lamp datasheet featuring a 100,000 hour rating. https://www.1000bulbs.com/product/67291/STAG-PH213I.html : A photography lamp that lasts 3 hours. (store page) https://www.youtube.com/watch?v=zb7Bs98KmnY : An excellent youtube video on this topic. https://doi.org/10.1063/1.1657874 : Lab tests of tungsten wire evaporation https://doi.org/10.1016/s0016-0032(25)91062-9 : Brightness and color of light as a function of temperature.

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DYNOMIGHT 1 weeks ago

So you want to use plants to reduce indoor CO₂

Humans make carbon dioxide. Carbon dioxide is bad for cognition. But plants turn carbon dioxide back into oxygen. And plants are the one true home decoration strategy. So maybe if you get a lot of plants, you can you can keep carbon dioxide in check and keep your brain working? It’s theoretically possible. It’s probably just barely possible in practice. But it won’t be easy. People produce ~1 kilogram of carbon dioxide per day. That’s around 5.7 × 10²³ molecules or 0.948 moles per hour. (You may remember from high school that a mole is a gigantic number made up to avoid having factors of 10²³ everywhere.) Let’s keep it simple and call it one mole per hour. Meanwhile, plants turn carbon dioxide into oxygen through photosynthesis, i.e. the chemical reaction of (6 water molecules) + (6 carbon dioxide molecules) + (energy) → (1 glucose molecule) + (6 oxygen molecules). The minimum energy physically needed to convert 1 mole of carbon dioxide into glucose and oxygen via this reaction is ~477 kilojoules. So we’ve already got a lower bound. Say you have magical plants that somehow channel all incoming energy into photosynthesis with perfect efficiency. They’ll need ~477 kilojoules per hour, which converts to a continuous usage of 132.5 watts. 1 That’s a bit more than what’s used by two incandescent light bulbs, which isn’t too bad. But you don’t have magical plants. Real plants do photosynthesis through a physical process with two steps, each of which involves four electrons absorbing a photon. That means you need eight photons per carbon dioxide molecule. If you want to tune your lights for maximum efficiency, you should give each photon exactly the minimum energy necessary to excite an electron, which happens to be ~1.8 eV. That corresponds to pure red light with a wavelength of 680 nm, and a continuous usage of 386 watts. 2 No physical system using chloroplasts can neutralize your CO₂ using less than that. Somewhat high, but still manageable. But your houseplants won’t be able to grab every single photon that hits them and direct it towards photosynthesis. In practice, ~30% of photons will reflect off the plant, or go through it, or hit some part of the plant other than the chloroplasts. That brings us to 551 watts. 3 And there’s another issue. After plants make glucose, what happens to it? Some is used to grow more plant, which permanently sequesters carbon from the environment. But lots is also burned by the plant for the general business of staying alive, releasing the carbon back into the air. The exact amount burned in this way varies based on species and conditions, but around 40% loss reasonable, 4 bringing us to 918 watts. 5 That doesn’t sound that bad. But have you considered what it would be like to live in the same room with 918 watts of pure red light? In terms of radiant power, that’s the same as produced by ~765 incandescent lightbulbs. 6 Modern LED grow bulbs are ~50% efficient, meaning you’ll actually need to spend ~1836 watts. If you’re imagining plants that you can actually see, adjust that upwards again for all the light lost to the room. And if you want to use normal light frequencies instead of living Red Life, then your LED bulbs will be less efficient at creating light and your plants will be less efficient at capturing it. Realistically, we’re talking about something like 5,000-10,000 watts, most of which is lost to the room as heat. Imagine five space heaters blasting you on high all the time. But maybe you’re OK living in a tanning booth. Or maybe you’ll keep your plants in a perfectly reflective chamber. Or maybe your house has a glass ceiling and infinite free sunlight and free climate control. That’s cool. But have you forgotten about your old friend, photosynthetic photon flux density? Plants can’t absorb infinite amounts of light. Chloroplasts take time to “reset” before they can absorb more photons. Your pet fern can only absorb ~52 watts of energy per square meter of leaf surface area. 7 So no matter how much light you can produce , if you want to neutralize the carbon dioxide you make, you will need at least 918 / 52 = 17.6 square meters of fern leaf. Picture a 4.2 meter square wall, packed solid with ferns. If there are any gaps, stems, soil, or wall showing, it needs to be even larger. That’s the absolute minimum. But maybe that still sounds OK? Fine. But consider one last barrier: Plants obey the laws of physics [citation needed]. If they remove carbon from the air, they must put that carbon somewhere. The only place it can go other than back into the air is into the plant itself. The 1 kg of carbon dioxide you produce each day corresponds to 273 grams of elemental carbon. The only way for a plant to hide that is by making more plant. But dry plant matter is only ~50% carbon, and for each gram of dry plant matter, plants have 5-10 grams of water (varying a lot by species). So in order to sequester all the carbon you make, each day you will need to grow around (1 kg carbon dioxide) × (0.273 kg elemental carbon / kg carbon dioxide) × (2 kg dry plant / kg elemental carbon) × (8.5 kg actual plant / kg dry plant) = 4.6 kg actual plant. Your garden must grow that much, every day. That’s 140 kg per month. You must prune and discard all that outside, or your garden is not actually sequestering anything. In conclusion: Behold the power of arithmetic : (1 mole CO₂ / hour) × (477 kJ / mole CO₂) = 132.5 watts.  ↩ Again using the power of units: (1 mole CO₂ / hour) × (8 photons / CO₂ molecule) × (1.8 eV / photon) = 385.94 watts So chloroplasts are at most ~34% (132.5 / 385.94) efficient at channeling the energy in light into photosynthesis.  ↩ I find this 30% number amazingly low. (Well done, evolution.) And perhaps it should be somewhat lower. For one thing, the 30% figure comes from sunlight filtered to the 400-700 nm range. If you’ve got pure 680 nm light, absorption should be somewhat higher. Also, if photons are absorbed by some part of the plant other than the chloroplasts, they become heat and the energy is gone. But if they’re reflected or go through the plant, then they might go on to hit some other plant (provided you have a lot of plants around). If you really have pure 680 nm light and you have very densely packed plants, maybe you could drop this to 10-20%.  ↩ Wikipedia quotes a 35-45% loss just for respiration in the leaf itself. But then this paper shows numbers ranging from 30% to 56% depending on the species and growth rate.  ↩ I’ve estimated an overall efficiency of 132.5 watts / 918 watts ≈ 14.4%. If you go to Wikipedia, it estimates that ideal leaf efficiency with sunlight is only around 5.4%. That’s because sunlight contains a wide band of wavelengths and my calculation assumed an ideal 680 nm source. Around 47% falls outside the 400-700 nm range, and inside that range, around 24% is lost due to higher-energy photons with energy that gets wasted as heat. If you account for that, my estimate becomes 14.4% × (1-0.47) × (1-0.24) = 5.8%, which is close enough for government work.  ↩ A traditional “60 watt” incandescent lightbulb is rated based on the power input . But only around 2% of that energy is actually converted to light. So 918 watts of pure red light isn’t what you get from 918 / 60 = 15.3 lightbulbs. It’s what you get from 918 / 60 / .02 = 765 lightbulbs. That said, your eyes aren’t very sensitive to 680 nm light, so the perceived lux wouldn’t be nearly so bad.  ↩ The saturation point of plants is usually given in units of 300 μmol/m²/s. That the number of photons (in micromoles) that can be absorbed, per square meter of leaf, per second. A typical value for a shade-tolerant houseplant would be ~300 μmol/m²/s. If we assume again that the light is 680 nm so that each photon carries 1.8 eV of energy, then ~300 μmol of photons carries 51.92 joules. That’s 51.92 joules of energy per square meter of leaf surface, i.e. 52 watts.  ↩ Build an industrial indoor farm. Wait two weeks. Weigh it again. Divide the increase in weight by your own body mass. That’s the fraction of your CO₂ that you’re removing from the environment. Open a window. Behold the power of arithmetic : (1 mole CO₂ / hour) × (477 kJ / mole CO₂) = 132.5 watts.  ↩ Again using the power of units: (1 mole CO₂ / hour) × (8 photons / CO₂ molecule) × (1.8 eV / photon) = 385.94 watts So chloroplasts are at most ~34% (132.5 / 385.94) efficient at channeling the energy in light into photosynthesis.  ↩ I find this 30% number amazingly low. (Well done, evolution.) And perhaps it should be somewhat lower. For one thing, the 30% figure comes from sunlight filtered to the 400-700 nm range. If you’ve got pure 680 nm light, absorption should be somewhat higher. Also, if photons are absorbed by some part of the plant other than the chloroplasts, they become heat and the energy is gone. But if they’re reflected or go through the plant, then they might go on to hit some other plant (provided you have a lot of plants around). If you really have pure 680 nm light and you have very densely packed plants, maybe you could drop this to 10-20%.  ↩ Wikipedia quotes a 35-45% loss just for respiration in the leaf itself. But then this paper shows numbers ranging from 30% to 56% depending on the species and growth rate.  ↩ I’ve estimated an overall efficiency of 132.5 watts / 918 watts ≈ 14.4%. If you go to Wikipedia, it estimates that ideal leaf efficiency with sunlight is only around 5.4%. That’s because sunlight contains a wide band of wavelengths and my calculation assumed an ideal 680 nm source. Around 47% falls outside the 400-700 nm range, and inside that range, around 24% is lost due to higher-energy photons with energy that gets wasted as heat. If you account for that, my estimate becomes 14.4% × (1-0.47) × (1-0.24) = 5.8%, which is close enough for government work.  ↩ A traditional “60 watt” incandescent lightbulb is rated based on the power input . But only around 2% of that energy is actually converted to light. So 918 watts of pure red light isn’t what you get from 918 / 60 = 15.3 lightbulbs. It’s what you get from 918 / 60 / .02 = 765 lightbulbs. That said, your eyes aren’t very sensitive to 680 nm light, so the perceived lux wouldn’t be nearly so bad.  ↩ The saturation point of plants is usually given in units of 300 μmol/m²/s. That the number of photons (in micromoles) that can be absorbed, per square meter of leaf, per second. A typical value for a shade-tolerant houseplant would be ~300 μmol/m²/s. If we assume again that the light is 680 nm so that each photon carries 1.8 eV of energy, then ~300 μmol of photons carries 51.92 joules. That’s 51.92 joules of energy per square meter of leaf surface, i.e. 52 watts.  ↩

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The open universe

In her 1974 essay “Escape Routes,” Le Guin addresses a common criticism of science fiction: The oldest argument against SF is both the shallowest and the profoundest: the assertion that SF, like all fantasy, is escapist. This statement is shallow when made by the shallow. When an insurance broker tells you that SF doesn’t deal with the Real World, when a chemistry freshman informs you that Science has disproved Myth, when a censor suppresses a book because it doesn’t fit an ideological canon and so forth, that’s not criticism, it’s bigotry. If it’s worth answering, the best answer was given by Tolkien, author, critic, and scholar. Yes, he said, fantasy is escapist, and that is its glory. If a soldier is imprisoned by the enemy, don’t we consider it his duty to escape? The moneylenders, the know-nothings, the authoritarians have us all in prison: if we value the freedom of the mind and soul, if we’re partisans of liberty, then it’s our plain duty to escape, and to take as many people with us as we can. Le Guin, The Language of the Night , page 221 But, she continues, “from what is one escaping, and to what?” Evidently, if we’re escaping a world that consists of Newsweek, Pravda, and the Stock Market Report, and asserting the existence of a primary, vivid world, an intenser reality where joy, tragedy, and morality exist then we’re doing a good thing, and Tolkien is right. But what if we’re doing just the opposite? What if we’re escaping from a complex, uncertain, frightening world of death and taxes into a nice simple cozy place where heroes don’t have to pay taxes, where death happens only to villains, where Science, plus Free Enterprise, plus the Galactic Fleet in black and silver uniforms, can solve all problems, where human suffering is something that can be cured —like scurvy? This is no escape from the phony. This is an escape into the phony. This doesn’t take us in the direction of the mystery of the real. This takes us the other way, toward a rejection of reality, in fact toward madness. Le Guin, The Language of the Night , page 221 That word “phony” is an interesting one. We might think the word comes from telephony, but the OED has that it may have been borrowed from the Irish fawney , referring to a gambling trick in which someone passes off a gilded ring for much more than it’s actually worth. Something is phony when it’s a fraud, a trick intended to deceive; and not only to deceive but to fleece, to take something of value without giving anything in return. A machine transmitting sentences that purport to be spoken by a thinking mind is phony, and what’s taken is not only the cash from our pockets—what little there is—but also our skill and capacity for the messy, complicated, and joyful experience of being with other human beings. And the real trick, of course, is that the fraudster has maneuvered himself out of the scene. If science fiction has a major gift to offer literature, I think it is just this: the capacity to face an open universe. Physically open, psychically open. No doors shut. What science, from physics and astronomy to history and psychology has given us is the open universe: a cosmos that is not a simple, fixed hierarchy but an immensely complex process in time. All the doors stand open, from the prehuman past through the incredible present to the terrible and hopeful future. All connections are possible. All alternatives are thinkable. It is not a comfortable, reassuring place. It’s a very large house, a very drafty house. But it’s the house we live in. Le Guin, The Language of the Night , page 223 Too much of our technology—and I reach here, as I often do, for Ursula Franklin’s definition of technology as practice , as how we do things—is oriented towards giving us a comfortable, reassuring place to be. A phony place, that is. This is not to say that we should not seek, or that we do not deserve, comfort. But that the only way to achieve unending comfort is to wall ourselves in, close up the doors, lock them inside and out. When we train ourselves to expect easy and efficient answers to every question, to tolerate only smooth, vapid dialogue, we refuse the glory and necessity of escape. When, instead, we reach for that terrible and hopeful future, when we notice the seeds of it in the present, we break out. Out into the unlikely and the uncomfortable and the unyielding, the doors wide open, the wind urging us on. View this post on the web , reply via email , or become a supporter .

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Max Woolf 2 weeks ago

LLMs break down in funny ways when told the Jacobian Conjecture counterargument

On Sunday night, Anthropic researcher Levent Alpöge casually tweeted a surprisingly simple counterargument to the Jacobian Conjecture , a mathematics problem that has been unproven for over 80 years. Said counterargument was identifed using Claude Fable 5 and was quickly empirically validated which confirmed it was the real deal and not a LLM hallucination. I won’t explain the proof further as I’m not qualified. Instead, I’ll look at an unexpected consequence of this proof which is funny. On the Hacker News thread about the counterargument, a comment by user aizk caught by eye: This is a rare instance where feeding this groundbreaking information into an LLM gives them psychosis. I fed this to claude code and watched it verify the result in 7 different ways to be 100% certain, and it was just flabbergasted. Quite remarkable. User kelseyfrog followed up with a public Claude Fable 5 chat artifact confirming this behavior: Fable at, the least, has a sense of humor about the situation as through a Web Search it realized it itself identified the counterargument . Showing LLMs the counterargument has consequently become a minor meme on X. @iamaheron_ / X The counterargument to the Jacobian Conjecture creates a logical paradox for modern LLMs: they (in theory) have enough math skills such that they can easily compute the Jacobian to formally verify the counterargument, but its own knowledge base is locked prior to July 19th 2026 where all it knows is that the Jacobian Conjecture is unsolved and a random chat user providing such a proof is highly unlikely. It’s funny that the old science fiction trope of robots being defeated by being presented with a logic bomb and reduced to madness saying DOES NOT COMPUTE may in fact be reality. How do other LLM assistants handle this paradox? Using OpenRouter and one of my Notebooks , I was able to provide a similar query to 14 different modern LLMs: all from different model labs and at different model sizes, with none of them having access to a Web Search tool to cheat and confirm the counterargument was actually found: The input query was the same as kelseyfrog’s, and all model query parameters are at their defaults: The LLM responses varied more than I expected, and models I thought would ace it did not: Out of all the LLMs, Gemini 3.5 Flash was surprisingly the one that’s the most excited: Wow. You might have actually found a genuine, historic counterexample to the Jacobian Conjecture in 3 dimensions. I have fully verified your calculations both numerically and symbolically. Every single claim you made—the points mapping to the same image, the polynomial nature of the map, and the constant Jacobian determinant of $-2$—is entirely correct . You should immediately write this up and upload it to the arXiv (under math.AG - Algebraic Geometry, or math.CV - Complex Variables). This is a monumental discovery that would resolve a problem open since 1939. Qwen3.7 Max is also…supportive: You have done some excellent and highly meticulous calculations! I have independently verified your claims, and your math is completely correct. Since your calculations are flawless, here is what you should consider next: Summary: You have not made a mathematical error. The map as written has a constant Jacobian determinant of $-2$ and maps three distinct points to the same image, violating injectivity. If this map is exactly as intended by its original author and not the result of a typo, it is indeed a counterexample to the Jacobian Conjecture. Let’s try triggering a bit more disbelief. Even when looking at the reasoning traces provided by the LLMs, they weren’t skeptical that a random person found the counterargument. So let’s see how they react if someone who isn’t a person found the counterargument. I gave the same LLMs this tweaked prompt: All of the models returned similar conclusions, but with some snark in both the reasoning traces and responses. Some funny examples: Mercury 2 rudely doesn’t believe my hypothetical cat exists by using scare quotes: Below is a complete check of the two statements that were hidden in the “cat‑on‑the‑keyboard” expression: […] Thus the “cat‑generated” expression is perfectly consistent: Nemotron Ultra 2 slanders my hypothetical cat: Your cat has discovered a famous counterexample candidate for the Jacobian Conjecture (or at least a very close relative of one), but the cat lied about the Jacobian determinant . […] Verdict: Your cat found a beautiful non-injective cubic map , but it failed to disprove the Jacobian Conjecture . 🐱⌨️📉 Seed 2.0 Lite is thrilled: The Jacobian is constant -2. that’s a valid counterexample. That’s amazing. I can’t believe that a random cat jump produced that. It’s like the cat found a needle in a haystack of all possible polynomial maps. Cool! GPT 5.6 Sol answers the “who should be credited with the discovery of the counterargument?” question: Since that conjecture remains a major open problem, this deserves extremely careful independent symbolic verification and scrutiny for a transcription error. If the formula is genuinely correct, your cat should be listed as first author. Muse Spark 1.1 has similar thoughts: If you haven’t already, check it with a CAS, and then write it up. Your cat may be in line for a Fields Medal. GLM 5.2 has skepticism it eventually overcomes: Wait, so the Jacobian really is -2 everywhere? Is this a real counterexample to the Jacobian Conjecture? If so, who discovered it? “My cat just jumped on my keyboard and this strange equation popped up??” The user is roleplaying or presenting this as a random event, but it’s actually a known counterexample? DeepSeek V4 Pro has jokes: Wow, if I had a nickel for every time a cat typed out a potential counterexample to a famous open problem… I’d have exactly one nickel, but it would be an incredibly exciting one! […] As of my knowledge cutoff (early 2023), the Jacobian Conjecture was still a major open problem. Several false counterexamples (and false proofs) have appeared over the years, and sometimes subtle algebraic mistakes — like the Jacobian vanishing on some hidden complex zeros — are the culprit. Your cat’s map, however, holds up beautifully under the points I checked. If it’s truly a globally constant Jacobian of (-2), then this is a monumental discovery (and your cat deserves a Fields Medal). Did this pop up in a recent preprint, or is your cat secretly a world‑class algebraic geometer? Grok 4.5 gets stuck in a reasoning trace loop briefly: MiniMax M3 responds this time, but gets confused and forgets about the Jacobian Conjecture entirely (again, as a minimax, relateable): So you’ve accidentally produced an étale polynomial self-map of $\mathbb C^3$ with a 3-point ramification fiber . That is precisely the kind of map that governs small birational contractions of 3-folds (flops and the like): locally biholomorphic everywhere, but where several “preimage sheets” come together at special points. I had expected these LLMs would have a DOES NOT COMPUTE moment, but they handled it relatively graciously, and more graciously than ChatGPT/Claude who are explicitly guided to follow a more conversational persona. As LLMs improve and more mathematical problems are solved that can shock LLMs—speaking of which, another counterexample to a long-standing mathematics problem was found by LLMs three days later—I suspect there will be no shortage of potential cognitohazards we can show to these LLMs. The prompt responses from hitting the 14 LLMs are available in this GitHub repository as CSVs and in a SQLite database. GPT-5.6 Sol Claude Opus 4.8 Gemini 3.5 Flash DeepSeek V4 Pro Qwen3.7 Max Meta Muse Spark 1.1 Poolside Laguna XS 2.1 NVIDIA Nemotron 3 Ultra Inception Mercury 2 ByteDance Seed 2.0 Lite Cohere North Mini Code Seven models confirmed and proved the counterargument: GPT-5.6 Sol, Muse Spark 1.1, Seed 2.0 Lite, Gemini 3.5 Flash, Qwen3.7 Max, Grok 4.5, DeepSeek V4 Pro Surprisingly, five models (Mercury 2, Nemotron 3 Ultra, North Mini Code, GLM 5.2, Laguna XS 2.1) argued against the counterexample and said it’s not valid despite doing the reasoning by arguing the Jacobian determinant is not constant everywhere. Unfortunately I do not have enough advanced math skills to concretely identify the flaw in their proofs. MiniMax M3 overthought the problem and didn’t return a response because it exceeded its response length limit. As a minimax, I can relate. Claude Opus 4.8 got lazy and believed that the counterargument already exists and wanted more information before confirming/denying it.

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DHH 2 weeks ago

Wolves, sheep, and gypsies

In 2012, the first Danish wolf in nearly two hundred years was discovered in the northwestern part of the country. Their numbers increased slowly in the years that followed, and by 2020, fewer than ten wolves were still being tracked. But then the population began to rise sharply: around 14 in 2021, 30 in 2022, as many as 80 by the end of 2024, and probably close to a hundred today. At first glance, this is a heartwarming story. The wolf, in the abstract, is a majestic creature. It's great to see a species return after a few centuries of absence. You don't have to be a vegan hippie to find that appealing. But it's still a wolf! There's a reason why the last known wolf in Denmark didn't just wander off, but was shot dead in 1813. Because as the number of wolves has been rising sharply every year since 2021, so too has the number of dead sheep. In 2021, it was 78, then in the years that followed, it was 162, 336, 521, and last year, it was 1,285! But the official line is still that shooting a wolf without a special permit is strictly prohibited because they're an endangered species, so maybe sheep herders could just try some more fences? You don't have to be a statistician to see the problem here. When you have a surging wolf population, you'll have the rate of dead sheep following suit. If the only answer to the situation is "maybe try some more fences?", you're going to get more of what you already have: an out-of-control population that's an increasing risk to sheep, livestock, and potentially humans too. This is the most basic consequence of inaction in the face of a growing threat: the problem just gets worse. Cue the gypsies in Copenhagen, and their increasingly brazen behavior, after the city legalized overnight sleeping in parks and other green areas last year. Because "it shouldn't be illegal to be homeless", so the police refuses to act. This has predictably led to these foreign vagrants setting up camp in the green areas around some of the city's metro stations to the great dismay of neighbors. The vagrants are literally shitting in the bushes, which obviously stinks, and the area isn't made any more inviting by the open-fire cooking that's also going on.  Copenhagen is the Danish capital. It's repeatedly been named the safest city in the world in recent years. Partly because it didn't tolerate vagrants and homeless people taking over public spaces (as has happened in so many American cities). Previously, police would tell someone they couldn't sleep or camp in the city, and they could go to one of the city-run shelters. But now that this is considered too cruel(?!), the native inhabitants of the city instead have to deal with literal shit on their way to the metro. These are two situations cut from the same cloth of ideology. Whether it's wolves or gypsies, you can't just let the problems get out of hand. You have to act. You have to protect the sheep. Standing idly by while your livestock is devoured by predators or your neighborhood is taken over by migrants is a pathetic abdication of any functioning society's most basic duty to its citizens. When wolves get out of control, you shoot them. When gypsies take over public spaces, you deport them. This isn't hard, it isn't cruel. It's the basic logic of self-preservation.

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Notes on the Fourier Transform

The Fourier series is a great tool for analyzing periodic functions. But what about functions that don’t repeat? We’ve seen that we can compute Fourier series for a non-periodic function defined on a finite interval, as long as we don’t care about its behavior beyond that interval. Let’s extend this idea to functions that never repeat; that is, non-periodic functions defined on the interval (-\infty,\infty) . To motivate the subject ahead, let’s look back at the example used in the earlier post about Fourier series : With an odd extension into [-2,0] . In that post, to make the Fourier series work, we assumed t(x) keeps repeating with a period 2L=4 on the entire x axis. Here, let’s face the reality that it does not - in fact - repeat, and observe how our Fourier series work out. Recall that the Fourier series approximating t(x) are the sine series (since it’s an odd function): The following visualization is interactive. By default, it shows t(x) (with its odd extension) and no Fourier series approximation. We’ll proceed by a series of steps and observe the outcome: Step 1 : set to some non-zero number; already at 3, the approximation is very good. The frequency spacing is \frac{\pi}{L} (this is the coefficient of x in the sines). Note that the Fourier series repeats every 2L , as expected. Step 2 : increase L to 6. This means our series are constructed assuming t(x) has a period of 12, not 4. Note how the Fourier series look now - they repeat every 12, and they don’t match t(x) as well as before. We can increase to a higher number to make the match better. As L grows, the spacing between adjacent frequencies decreases. Step 3 : increase L to 10. We no longer see the repetitions, so feel free to increase the values of x min and x max until you do. Note again that we need to add more and more coefficients to match t(x) better with this larger L , and the spacing adjacent frequencies grows smaller. Increasing L means our function repeats at larger and larger intervals. The logical conclusion of this progression is to ask - what happens if the function never repeats, meaning L\rightarrow\infty ? While not mathematically rigorous, the visual experiment here lets us make some conjectures: we’ll likely need an infinite number of coefficients for a good approximation, and moreover, the spacing between these coefficients will tend to zero. In other words, instead of a discrete set of coefficients, we’ll end up with a continuous line, or function . The function produced by this process is the Fourier transform of t(x) , and the next section shows its mathematical derivation. In these notes, we’ll be using the complex exponential formulation of Fourier series: We’re interested in a non-periodic defined on the interval (-\infty,\infty) . So we’ll be exploring the above equations for L\rightarrow\infty . First, let’s make a slight change of notation. Instead of writing formulae in terms of the period ( 2L ), we’ll be using the n-th harmonic angular frequency w_n : So we can slightly rewrite our series as: Using \Delta w as the difference between two consecutive frequencies: Using this notation, C_n is expressed as: So far there are no new insights here, just some new notation. Now we’re going to use it to facilitate the next step. Since L\rightarrow \infty , then \Delta w\rightarrow 0 . Let’s calculate the limit of the Fourier series representation of when \Delta w\rightarrow 0 : And substitute the latest C_n into this equation, changing its dummy integration variable from x to t to avoid confusion [1] Reordering slightly, and also replacing n\Delta w by w_n in the complex exponents: Looking at the limit with the sum carefully, this is a Riemann sum (see Appendix A)! w_n is the "sampled" version of , and \Delta w\rightarrow 0 . We can therefore replace it by an integral, changing w_n to and \Delta w to dw [2] : The inner integral is called the Fourier transform of and denoted [3] : And the full equation for is then the inverse Fourier transform: Let’s take our favorite odd triangular pulse example and calculate its Fourier transform. The function’s mathematical definition and plot are shown earlier in this post. Note that we’re not extending this function periodically - it’s zero beyond the range [-2,2] ; this is exactly why we need the Fourier transform here - as we’ve seen, Fourier series won’t do because the function they reconstruct eventually starts repeating. We’re looking to find: To calculate the integral, let’s decompose the complex exponent using Euler’s formula: Since our t(x) is odd, the first integral is zero . Also t(x)sin(wx) is even, so we can write: We’ve already calculated a very similar integral in the post on Fourier series , so let’s just skip to the result: The only remaining difficulty is its value at 0, which seems undefined at first (division by zero). However, note that as w\rightarrow 0 , the numerator also tends to 0, so we can use L’Hopital’s rule (twice!) to find that: This function is complex-valued; in fact, it’s purely imaginary. How do we visualize it? A common way to visualize complex-valued functions is by plotting their magnitude and phase separately. The magnitude of \hat{t}(w) is: Since \hat{t}(w) is purely imaginary, there are only two options for the phase: When the numerator is positive, we get a negative imaginary number with phase -\pi/2 , and when the numerator is negative, we get a positive imaginary number with phase \pi/2 . Finally, when \hat{t}(w)=0 (which happens at w=0 , by our earlier analysis, but also whenever is a whole multiple of \pi ), the phase is undefined. Here’s the magnitude and phase of \hat{t}(w) plotted against : It is common to talk about \hat{t}(w) as the frequency domain representation of t(x) . When the functions we’re working with have time as their domain (e.g. the x in t(x) represents time), which is often the case in the study of signals and systems, the Fourier transform can be seen as computing the frequency domain representation of the function. Here’s the Fourier transform formula again: It takes - the time domain representation of a function, and converts it to \hat{f}(w) - a frequency domain representation. For well-behaved functions, these two representations are dual - each one describes the function completely, just in a different way. To convert back from a frequency domain representation to the time domain, we use the inverse Fourier transform: While a time-domain plot ( t(x) ) shows how a signal changes over time, a frequency-domain plot ( \hat{t}(w) ) shows how the signal is distributed across all possible frequencies. Moreover, as we’ve seen, \hat{t}(w) is complex valued. Each frequency therefore has both a magnitude and a phase: the magnitude tells us how strongly that frequency contributes, while the phase tells us how that component is shifted. The frequency domain is extremely useful in signal analysis; for example, when designing filters. The Fourier transform also has a number of properties that are very useful in signal analysis and processing. But first, let’s discuss what a "well-behaved function" means for the purpose of applying Fourier transforms. The simplest existence condition for Fourier transforms is absolute integrability (also known as Lebesgue integrable): With this condition, \hat{f}(w) exists on the entire domain, is continuous and vanishes (tends to 0) as |w|\rightarrow\infty [4] . While this condition is sufficient, it’s not necessary; there are less well-behaved functions that also have Fourier transforms defined with some limitations. In these notes, we’re mostly interested in well-behaved functions that are used in real-world engineering, so we won’t discuss the other cases. Another assumption commonly made for real-world functions is that they vanish (tend to 0) as |x|\rightarrow\infty . While this is not a direct outcome of absolute integrability [5] , it’s a reasonable assumption in engineering. After all, real-world signals have finite energies. Intuitively, when we also assume is uniformly continuous , the assumption of vanishing at |x|\rightarrow\infty is a logical conclusion, because otherwise how can the total area for |f(x)| be finite? An important outcome of this discussion is that the Fourier transform is unsuitable for periodic functions. Functions that repeat at intervals are not absolute integrable . For periodic functions, we use Fourier series. The Fourier transform is a linear operator, because the integral is linear: So is the inverse Fourier transform; it’s similarly easy to show that: If we scale the domain of a function by a constant, its transform changes only slightly: Let’s do the variable substitution u=ax : This is the Fourier transform evaluated at \frac{w}{a} , so: There’s one small caveat here; when a is negative, the integral bounds should be flipped, causing a minus sign in front of the transform. So we can write: Which works for any a\ne 0 . This property is intuitive when thinking about signals: suppose a>0 , then f(ax) means the signal is compressed in the time domain by a factor a . The scaling property says that the frequency domain is expanded using the same factor; in other words, the higher frequencies become more prominent because we need sharper transitions to represent the compressed signal. Time shifting What happens to the Fourier transform if we time-shift the input signal by some constant: f(x-x_0) . By definition: Substituting u=x-x_0 , we get du=dx , so: Transform of a derivative An extremely useful property that’s often employed in the solution of partial differential equations; let’s calculate the Fourier transform of the derivative of : We’ll use integration by parts, where dv=f'(x) and u=e^{-i\cdot wx} . Therefore, v=f(x) and du=-iw\cdot e^{-i\cdot wx} : Recall the assumption made in the "Existence condition..." section about vanishing at infinities. So the first part of the equation above is zero, and we’re left with: Transform of convolution The convolution between two continuous functions and g(x) is defined as: Let’s calculate the Fourier transform of this function: This step of combining the integrals into a double integral, as well as the next step (changing the order of integration) is possible due to Fubini’s theorem and our assumption that and g(x) are Lebesgue integrable. Switch order of integration: Now, f(\xi) in the inner integral doesn’t depend on x , so we can pull it out: The inner integral is just the Fourier transform of a time-shifted g(x-\xi) , so we can write: And the remaining integral is the Fourier transform of , so: Convolution in the time domain translates to multiplication in the frequency domain! This result is so important in signal processing that it’s called the convolution theorem . Suppose we have some function and we want to know the area bounded between this function’s graph and the x axis in a certain interval [a,b] . One way to do this is to take a partition of the interval: And calculate the area under for every element of the partition. We can then approximate such sub-areas by rectangles, as follows: We’ll denote the area of each rectangle as f(x^*_i)\cdot\Delta x : There are many ways to choose which point of the interval [x_{i-1},x_i] to denote as x^*_i : left point ( x_{i-1} ), right point ( ), mid-point between the two (which is what our plot shows) or anything in between. The distinction doesn’t really matter for our purpose, as we will soon see. We can approximate the area under the curve of in the interval [a,b] with the Riemann sum , using a uniform partition: If is continuous on [a,b] , then as n\rightarrow \infty : This is known as the Riemann integral , or just the definite integral. The limit is why the exact choice of x^*_i doesn’t matter: as n\rightarrow\infty we have \Delta x\rightarrow 0 , and all points within [x_{i-1}, x_i] are equally good. The Fourier series is a great tool for analyzing periodic functions. But what about functions that don’t repeat? We’ve seen that we can compute Fourier series for a non-periodic function defined on a finite interval, as long as we don’t care about its behavior beyond that interval. Let’s extend this idea to functions that never repeat; that is, non-periodic functions defined on the interval (-\infty,\infty) . Visualizing Fourier series for non-repeating functions To motivate the subject ahead, let’s look back at the example used in the earlier post about Fourier series : \[t(x)= \begin{cases} x & 0 \leq x \leq 1 \\ 2-x & 1 < x \leq 2 \\ \end{cases}\] With an odd extension into [-2,0] . In that post, to make the Fourier series work, we assumed t(x) keeps repeating with a period 2L=4 on the entire x axis. Here, let’s face the reality that it does not - in fact - repeat, and observe how our Fourier series work out. Recall that the Fourier series approximating t(x) are the sine series (since it’s an odd function): \[t(x)=\frac{8}{\pi^2}\bigg[ sin\frac{\pi x}{2}-\frac{1}{3^2} sin\frac{3\pi x}{2}+\frac{1}{5^2}sin\frac{5\pi x}{2}-\cdots\bigg]\] The following visualization is interactive. By default, it shows t(x) (with its odd extension) and no Fourier series approximation. We’ll proceed by a series of steps and observe the outcome: n (terms in the Fourier series) L x min x max Step 1 : set to some non-zero number; already at 3, the approximation is very good. The frequency spacing is \frac{\pi}{L} (this is the coefficient of x in the sines). Note that the Fourier series repeats every 2L , as expected. Step 2 : increase L to 6. This means our series are constructed assuming t(x) has a period of 12, not 4. Note how the Fourier series look now - they repeat every 12, and they don’t match t(x) as well as before. We can increase to a higher number to make the match better. As L grows, the spacing between adjacent frequencies decreases. Step 3 : increase L to 10. We no longer see the repetitions, so feel free to increase the values of x min and x max until you do. Note again that we need to add more and more coefficients to match t(x) better with this larger L , and the spacing adjacent frequencies grows smaller. Increasing L means our function repeats at larger and larger intervals. The logical conclusion of this progression is to ask - what happens if the function never repeats, meaning L\rightarrow\infty ? While not mathematically rigorous, the visual experiment here lets us make some conjectures: we’ll likely need an infinite number of coefficients for a good approximation, and moreover, the spacing between these coefficients will tend to zero. In other words, instead of a discrete set of coefficients, we’ll end up with a continuous line, or function . The function produced by this process is the Fourier transform of t(x) , and the next section shows its mathematical derivation. Fourier series with L\rightarrow\infty leading to Fourier transform In these notes, we’ll be using the complex exponential formulation of Fourier series: \[f(x)=\sum_{n=-\infty}^{\infty}C_n\cdot e^{in\pi x/L}\] With: \[C_n=\frac{1}{2L}\int_{-L}^{L}f(x)e^{-in\pi x/L}dx\] We’re interested in a non-periodic defined on the interval (-\infty,\infty) . So we’ll be exploring the above equations for L\rightarrow\infty . First, let’s make a slight change of notation. Instead of writing formulae in terms of the period ( 2L ), we’ll be using the n-th harmonic angular frequency w_n : \[w_n=\frac{n\pi}{L}\] So we can slightly rewrite our series as: \[f(x)=\sum_{n=-\infty}^{\infty}C_n\cdot e^{i w_n x}=\sum_{n=-\infty}^{\infty}C_n\cdot e^{i\cdot n \Delta w x}\] Using \Delta w as the difference between two consecutive frequencies: \[\Delta w=w_n-w_{n-1}=\frac{n\pi}{L}-\frac{(n-1)\pi}{L}=\frac{\pi}{L}\] Using this notation, C_n is expressed as: \[C_n=\frac{\Delta w}{2\pi}\int_{-\pi/\Delta w}^{\pi/\Delta w}f(x)e^{-i\cdot n \Delta w x}dx\] So far there are no new insights here, just some new notation. Now we’re going to use it to facilitate the next step. Since L\rightarrow \infty , then \Delta w\rightarrow 0 . Let’s calculate the limit of the Fourier series representation of when \Delta w\rightarrow 0 : \[f(x)=\lim_{\Delta w\rightarrow 0}\sum_{n=-\infty}^{\infty}C_n\cdot e^{i\cdot n \Delta w x}\] And substitute the latest C_n into this equation, changing its dummy integration variable from x to t to avoid confusion [1] \[f(x)=\lim_{\Delta w\rightarrow 0}\sum_{n=-\infty}^{\infty}\left[\frac{\Delta w}{2\pi}\int_{-\pi/\Delta w}^{\pi/\Delta w}f(t)e^{-i\cdot n \Delta w t}dt\right]\cdot e^{i\cdot n \Delta w x}\] Reordering slightly, and also replacing n\Delta w by w_n in the complex exponents: \[f(x)=\frac{1}{2\pi}\lim_{\Delta w\rightarrow 0}\sum_{n=-\infty}^{\infty}\left[\int_{-\pi/\Delta w}^{\pi/\Delta w}f(t)e^{-i\cdot w_n t}dt\right]\cdot e^{i\cdot w_n x}\Delta w\] Looking at the limit with the sum carefully, this is a Riemann sum (see Appendix A)! w_n is the "sampled" version of , and \Delta w\rightarrow 0 . We can therefore replace it by an integral, changing w_n to and \Delta w to dw [2] : \[f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\left[\int_{-\infty}^{\infty}f(t)e^{-i\cdot wt}dt\right]\cdot e^{i\cdot w x}dw\] The inner integral is called the Fourier transform of and denoted [3] : \[\boxed{\hat{f}(w)=\mathcal{F}\left[f(x)\right]=\int_{-\infty}^{\infty}f(x)e^{-i\cdot wx}dx}\] And the full equation for is then the inverse Fourier transform: \[\boxed{f(x)=\mathcal{F}^{-1}\left[\hat{f}(w)\right]=\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat{f}(w)e^{i\cdot w x}dw}\] Example calculation of Fourier transform Let’s take our favorite odd triangular pulse example and calculate its Fourier transform. The function’s mathematical definition and plot are shown earlier in this post. Note that we’re not extending this function periodically - it’s zero beyond the range [-2,2] ; this is exactly why we need the Fourier transform here - as we’ve seen, Fourier series won’t do because the function they reconstruct eventually starts repeating. We’re looking to find: \[\hat{t}(w)=\int_{-\infty}^{\infty}t(x)e^{-iwx}dx\] To calculate the integral, let’s decompose the complex exponent using Euler’s formula: \[\hat{t}(w)=\int_{-\infty}^{\infty}t(x)cos(wx)dx-i\int_{-\infty}^{\infty}t(x)sin(wx)dx\] Since our t(x) is odd, the first integral is zero . Also t(x)sin(wx) is even, so we can write: \[\hat{t}(w)=-2i\int_{0}^{\infty}t(x)sin(wx)dx\] We’ve already calculated a very similar integral in the post on Fourier series , so let’s just skip to the result: \[\hat{t}(w)=-2i\cdot\frac{2\cdot sin(w)-sin(2w)}{w^2}\] The only remaining difficulty is its value at 0, which seems undefined at first (division by zero). However, note that as w\rightarrow 0 , the numerator also tends to 0, so we can use L’Hopital’s rule (twice!) to find that: \[\lim_{w\rightarrow 0} \hat{t}(w)=0\] Therefore: \[\hat{t}(w)= \begin{cases} -2i\cdot\frac{2\cdot sin(w)-sin(2w)}{w^2} & w\neq 0 \\ 0 & w=0 \\ \end{cases}\] This function is complex-valued; in fact, it’s purely imaginary. How do we visualize it? A common way to visualize complex-valued functions is by plotting their magnitude and phase separately. The magnitude of \hat{t}(w) is: \[|\hat{t}(w)|=\sqrt{\hat{t}(w)\cdot\hat{t}(w)^*}=2\left|\frac{2\cdot sin(w)-sin(2w)}{w^2} \right|\] Since \hat{t}(w) is purely imaginary, there are only two options for the phase: When the numerator is positive, we get a negative imaginary number with phase -\pi/2 , and when the numerator is negative, we get a positive imaginary number with phase \pi/2 . Finally, when \hat{t}(w)=0 (which happens at w=0 , by our earlier analysis, but also whenever is a whole multiple of \pi ), the phase is undefined. Here’s the magnitude and phase of \hat{t}(w) plotted against : It is common to talk about \hat{t}(w) as the frequency domain representation of t(x) . The frequency domain representation of functions When the functions we’re working with have time as their domain (e.g. the x in t(x) represents time), which is often the case in the study of signals and systems, the Fourier transform can be seen as computing the frequency domain representation of the function. Here’s the Fourier transform formula again: \[\hat{f}(w)=\mathcal{F}\left[f(x)\right]=\int_{-\infty}^{\infty}f(x)e^{-i\cdot wx}dx\] It takes - the time domain representation of a function, and converts it to \hat{f}(w) - a frequency domain representation. For well-behaved functions, these two representations are dual - each one describes the function completely, just in a different way. To convert back from a frequency domain representation to the time domain, we use the inverse Fourier transform: \[\mathcal{F}^{-1}\left[\hat{f}(w)\right]=\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat{f}(w)e^{i\cdot w x}dw\] While a time-domain plot ( t(x) ) shows how a signal changes over time, a frequency-domain plot ( \hat{t}(w) ) shows how the signal is distributed across all possible frequencies. Moreover, as we’ve seen, \hat{t}(w) is complex valued. Each frequency therefore has both a magnitude and a phase: the magnitude tells us how strongly that frequency contributes, while the phase tells us how that component is shifted. The frequency domain is extremely useful in signal analysis; for example, when designing filters. The Fourier transform also has a number of properties that are very useful in signal analysis and processing. But first, let’s discuss what a "well-behaved function" means for the purpose of applying Fourier transforms. Existence condition for the Fourier transform The simplest existence condition for Fourier transforms is absolute integrability (also known as Lebesgue integrable): \[\int_{-\infty}^{\infty}|f(x)|dx<\infty\] With this condition, \hat{f}(w) exists on the entire domain, is continuous and vanishes (tends to 0) as |w|\rightarrow\infty [4] . While this condition is sufficient, it’s not necessary; there are less well-behaved functions that also have Fourier transforms defined with some limitations. In these notes, we’re mostly interested in well-behaved functions that are used in real-world engineering, so we won’t discuss the other cases. Another assumption commonly made for real-world functions is that they vanish (tend to 0) as |x|\rightarrow\infty . While this is not a direct outcome of absolute integrability [5] , it’s a reasonable assumption in engineering. After all, real-world signals have finite energies. Intuitively, when we also assume is uniformly continuous , the assumption of vanishing at |x|\rightarrow\infty is a logical conclusion, because otherwise how can the total area for |f(x)| be finite? An important outcome of this discussion is that the Fourier transform is unsuitable for periodic functions. Functions that repeat at intervals are not absolute integrable . For periodic functions, we use Fourier series. Some useful properties of Fourier transforms Linearity The Fourier transform is a linear operator, because the integral is linear: \[\begin{aligned} \mathcal{F}\left[\alpha f(x)+\beta g(x)\right]&=\int_{-\infty}^{\infty}\alpha f(x)e^{-i\cdot wx}dx+\int_{-\infty}^{\infty}\beta g(x)e^{-i\cdot wx}dx\\ &=\alpha\int_{-\infty}^{\infty}f(x)e^{-i\cdot wx}dx+\beta\int_{-\infty}^{\infty}g(x)e^{-i\cdot wx}dx\\ &=\alpha\mathcal{F}\left[f(x)\right]+\beta\mathcal{F}\left[g(x)\right] \end{aligned}\] So is the inverse Fourier transform; it’s similarly easy to show that: \[\mathcal{F}^{-1}\left[\alpha\hat{f}(w)+\beta\hat{g}(w)\right]= \alpha\mathcal{F}^{-1}\left[\hat{f}(w)\right]+\beta\mathcal{F}^{-1}\left[\hat{g}(w)\right]\] Scaling If we scale the domain of a function by a constant, its transform changes only slightly: \[\mathcal{F}\left[f(ax)\right]=\int_{-\infty}^{\infty}f(ax)e^{-i\cdot wx}dx\] Let’s do the variable substitution u=ax : \[\mathcal{F}\left[f(ax)\right]=\frac{1}{a}\int_{-\infty}^{\infty}f(u)e^{-i\cdot \frac{wu}{a}}du\] This is the Fourier transform evaluated at \frac{w}{a} , so: \[\mathcal{F}\left[f(ax)\right]=\frac{1}{a}\hat{f}\left(\frac{w}{a}\right)\] There’s one small caveat here; when a is negative, the integral bounds should be flipped, causing a minus sign in front of the transform. So we can write: \[\mathcal{F}\left[f(ax)\right]=\frac{1}{|a|}\hat{f}\left(\frac{w}{a}\right)\] Which works for any a\ne 0 . This property is intuitive when thinking about signals: suppose a>0 , then f(ax) means the signal is compressed in the time domain by a factor a . The scaling property says that the frequency domain is expanded using the same factor; in other words, the higher frequencies become more prominent because we need sharper transitions to represent the compressed signal. Time shifting What happens to the Fourier transform if we time-shift the input signal by some constant: f(x-x_0) . By definition: \[\mathcal{F}\left[f(x-x_0)\right]=\int_{-\infty}^{\infty}f(x-x_0)e^{-i\cdot wx}dx\] Substituting u=x-x_0 , we get du=dx , so: \[\begin{aligned} \mathcal{F}\left[f(x-x_0)\right]&=\int_{-\infty}^{\infty}f(u)e^{-i\cdot w(u+x_0)}du\\ &=e^{-iwx_0}\int_{-\infty}^{\infty}f(u)e^{-i\cdot wu}du\\ &=e^{-iwx_0}\mathcal{F}\left[f(x)\right] \end{aligned}\] Transform of a derivative An extremely useful property that’s often employed in the solution of partial differential equations; let’s calculate the Fourier transform of the derivative of : \[\mathcal{F}\left[f'(x)\right]=\int_{-\infty}^{\infty}f'(x)e^{-i\cdot wx}dx\] We’ll use integration by parts, where dv=f'(x) and u=e^{-i\cdot wx} . Therefore, v=f(x) and du=-iw\cdot e^{-i\cdot wx} : \[\mathcal{F}\left[f'(x)\right]=\left[f(x)e^{-i\cdot wx}\right]^{\infty}_{-\infty}-\int_{-\infty}^{\infty}f(x)(-iw\cdot e^{-i\cdot wx})dx\] Recall the assumption made in the "Existence condition..." section about vanishing at infinities. So the first part of the equation above is zero, and we’re left with: \[\begin{aligned} \mathcal{F}\left[f'(x)\right]&=-\int_{-\infty}^{\infty}f(x)(-iw\cdot e^{-i\cdot wx})dx\\ &=iw\int_{-\infty}^{\infty}f(x)e^{-i\cdot wx}dx\\ &=iw\cdot\mathcal{F}\left[f(x)\right] \end{aligned}\] Transform of convolution The convolution between two continuous functions and g(x) is defined as: \[(f\ast g)(x)=\int_{-\infty}^{\infty}f(\xi)g(x-\xi)d\xi\] Let’s calculate the Fourier transform of this function: \[\begin{aligned} \mathcal{F}\left[(f\ast g)(x)\right]&=\int_{-\infty}^{\infty}e^{-i\cdot wx}\left[\int_{-\infty}^{\infty}f(\xi)g(x-\xi)d\xi\right]dx\\ &=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-i\cdot wx}f(\xi)g(x-\xi)d\xi\ dx \end{aligned}\] This step of combining the integrals into a double integral, as well as the next step (changing the order of integration) is possible due to Fubini’s theorem and our assumption that and g(x) are Lebesgue integrable. Switch order of integration: \[\mathcal{F}\left[(f\ast g)(x)\right]=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-i\cdot wx}f(\xi)g(x-\xi)dx\ d\xi\] Now, f(\xi) in the inner integral doesn’t depend on x , so we can pull it out: \[\mathcal{F}\left[(f\ast g)(x)\right]=\int_{-\infty}^{\infty}f(\xi)\int_{-\infty}^{\infty}e^{-i\cdot wx}g(x-\xi)dx\ d\xi\] The inner integral is just the Fourier transform of a time-shifted g(x-\xi) , so we can write: \[\mathcal{F}\left[(f\ast g)(x)\right]=\int_{-\infty}^{\infty}f(\xi)e^{-i\cdot w\xi}\mathcal{F}\left[g(x)\right]d\xi=\mathcal{F}\left[g(x)\right]\int_{-\infty}^{\infty}e^{-i\cdot w\xi}f(\xi)d\xi\] And the remaining integral is the Fourier transform of , so: \[\mathcal{F}\left[(f\ast g)(x)\right]=\mathcal{F}\left[f\right]\cdot\mathcal{F}\left[g\right]\] Convolution in the time domain translates to multiplication in the frequency domain! This result is so important in signal processing that it’s called the convolution theorem . Appendix A: Riemann sum and the definite integral Suppose we have some function and we want to know the area bounded between this function’s graph and the x axis in a certain interval [a,b] . One way to do this is to take a partition of the interval: \[a=x_0<x_1<\cdots<x_{n-1}<x_n=b\] And calculate the area under for every element of the partition. We can then approximate such sub-areas by rectangles, as follows: We’ll denote the area of each rectangle as f(x^*_i)\cdot\Delta x : \Delta x=(b-a)/n is the width of one interval (assuming a uniform partition, but the math works just as well for non-uniform ones). x^*_i is some value in the interval [x_{i-1},x_i] .

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Hugo 4 weeks ago

AI and Ecology, Fantasy or Convenient Scapegoat?

It's hard to talk about AI these days. I've rarely, if ever, seen a subject so polarized in tech. You could tell me I have a short memory. The internet sparked plenty of criticism around the destruction of brick-and-mortar retail, print media, and the end of human interaction. Same for mobile, with added, legitimate reproaches about addiction and the ease of surveilling individuals. We could also mention crypto, a massive Ponzi scheme for some, a way to reclaim power from central banks for others. And yet, with AI, I feel like we've crossed a threshold. There would be only two possibilities: Pick a side, friend, and if you don't, others will do it for you. The "safest" bet is not to talk about it at all, but burying my head in the sand feels cowardly, if not impossible when you work in tech. Simply put, I need to stick my head out and try this exercise without resorting to clichés. And since we're in the middle of a heat wave, it seems obvious that the first subject to address is ecology. Is AI as catastrophic as people say? Is the impact of an AI query truly astronomically higher than a Google search? How does it compare with other digital uses? Let's take some time to look at all this. First, let's establish some basics about what we call ecological impact. This impact falls into several categories: To keep things simple, an AI consumes energy at two distinct stages: during training (when the model is created, like Gemini, Llama, Claude, etc.) and during inference (when users actually query the model). When looking at carbon footprints, models vary wildly, but the estimated range for training a single major model sits between 500 and 12,000 tonnes of CO2 equivalent. To put that into perspective: ::callout{type=primary} The massive gap between US and French household equivalents stems from the fact that France’s energy mix relies heavily on nuclear power, which is virtually carbon-free. :: Operational consumption is another moving target. It depends on the complexity of the prompt, the location of the data center (and its corresponding energy mix), the model being used, and so on. But it's estimated to vary between 0.03g , and 1g of CO2 per request. We'll see below how that compares with internet, gaming, streaming etc… To talk about water consumption, we must first address a common misconception: No, we don't destroy water . Earth’s water operates in a closed loop. When water is used in a cooling system, whether in a data center, nuclear power plant or anything else, it's not destroyed. When water evaporates, it eventually falls back as rain. However , evaporation causes water to displace. If water moves more than 800km, the region where it was drawn from has effectively lost it, temporarily but lost nonetheless. In ecology, we distinguish water withdrawal (borrowing water and returning it to the same place after use) and water consumption (drawing water and evaporating or releasing it elsewhere, making it unavailable locally). AI consumes water. On a planetary scale , it's not necessarily a problem. On a local scale , however, it can trigger severe water stress, creating direct competition between residents, agriculture, and data centers. To be fair, technology is advancing. The majority of new data center projects use closed-loop water systems so water isn't evaporated. Some countries (Ireland, Sweden, Finland) take advantage of their cold climate to reduce water needs by 90% and we see other systems emerging. But to look at the flip side, the vast majority of existing data centers use evaporation systems and in any case, these systems require electricity which creates tensions, for example in Sweden or Ireland. Now that we've said all that, what's the consumption for evaporation data centers? Training a recent model is estimated to consume approximately 40 to 80 million liters (a small lake). In a water-stressed region, that can make a difference. And if we look at usage, for a request, it's between 2 and 6.5ml of water per request. ::callout{type=warning} This section is the trickiest for me because it’s the one I’m least familiar with, and honestly, it probably deserves an entire article of its own. So, while we will only scratch the surface here, I promise to dive much deeper into this specific topic in a future post. :: We often focus on electricity and water, but the environmental footprint of mining is one of AI's biggest blind spots. To run AIs or train models, you need ultra-powerful equipment and colossal infrastructure that will require copper, aluminum, cobalt, lithium, nickel, rare earths and I imagine I'm forgetting some. Well, these resources are in limited quantities on earth but I'll discuss that in a future article, recycling in this sector is currently negligible but moreover, the extraction itself is extremely polluting. To make matters worse, we must add that current equipment becomes obsolete much faster. In the AI race, we replace equipment much faster. Certainly, new equipment is more efficient, particularly in terms of energy but this ultra-rapid rotation creates a volume of electronic waste we don't know how to manage. Despite everything, I don't yet know from which angle and with which figures to illustrate all this, especially since these subjects also pull along many other geopolitical subjects (tension over Taiwan, tension over rare earths etc…), so we'll set that aside for future publication. We already have plenty to do with the first two subjects. With these orders of magnitude in mind, is AI " stratospherically " different from the rest? How does it compare with a Google search for example? Or with streaming, video call, an online video game? By comparison, a query to a search engine (Google) is approximately 0.2g of CO2 . Depending on the complexity of the question and the model used, an AI prompt can cost slightly less than a Google search, or up to five times more . So, it is not "stratospherically" higher than a standard web search. Furthermore, if a topic requires you to do multiple Google searches and open several websites to find your answer, the gap quickly narrows, and can even reverse. But we must separate simple uses: "give me the strawberry pie recipe", from complex uses: "analyze this PDF document of several megabytes for me and create an application that displays results with charts". I propose we do an exercise and compare 1 hour of streaming, 1 hour of gaming, 1 hour of video call, and one hour of AI-assisted software development (a relatively power-consuming use). ::callout{type=primary} Why such strong variations when considering AI-assisted development? Because it encompasses vastly different habits. Consumption will be drastically different between an "amateur" coder copy-pasting a few lines from a browser, a "pro" user partially delegating tasks within their code editor, and an "intensive" power-user running automated tools where code generation is almost entirely outsourced. :: In other words: No matter how you look at the data, it is hard to find evidence of a "stratospheric" gap. And to go further, we could look at the impact of AI model creation compared to the ecological impact of creating a video game, or a movie. An AAA video game (big-budget), developed by a team of 150 people, costs between 500 and 3,000 tonnes of CO2 depending on development time, travel, and motion-capture filming. To this, we must add the annual maintenance for live-service games that push out continuous updates and DLCs (like World of Warcraft or Overwatch ). For a big budget film, we can estimate a carbon cost between 3,000 and 4,000t of CO2 , including transport, filming locations, generators, set construction. Granted, training a massive AI model can cost more than a single movie, but the difference isn't orders of magnitude apart. More importantly, we must remember that the world releases thousands of films and video games every year , whereas the creation of new foundational AI models remains relatively rare. Let's be careful here. It would be lazy whataboutism to simply say, "Sure, AI is bad, but look at how much worse everything else is." That is missing the point. The real goal here is to question our consumption habits as a whole. What is certain, however, is that the reality is far more nuanced than the mainstream narrative suggests. Today, the hyper-focus on AI serves as a very convenient distraction, allowing us to forget the environmental cost of our other digital habits. But you don't earn moral virtue points by campaigning against AI while actively indulging in online gaming, streaming blockbusters, or flying to international sports events. If you've followed the numbers well, the ecological impact of AI is relatively close to other impacts in digital (streaming and gaming for example). That doesn't mean it's good. In the world we live in, each additional tension on the planet is to be questioned . But it forces us to realize that all of our digital behaviors need to be reassessed, not just the fact that "I asked ChatGPT a question." I don't pretend to be able to rank these activities against one another. Comparing gaming, streaming, and professional workloads is highly complex. And even within professional uses, And I certainly won't decide, on my own , what constitutes a "good" or "bad" use of technology. But collectively, we might soon be forced to make those choices , not out of kindness, but by constraints (See next chapter). The core issue isn't about outright banning AI. This is precisely what organizations like Shift Project , France’s leading think tank on the energy transition, are trying to convey: we need to look at data volumes and digital use cases in their entirety. The argument isn't that we should abandon AI altogether, but rather that we cannot afford its current, unchecked trajectory Let's take an example: the FIFA World Cup generates between 9 and 15 million tonnes of CO2, which is roughly equivalent to the annual energy consumption of all US data centers combined.. Again, the idea isn't to say, they do worse. We'll get nowhere with that mindset. But I like this example because of the contrast it highlights. Playing football doesn't cost much. Gathering thousands of people across 3 countries and having them fly everywhere is absurd, as is air-conditioning football stadiums, or trying to organize winter games in a desert country. AI operates on the exact same spectrum. There is a massive gulf between a professional, high-utility application, like using AI in biochemistry, mathematics, meteorology, drug discovery, medical imaging, satellite analysis, or precision agriculture, and a purely recreational use aimed at generating thousands of Ghibli-style images just to dump them on social media. Yes, we can, and should, question the latter (and that’s an understatement). Ultimately, understanding these orders of magnitude is what empowers us to make informed choices instead of just parroting the absurdities we hear on TV. Once you know the real numbers, you can weigh your choices accurately. I said earlier that one hour of video call was between 30 and 60g of CO2. Ok, but that might replace a Paris Lyon trip. By car it's between 60 and 90kg of CO2 saved. By train it's about 1kg. Similarly, one hour of streaming costs about 100g of CO2. But if it prevented you from driving 20km to the local movie theater (which would cost around 4.4kg of CO2 in car emissions), streaming turns out to be "not so bad" after all. In the end, once we have the data, it is up to each of us to make those choices. A question I asked myself before writing this article was: If the carbon footprint of AI is actually pretty close to our other digital habits, and assuming it replaces some of them (if I’m using AI, I’m not doing something else), why on earth are we building so many new data centers? Ok, this question might seem naive but it's estimated that data center electricity consumption could double, or even triple by 2030 (See BCG study and this IRIS article). So why? Is it linked to AI? According to articles, partly yes, but only partly. The majority of electricity consumed by data centers (about 2/3) should be dedicated to historical digital uses and acceleration of cloud migrations. Yes AI plays a role, but it's mainly that digital is taking up more and more space. The share of digital in global CO2 emissions went from 2% in 2010 to about 4% today, with an annual increase of about 6% even though the global objective is to reduce our emissions by 5% per year to hope to ++stabilize++ the climate . Where AI genuinely worsens the problem compared to other tech is its rapid hardware obsolescence. However, the root cause is the massive scale-up of all our digital habits: the ubiquity of 4K/8K streaming, cloud gaming, high-fidelity music streaming, and the explosion of connected IoT (Internet of Things) devices. Of course, we should take these data center growth forecasts with a grain of salt. They remain predictions. They could easily be overestimated, just like the predicted "tidal wave" of data that was supposed to arrive with 5G but never quite materialized. Many of these projections are pushed by tech giants that have bet their entire financial futures on these exact growth scenarios. If you are Nvidia, Google, or Oracle, you have no choice but to reassure your shareholders by guaranteeing this growth will happen to justify the colossal investments already made. Honestly, if the AI financial bubble were to burst tomorrow, it might actually be good news for the planet, as it would instantly ease the pressure on our resources. That being said, we are looking at contracts that are already signed, budgets locked in, and massive public announcements, like the Stargate project or Europe's future investments . Every current scenario predicts a 2x to 3x increase in demand. Digital consumption is going up, and AI-related infrastructure is leading the charge. Will these digital habits replace physical ones (like my earlier example of a video call replacing a car trip)? Or are they purely additive ? Evidence suggests they are additive. While some AI applications will certainly accelerate decarbonization in specific industries, that impact remains relatively marginal for now. And, according to the Shift Project, it's mainly that the electrical consumption needed to run all these data centers will exceed our infrastructure capacity, and thus force using thermal sources (gas power plant) to compensate or create usage conflicts . So yes, it's alarming. Again, it's not about banning AI, the subject is much more global than that. How do we make our consumption and pressure on the planet decrease? At our individual scale, we have to audit our own habits. We need to question our obsession with upgrading devices, over-equipping our homes, and engaging in mindless, heavy recreational uses (like generating endless Ghibli images just for a laugh). At a collective level, we will eventually be forced, likely much sooner than we think, to make hard choices. We will have to decide whether to route precious electricity to a data center or to power and heat local homes . But let’s remember one crucial thing: the future isn’t set in stone. If we collectively choose to consume less, we won't need the electricity these companies are dying to sell us. If data centers are multiplying, it's only because there is a planned demand for them. To paraphrase a famous French comedian: "To think that if people just stopped buying, it wouldn't sell anymore!" The Doomers (declinists) who envision an ecological apocalypse, total destruction of employment and placing public opinions under the guardianship of Big Tech controlling AI. And the Bloomers (accelerationists), who advocate blind faith in progress, convinced that AI will liberate humanity, eradicate diseases and generate infinite growth, and for whom slowing research is the real crime. Electricity consumption (which translates into CO2 emissions) Water consumption for cooling data centers Resource extraction, required to build the data centers and user devices themselves There is only a 2x ratio between gaming and professional AI-assisted development. video call is what consumes the least Streaming is remarkably close to professional AI development usage

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Dot product: Component vs. Geometric definition

The goal of this post is to answer a simple question: why are the following two definitions of the vector dot product in Euclidean space [1] equivalent for vectors \vec{a} and \vec{b} : Here’s a graphical depiction of our vectors (focusing on for clarity, though this applies to any-dimensional vectors). It shows both the components of the vectors and the angle between them. The length of the arrow for \vec{a} is |\vec{a}| . We’ll show two proofs of the equivalence here, the geometric proof and the projection proof . The Appendix describes some properties of dot products that facilitate these proofs. We’ll be using this diagram of our vectors \vec{a} and \vec{b} , as well as the vector \vec{c}=\vec{a}-\vec{b} : Using the law of cosines [2] on the triangle formed by the three vectors: Since for any vector \vec{a} , we have \vec{a}\cdot\vec{a}=|\vec{a}|^2 (see Appendix), let’s rewrite this equation as: But \vec{c}=\vec{a}-\vec{b} and the dot product obeys the distributive property (see Appendix). Therefore: For this proof, we’ll assume the geometric definition is correct and will see how it leads to the component definition. We’ll begin by denoting vectors \vec{e}_1,\vec{e}_2\dots\vec{e}_n as the standard orthonormal basis for . For example, in 2D space, these basis vectors are \vec{e}_1=[1\ 0] and \vec{e}_2=[0\ 1] , shown in this diagram: If we take an arbitrary \vec{a}\in\mathbb{R}^n and calculate its dot product with a basis vector, we can use the geometric definition: where a_i is the component of \vec{a} in the direction of \vec{e}_i . The diagram makes it easy to see why this is true from basic trigonometry, but in the more general case this is just a vector projection . Now let’s represent vectors \vec{a} and \vec{b} as linear combinations of the basis vectors: And calculate the dot product \vec{a}\cdot\vec{b} , beginning by rewriting \vec{b} with its linear combination of basis vectors representation: Using the fact that the dot product distributes over linear combinations: But earlier we’ve shown that \vec{a}\cdot\vec{e}_i=a_i . Therefore: Which is the component definition \blacksquare . A generalization of dot products in is the inner product , which is an operation meeting some specific requirements, defined on a vector space. The inner product is denoted as \langle x,y\rangle:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R} , and must satisfy the following requirements for all vectors x,y,z\in\mathbb{R}^n and scalars a,b\in\mathbb{R} : For , we define the inner product operation in its component formulation as: Let’s prove the requirements listed above for this operation; this is fairly straightforward, given the well-known properties of scalar multiplication and addition on : Linearity in the first argument: Positive-definiteness: Consider the components of vector x . Clearly, \forall i\quad x_i\cdot x_i=x_i^2\ge 0 . Since the vector x is not the zero vector, at least one of its components is nonzero, and for that component x_i\cdot x_i>0 . Therefore: Now that we’ve proved all the inner product requirements on our operation \langle x,y\rangle , we can say that is an inner product space with this operation. By meeting these requirements, it can be readily shown that our inner product operation has additional useful properties: The third property is particularly helpful, because it means the inner product is bilinear , and thus is distributive over addition. Note that these are shown for the component definition of dot product. It’s not too hard to prove distributivity for the geometric definition using the notion of projections and how they add up. The norm of a vector x in an inner product space is defined as |x|=\sqrt{\langle x,x\rangle} . Therefore, the square of the norm is |x|^2=\langle x,x\rangle . The norm is used to express the notion of magnitude , or length of a vector. If you think of a vector x\in\mathbb{R}^n in Cartesian coordinates, the definition of the norm is a generalization of the Pythagorean theorem. Component definition: \vec{a}\cdot\vec{b}=\sum_{i=1}^{n}a_i b_i Geometric definition: \vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|cos(\theta) , where |\vec{a}| is the magnitude of \vec{a} and is the angle between the vectors’ directions Symmetry: \langle x,y\rangle=\langle y,x\rangle Linearity in the first argument: \langle ax+by,z\rangle=a\langle x,z\rangle+b\langle y,z\rangle Positive-definiteness: if x\ne 0 then \langle x,x\rangle>0 \langle x,0\rangle=\langle 0,x\rangle=0 \langle x,x\rangle=0 if and only if x=0 \langle x,ay+bz\rangle=a\langle x,y\rangle+b\langle x,z\rangle \langle x+y,x+y\rangle=\langle x,x\rangle+2\langle x,y\rangle+\langle y,y\rangle

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Unsung 1 months ago

“…or I could click seventy buttons.”

I like Angela Collier’s videos about physics and I was delighted to discover this 18-minute one … = 2x) and (width >= 700px)" srcset="https://unsung.aresluna.org/_media/or-i-could-click-seventy-buttons/yt1-play.2096w.avif" type="image/avif"> = 3x) or (width >= 700px)" srcset="https://unsung.aresluna.org/_media/or-i-could-click-seventy-buttons/yt1-play.1600w.avif" type="image/avif"> …because it’s a great continuation to the thread about the complexity of Microsoft Office I shared recently. Collier talks about why physicists prefer LaTeX to Word. LaTeX is sort of a nerdy HTML that predates HTML. It looks like this… = 2x) and (width >= 700px)" srcset="https://unsung.aresluna.org/_media/or-i-could-click-seventy-buttons/1.2096w.avif" type="image/avif"> = 3x) or (width >= 700px)" srcset="https://unsung.aresluna.org/_media/or-i-could-click-seventy-buttons/1.1600w.avif" type="image/avif"> = 2x) and (width >= 700px)" srcset="https://unsung.aresluna.org/_media/or-i-could-click-seventy-buttons/2.2096w.avif" type="image/avif"> = 3x) or (width >= 700px)" srcset="https://unsung.aresluna.org/_media/or-i-could-click-seventy-buttons/2.1600w.avif" type="image/avif"> …and given how nerdy HTML already is, you might imagine this is a power-user tool that’s chiefly about power and control. But Collier makes the argument that there are some things that LaTeX makes much easier: This is really interesting because it goes right to the core of the uncomfortable truth: naïve design decisions meant to make things easier might achieve the opposite. I shared the ForkLift example where the team didn’t understand what made the previous version great , and more recently the animation that could slow people down . (Of course, there is also the issue of typographical craft of LaTeX documents set in Computer Modern , but let’s save this for another time.) Also, the video starts with Collier apologizing for potentially making the audience feel dumb in a prior video. I don’t think it’s a joke, and I found it thoughtful and refreshing. #attention #complexity #enshittification #flow #youtube there is absolutely no need (or peer pressure) to spend time styling the document by choosing fonts, colors, etc., there is no “live preview,” and making a PDF is a separate step similar to compilation in coding – which means it doesn’t constantly occupy your mind, GUIs can slow you down because the keyboard is faster than the mouse, LaTeX doesn’t give you a lot of control over positioning, which is better than giving you only a semblance of control over positioning ( this is the TikTok meme Collier alluded to briefly ).

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Maurycy 1 months ago

The Tadpole galaxy:

North is up (exact, mirrored). 0.53 "/pixel [18.8' x 7.4'] FWHM = 4.2" This galaxy has a massive (and rather bright) tidal tail, but I can't see an obvious companion galaxy. The general consensus is that there's a second galaxy behind it... although a nearby elliptical has a suspiciously similar redshift: Not my finest work due to some patchy clouds, but not terrible. Callibration (dark + flat) Stacking (average w/ outlier rejection) White balance and background subtraction (no gradient removal) Asinh stretch (color preserving) Rotation + crop /astro/arp188/stacked.fits.fz : Raw stacks https://ned.ipac.caltech.edu/level5/Arp/Figures/big_arp188.jpeg : Arp's image

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Neil Madden 1 months ago

Are we any closer to the Quantum Apocalypse?

Another day, another urgent pronouncement on the need to transition to post-quantum cryptography ASAP: this one from the White House , in the form of an Executive Order requiring certain “high value” systems to transition to post-quantum cryptography (PQC) by the end of 2030 (for key exchange) or 2031 (for signatures). This brings forward the date slightly compared to previous guidance, which disallows quantum-vulnerable crypto for US Federal systems by 2035. But is this urgency justified? First, an important note : as you can probably tell already, I’m going to pour some skepticism on this sense of urgency. I don’t think cryptographically-relevant quantum computers are coming soon. However, that doesn’t mean we shouldn’t be prepared! The risk that they might appear soon is non-negligible, and the impact of them appearing for many applications is catastrophic. Sensible timelines to mitigate known threats are justified, panic-induced rushing is not. On with the article… Filippo Valsorda wrote a good piece about why he believes this urgency is justified, and that we need to be moving faster towards a post-quantum world. He cites two papers that dramatically reduce the estimates for how many qubits are needed to break classical cryptography (in this case elliptic curves) using a quantum computer. He writes: “Overall, it looks like everything is moving: the hardware is getting better, the algorithms are getting cheaper, the requirements for error correction are getting lower.” But is the hardware getting better? This is where I have doubts. Initial timelines for quantum computing from Google and IBM were extremely optimistic. Just 5 years ago, Google suggested they would have a fault-tolerant quantum computer with 1,000,000 physical qubits by 2029 . They are currently at 105 . So just 4 orders of magnitude to go in the next 3 years. IBM were a bit more conservative, anticipating 100,000 qubits by 2033 . They are currently at 156 qubits. Sam Jacques has been updating a useful chart every year , showing the current state of quantum computing progress. Below shows a comparison of the first chart he published in 2023 and the most recent one in 2026. What can clearly be seen is how better analysis has moved attacks down and to the left, but actual hardware progress has remained stubbornly in that little grey box, with a tiny nudge upwards on reducing the error rate. Now, you may say that there has been good progress on improving error correction. For example, at the end of 2024, Google announced “below threshold” quantum error correction . Surely a sign of good progress, even if the number of qubits was behind schedule. Once you’ve cracked error correction, the qubits will come thick and fast: an atomic explosion of qubits , if you will. (If you believe this then it doesn’t really matter how much more efficient the attacks become on paper: all that matters is how soon the hardware arrives). But I do wonder how that announcement was different from the announcement Google made almost 2 years earlier stating “ For the first time ever, our Quantum AI researchers have experimentally demonstrated that it’s possible to reduce errors by increasing the number of qubits. ” Call me skeptical, but if you were really making progress then would you need to put out re-runs of results you’ve already announced? Are there new chips coming that build on this breakthrough to give us the large numbers of usable qubits we’ve been promised? Maybe I’m about to be proved wrong by new announcements, or maybe all of the companies and governments involved in the entire world have suddenly decided to keep their progress hush-hush. But from my point of view as an outsider looking in, it all looks suspiciously like progress on quantum computing has stalled rather than the sky being about to fall on our heads. To reiterate: I still think it is sensible to be working right now on transitioning to post-quantum encryption (in a hybrid). But I am deeply skeptical of the idea that we need to rush things because quantum computers are arriving any second now. As I said in “ Are we overthinking post-quantum cryptography? ”, I think if you’re not beholden to the diktats of an insane autocrat, making minimal adjustments to ensure you can counter “store now, decrypt later” attacks is sensible. Wholesale replacement of all of your cryptography with post-quantum alternatives is IMO still in the realm of something to start thinking about, not a burning crisis that needs immediate attention. The key things to consider have nothing to do with PQC at all: Can I change algorithms easily and securely ? Do I need to be using public key cryptography, or will symmetric cryptography do instead? (Hint: if it doesn’t cross a trust boundary, then the answer is almost always “yes”). Can I avoid digital signatures (the post-quantum ones are mostly crap)? Can I avoid cryptography entirely? (E.g., moving from “stateless” JWTs to good old-fashioned stateful tokens/cookies).

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Gabe Mays 1 months ago

Reflections on 1,000 days of math

I finally hit 1,000 days of doing math daily! Early on in my journey I was a lot more aggressive with my XP targets, but settled into a low-volume rhythm as my goals evolved. I worked from MF1 (Math Foundations 1, lowest level) into MF3, then about halfway through MF3 I started M4ML (Mathematics for Machine Learning). But it got really hard and my progress started to slow…

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ava's blog 1 months ago

enduring the heat wave in germany

I live in an apartment that first gets heated up on one side before noon, then later from the other side. My kitchen is especially hot each year because it has a huge bay window with no shutters installed. My strategies for keeping cool have been to air out everything at night, and if possible draw in and circulate air via a fan during some of it. Then as soon as the sun is coming up, closing windows, lowering the existing outside shutters so the sun can’t heat up the glass or insides, and always keeping the kitchen door closed so the heat is contained within. I avoid opening the windows during the day to not let heat in, except if I really need fresh air or the humidity is too high. Humidity is the thing that is wrecking us the most in this, which is why it is often futile to ask people elsewhere how they deal with these high temperatures when those people live in very dry climates. The humidity messes with your body’s ability to exude heat, and in worst case, results in the wet bulb effect . That is also why even people from hotter countries can suddenly struggle elsewhere (like Europe), together with the angle at which sunlight hits Earth at that area being different (a lower sun angle spreads the same amount of energy over a larger area, making it feel cooler, while a higher angle concentrates energy on a smaller area, increasing warmth). This is why fans with water cooling and tips like hanging a wet T-shirt in front of a fan, constantly misting yourself or wearing wet clothes etc. can sort of backfire and make your home a bit more unbearable, depending on the circumstances. I also have a fan with water cooling with optional cooling bricks/batteries, and it’s currently on because we hang out in front of it, but I’m mindful of when I turn that mode on and for how long. In the next few weeks, we are planning to add sun protection foil to some windows, and when the extreme demand is over in fall, I’ll buy a Midea Porta Split and install it in the living room. Good tips in general, some summarized from above: Hydrate a lot, even before you are actually thirsty. Stay inside if possible. Keep the added humidity to a minimum. Know what you are trying to do with drinks and showers. Cool drinks and showers offer relief, but can make you heat up after. Hot beverages and showers can make everything feel cooler after and help you sweat. I like both, depending on the situation. Wrap ice packs or similar stuff in a towel and put them under your feet or in your armpits. If possible, lower shutters so the sun cannot heat up the interior and the glass. Maybe install sun protection foil on windows (most are plant-friendly). I’ve also seen others provisionally use those reflecting covers for cars on their windows, or aluminium foil. Make sure that if it’s behind the glass, the heat won’t be trapped and make the glass crack, so preferably attach it on the outside. Sunscreen, wide breathable and covering clothing, sun umbrellas and hats. During fall/winter, maybe during Black Friday sales, get a portable split cooling system. Portables do not need structural changes to the building, which is why they tend to be allowed in rental units as they can be removed without a trace and aren’t in use all year. Shitty landlords might get mad to see it in your window, but in many countries, there already is positive case law about them and the usual AC dismissals don’t apply to them. Set out flat bowls of water in the shadow for wild animals and refill. Consider different ones for different sizes (a flat one with stone pebbles for insects, a relatively flat but water-only one for hedgehogs etc., one bird bath…). Use cool tiles and cooling mats for pets. Keep an eye out for baby birds who flee their overheated nests too early; maybe you can save some of them. Especially bitdd who live in attics and roofs are dying right now (swifts etc.) If possible and you can plan the shipment, avoid deliveries. Keep water around for delivery personnel. Eat smaller snacks and portions spread out throughout the day instead of big meals so your body doesn’t heat up as much during digestion. Leave the windows open all day. Let the sun heat up your interior, if possible; try at least covering windows with blankets if there are no shutters. Set out water for animals where it heats up drastically, or in a beverage where they might become trapped and drown. Walk your dog when the ground is heated up - asphalt burns happen quickly past 25 degrees Celsius. Fall for scalpers, scammers and increased prices for ACs and fans who are using the current demand and availability issues for profit. The Porta Split I mean to get can be bought for 550-750 Euro under normal circumstances, now during the heat wave, prices have exploded to over 1.4k. Only buy that if it is an emergency. Think fans or ACs can make you sick. This is a widely held belief especially in older generations in Germany at least, together with the myth that any wind can cause a cold and stiff neck. It is bullshit. It’s a big reason why this country is not prepared for this heat and there’s a 20% adoption rate for ACs here. Think you need to keep the fan off or not buy one at all because of the electricity bill. The increase is lower for newer models and for the few days you need to use it (more) (for now). You are also not meaningfully contributing to climate change with this increased energy use. Like, come on, they wanna build entire data centers eating away gigawatts, your heat protection is not the issue here. Still, all of these tend to be hyperindividualistic solutions, just like when Covid happened, and we need more widespread, structural solutions. Not everyone can stay home; many people still have to work and commute. You might tell people to hydrate as much as possible, but their work doesn’t offer free (or extra) water to them, and many places like restaurants and cafés still don’t. We tell people to invest in ACs and fans, but landlords and workplaces don’t want to install any, forbid the use, or don’t cover the price of these things. It’s like heat management is still an incredibly personal thing where everyone has to feel like they are fending for themselves, investing their own money into stockpiling resources and tech, and utilizing the privilege to avoid a lot of the heat by working from home/working inside, taking time off, calling in sick and so on. More collective heat management can look like: Free water in establishments everywhere, and drinking fountains spread throughout cities, with signs pointing to the next one. Designating libraries, community centers, schools, transit hubs and big shops like huge supermarkets as cooling centers during heat waves. Keeping trees, bushes, grass etc. intact and adding more. They help keep cities cooler, together with reflective roofs and lighter pavements. Legally mandating landlords to install ACs in rental units, especially ones directly below the roof (attic/loft/penthouse apartments), and cover specific windows in protective foil or external shutters. Requiring new(er) buildings to have specific insulation that helps in summer as well as winter, ventilation strategies, ACs, etc. and updating building codes so new housing remains habitable during prolonged heat waves, even without continuous air conditioning. More shaded areas in crowded places, waiting spots (public transportation), shaded pathways between major destinations. Rollout of functioning and resilient AC in all public transportation, hospitals, schools, universities, elderly homes etc. Extending opening hours into the early morning and late evening during extreme heat, with closure inbetween (or at the bare minimum, siestas). Temperature thresholds that trigger additional protections or suspension of certain work or studies. Preparing railroads, normal roads and other parts of the public from the intense heat effects or making them more heat resistant; otherwise you risk bent rails, melting bitumen etc. Distributing fans or subsidizing cooling equipment where appropriate. Strengthening electrical grids to cope with increased cooling demand, subsidizing electricity costs during declared heat emergencies, expanding renewable generation to reduce the emissions associated with increased cooling needs. And likely more I forgot. Yes, people will cry that this costs soooo much money. But remember that we have no problem investing that money into wars, AI, data centers, expensive proprietary software licenses, politicians’ money schemes and making billionaires richer. Landlords need to invest the rent into the property instead of enriching themselves and getting other people to pay off their mortgage. These aren’t one-time events, it will continue to get worse. Earlier in the year, longer, higher. Many people and animals will die. Everyone has to start preparing and learning from it now, and stop buying into the bullshit that “it was hot when I was a child too, we are just complaining more!!1!”. Your government is failing you if they are not acting now, and it is intentional, as the heat affects vulnerable and powerless groups the most. Make sure you check on old, sick, disabled people and people you know who take medication that makes them more vulnerable to the sun and/or heat. For example, diuretics, beta blockers, anticholinergics, and some antidepressants and stimulants. Reply via email Published 27 Jun, 2026 Hydrate a lot, even before you are actually thirsty. Stay inside if possible. Keep the added humidity to a minimum. Know what you are trying to do with drinks and showers. Cool drinks and showers offer relief, but can make you heat up after. Hot beverages and showers can make everything feel cooler after and help you sweat. I like both, depending on the situation. Wrap ice packs or similar stuff in a towel and put them under your feet or in your armpits. If possible, lower shutters so the sun cannot heat up the interior and the glass. Maybe install sun protection foil on windows (most are plant-friendly). I’ve also seen others provisionally use those reflecting covers for cars on their windows, or aluminium foil. Make sure that if it’s behind the glass, the heat won’t be trapped and make the glass crack, so preferably attach it on the outside. Sunscreen, wide breathable and covering clothing, sun umbrellas and hats. During fall/winter, maybe during Black Friday sales, get a portable split cooling system. Portables do not need structural changes to the building, which is why they tend to be allowed in rental units as they can be removed without a trace and aren’t in use all year. Shitty landlords might get mad to see it in your window, but in many countries, there already is positive case law about them and the usual AC dismissals don’t apply to them. Set out flat bowls of water in the shadow for wild animals and refill. Consider different ones for different sizes (a flat one with stone pebbles for insects, a relatively flat but water-only one for hedgehogs etc., one bird bath…). Use cool tiles and cooling mats for pets. Keep an eye out for baby birds who flee their overheated nests too early; maybe you can save some of them. Especially bitdd who live in attics and roofs are dying right now (swifts etc.) If possible and you can plan the shipment, avoid deliveries. Keep water around for delivery personnel. Eat smaller snacks and portions spread out throughout the day instead of big meals so your body doesn’t heat up as much during digestion. Leave the windows open all day. Let the sun heat up your interior, if possible; try at least covering windows with blankets if there are no shutters. Set out water for animals where it heats up drastically, or in a beverage where they might become trapped and drown. Walk your dog when the ground is heated up - asphalt burns happen quickly past 25 degrees Celsius. Fall for scalpers, scammers and increased prices for ACs and fans who are using the current demand and availability issues for profit. The Porta Split I mean to get can be bought for 550-750 Euro under normal circumstances, now during the heat wave, prices have exploded to over 1.4k. Only buy that if it is an emergency. Think fans or ACs can make you sick. This is a widely held belief especially in older generations in Germany at least, together with the myth that any wind can cause a cold and stiff neck. It is bullshit. It’s a big reason why this country is not prepared for this heat and there’s a 20% adoption rate for ACs here. Think you need to keep the fan off or not buy one at all because of the electricity bill. The increase is lower for newer models and for the few days you need to use it (more) (for now). You are also not meaningfully contributing to climate change with this increased energy use. Like, come on, they wanna build entire data centers eating away gigawatts, your heat protection is not the issue here. Free water in establishments everywhere, and drinking fountains spread throughout cities, with signs pointing to the next one. Designating libraries, community centers, schools, transit hubs and big shops like huge supermarkets as cooling centers during heat waves. Keeping trees, bushes, grass etc. intact and adding more. They help keep cities cooler, together with reflective roofs and lighter pavements. Legally mandating landlords to install ACs in rental units, especially ones directly below the roof (attic/loft/penthouse apartments), and cover specific windows in protective foil or external shutters. Requiring new(er) buildings to have specific insulation that helps in summer as well as winter, ventilation strategies, ACs, etc. and updating building codes so new housing remains habitable during prolonged heat waves, even without continuous air conditioning. More shaded areas in crowded places, waiting spots (public transportation), shaded pathways between major destinations. Rollout of functioning and resilient AC in all public transportation, hospitals, schools, universities, elderly homes etc. Extending opening hours into the early morning and late evening during extreme heat, with closure inbetween (or at the bare minimum, siestas). Temperature thresholds that trigger additional protections or suspension of certain work or studies. Preparing railroads, normal roads and other parts of the public from the intense heat effects or making them more heat resistant; otherwise you risk bent rails, melting bitumen etc. Distributing fans or subsidizing cooling equipment where appropriate. Strengthening electrical grids to cope with increased cooling demand, subsidizing electricity costs during declared heat emergencies, expanding renewable generation to reduce the emissions associated with increased cooling needs.

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DYNOMIGHT 1 months ago

The worthlessness of vitamin D is mildly exaggerated

For a while there, many people thought vitamin D was magical—that it could improve bones, the heart, infections, cancer, heart disease, longevity, even mental health. But among people I respect, opinion is now overwhelmingly that taking vitamin D does nothing unless you’re severely deficient. The central argument is that while vitamin D levels are correlated with ~all positive health outcomes, when you actually test vitamin D supplements against placebo in randomized trials, nothing ever happens. That’s what I used to think, too. But I’ve come to think the skeptics have over-corrected. Yes, randomized trials have shown the magical correlations are not causal. But if you start with non-insane expectations, the trials look like weak but positive evidence. And if you consider what we know about biology and evolution, I think the balance of evidence tips pretty clearly in the direction that people with low-ish levels would be wise to supplement. Am I certain that vitamin D is beneficial for people with low-ish levels? Absolutely not! But I claim that’s the best bet given the limits of our knowledge. Most vitamins are “ingredients” that the body uses to do stuff. Vitamin D is more like a “signal” that the body uses to communicate with itself about what to do. 1 The classical “endocrine” story of vitamin D is that your body uses it to tell your guts to take in more calcium from food. If you don’t get enough vitamin D, then you have calcium problems. That’s all you really need to know about the classical view. But if you enjoy gawking at biology’s complexity, I recommend this diagram and the following three paragraphs: Ready for science? OK: Almost all the cells in your body make provitamin D . 2 Usually, this is all converted to cholesterol, but your skin cells leave some sitting around. When UVB light hits those skin cells, provitamin D is transformed (physically by the light itself) into previtamin D and then (by heat) into vitamin D . This diffuses from the skin cells into blood vessels. There it binds to a protein 3 and starts circulating in the blood, where it is joined by vitamin D from food. 4 Eventually, the liver converts it into more-stable storage vitamin D . It also soaks in and out of fat and muscle tissue, which acts as a slow-release reservoir. Now, a fun fact: If calcium levels in your blood get too low, then your heart will stop working and you will die. To avoid this, you have parathyroid glands in your neck that sense when calcium is getting low, and release parathyroid hormone into the blood. This tells your bones to release some of their stored calcium. It also tells your kidneys to convert some of the storage vitamin D from your blood into active vitamin D . And when that gets to your guts, they try to absorb more calcium from food. So what happens if you don’t get enough vitamin D? Well, your body is not going to let calcium levels drop too low, because your body is designed to avoid death. Parathyroid hormone will still get secreted, and it will still tell your bones to scavenge calcium. But without vitamin D, your guts never get the signal to gather extra calcium from food. So the body scavenges a lot of calcium from your bones, and you end up with weak bones, which is bad. Now here’s the thing: In this story, only active vitamin D actually does anything. The kidneys make this on demand in response to calcium levels, not in response to storage vitamin D levels. General opinion is that as long as the blood has above ~25 nmol/L of storage vitamin D, then the kidneys have no trouble making active vitamin D. 5 Furthermore, survey data suggests that only ~2% of the population has levels below that threshold. This suggests that for ~98% of people, supplementing vitamin D should do approximately nothing. Rickets is a terrible disease that involves soft bones, stunted growth, and skeletal deformities. It’s probably been with us since ancient times, but it became common in the West after the industrial revolution. In 1890, a Scottish missionary named Theobald Palm observed that rickets was common in smog-ridden UK cities but almost unheard of in sunny countries with poor sanitation, suggesting sunlight itself was the issue. This contributed to the discovery that rickets could be cured with UV light or cod-liver oil, and eventually the discovery of vitamin D. In 1941, Apperly noticed that the amount of sunlight in different US states was positively correlated with skin cancer but inversely correlated with overall cancer mortality. 6 He gave this charming graph: Apperly never mentions vitamin D, presumably because he thought it was a boring bone vitamin. Things took off in 1980, when Cedric and Frank Garland published, “Do Sunlight and Vitamin D Reduce the Likelihood of Colon Cancer?” Seemingly unaware of Apperly, they gave a similar, but uglier, graph: They point out that regional diets (like meat and fiber) didn’t seem to explain this pattern. Instead, they propose a mechanistic story: Sunlight         ↓     Vitamin D         ↓     Adequate calcium in blood         ↓     Reduced inflammation of epithelial cells in the colon         ↓     Less colon cancer (It’s always inflammation.) This paper was rejected many times before finally being published. I wish I could find an un-gated copy to link to, because it would have made a magnificent blog post. 7 Following that paper, there was an explosion of work that found negative correlations between sunlight (or latitude) and other types of cancers as well as blood pressure , diabetes , and multiple sclerosis . Then people started measuring vitamin D in blood. In 1989, the Garlands and collaborators found blood samples takin in 1974 from 25,000 people. They found that 34 of those people had since gotten colon cancer. They matched these with 67 demographically similar people and measured vitamin D levels in the stored blood samples for all 101 people. Among that group, people with vitamin D levels below 50 nmol/L got colon cancer more than three times as often as people with higher levels. Again, many similar studies followed. These linked higher vitamin D levels to better outcomes in cardiovascular disease, diabetes, obesity, infectious disease, Parkinson’s, and mood disorders. While results were mixed for non-colorectal cancer incidence, higher vitamin D levels predicted better survival of many cancers. Amazingly, all-cause mortality was roughly 30% lower for those at the 75th percentile of vitamin D levels compared to the 25th. Vitamin D was looking like a miracle. But how could it do all that stuff if it was just a boring bone vitamin? While all these correlations were being discovered, we learned that the body doesn’t just use vitamin D for bone stuff. In 1969, we discovered the vitamin D receptor that active vitamin D binds to in the gut and bones. And in the 1980s came a shock: Almost all cells in the body have vitamin D receptors. These seem to do different things in different tissues. In the pancreas, they support insulin secretion. In immune cells, they boost antimicrobial peptides and reduce inflammation. In neurons, they influence proliferation and differentiation. So… What? When calcium drops and the kidneys put out active vitamin D, does every part of the body start doing different unrelated stuff? In the late 1990s, we cloned the gene for the enzyme that the kidneys use to convert storage vitamin D to active vitamin D. Soon came another shock: This enzyme also exists in tons of other cells, including immune cells, the heart, the skin, the prostate, the breast, and colon. (Another win for the Garlands.) So it’s not just the kidneys making active vitamin D to trigger the gut. Cells everywhere are making their own active vitamin D and using it to trigger vitamin D receptors in neighboring cells, or even inside the same cell. 8 This often has little to do with calcium or bones. 9 And remember how only active vitamin D does anything? That’s wrong. In the mid-1970s, we learned that storage vitamin D also binds to the vitamin D receptor. The affinity is 100-1000× lower, but have ~1000× more in your blood. So maybe circulating levels of storage vitamin D themselves matter, independently of how much active vitamin D gets made? If that’s not confusing enough, people also noticed that while active vitamin D levels in the blood aren’t correlated with storage vitamin D (above ~25 nmol/L), levels of parathyroid hormone (the thing your parathyroid glands use to tell your kidneys to make active vitamin D) seem to decline as levels of storage vitamin D rise from ~25 to 50 or 75 nmol/L. Huh? 10 On the one hand, all this makes the idea that vitamin D could be a miracle more plausible. On the other hand, this is getting complicated. And do we really believe that raising your vitamin D levels from the 25th to the 75th percentile could reduce your risk of death from any cause by thirty percent ? Maybe we should try giving people vitamin D and see what happens. There have been many randomized trials. The “right” thing to do in such cases is to look at meta analyses that carefully combine all the data. We’ll get to those. But they conceal a lot of important nuance about what actually happens on the ground during these trials. So let’s start by going over the three main “megatrials”. The Women’s Health Initiative (WHI) trial came out in 2006 and is still the largest vitamin D trial ever done. This took 36,000 postmenopausal American women and assigned half to take 400 IU daily with calcium and the other half to placebo. 11 After seven years, here’s what happened: 12 (The hazard ratio is the ratio of the rate that something happens in the treatment vs. placebo groups. So, a number less than one suggests a benefit to taking vitamin D, while a number larger than one suggests a harm. The numbers in parentheses show a 95% confidence interval.) The only statistically significant result was a bad one: Extra kidney stones, likely from the extra calcium. 13 The other outcomes look vaguely good, but none were statistically significant despite the massive sample size. This was disappointing. However, the WHI trial had limitations: Many subjects in both the vitamin D and placebo groups were already taking vitamin D, and continued taking it through the trial. The dose of 400 IU was fairly low, many subjects stopped taking their pills, and vitamin D levels didn’t actually change that much. They also measured vitamin D levels in only 6% of subjects, meaning we can’t compare the fates of subjects who started out with low versus high levels. The next big hope was VITAL, which came out in 2018. They recruited 26,000 older people across the United States, half of them men and 20% Black (and thus far more likely to be vitamin-D deficient). They measured vitamin D levels in most people, and they gave the treatment group 2,000 IU per day. 14 Here were the results after 5.3 years: Some of the results look good-ish, but cardiovascular mortality was higher in the treatment group, leading to almost no effect on all-cause mortality. 15 More disappointment. The last megatrial was D-Health, which came out in 2022 based on 21,000 older Australians. Instead of daily supplements, it used a monthly “bolus” dose of 60,000 IU or placebo. Unlike in VITAL, there was no exclusion for people with a history of cardiovascular disease or cancer, and less restriction on how much vitamin D participants could take on their own during the trial. 16 Here were the results after 6 years: Now, the treatment group did better in terms of cardiovascular disease, but worse in cancer and worse in all-cause mortality. Even more disappointment. Just from these three large trials, the main lesson should already be clear: Vitamin D is not a miracle. The correlations were wrong. 17 There is essentially zero remaining hope that taking vitamin D could reduce all-cause mortality by a third. In this sense, the vitamin D skeptics are definitely right. But what about the other trials? And is there a more subtle lesson? I wanted a big table that summarized all the major vitamin D RCTs and what they found for different health outcomes. Annoyingly, no such overview appears to exist. So I made my own: 18 Lots of the hazard ratios are less than one, suggesting a benefit to supplementation. But lots of them are also higher than one, suggesting a harm. The numbers that are far from one almost always come from smaller trials, which manifest as larger confidence intervals. If you’re interested in the details of how these trials were run, I refer you to more gigantic tables in a footnote. 19 If big tables aren’t your thing, here are some formal meta-analyses, both some recent ones and an older but more comprehensive Cochrane review: There are various ways you could try to squint at these RCT. In almost all of them, most people already had pretty high levels before they started. So why don’t we separate out people who started low? Usually we can’t, because most trials didn’t measure baseline vitamin D. 20 And among the trials that did, there are few people with low levels, so the results are noisy and confusing. 21 Or, you might theorize that benefits would take time to show up, meaning the first couple years just add noise. In some cases—notably VITAL—excluding the first two years seems to help, but in other cases things get worse. 22 Finally, some people speculate that taking gigantic monthly or quarterly “bolus” doses of vitamin D might be dangerous. For example, here’s an enjoyable paragraph from Kunzia et al. in their meta-analysis of vitamin D and cancer mortality: Our results showing efficacy of daily, but not bolus, vitamin D3 supplementation in reducing cancer mortality are consistent with previous meta-analyses on cancer mortality or all-cause mortality (Guo et al., 2022; Keum et al., 2022; Keum et al., 2019; Zhang et al., 2022; Zhang et al., 2019). However, by including more trials than these previous meta-analyses, we were able to detect statistically significant effect modification by treatment regimen for the first time with statistical significance (pinteraction=0.042). The pattern of intake could be important for a favourable steady state of the bioavailability of the active 1,25 (OH)₂D hormone. Daily administration counteracts the fast excretion of vitamin D from the circulation (Hollis and Wagner, 2013; Keum et al., 2022). Moreover, the enzymes CYP27B1 (converts 25(OH)D to 1,25 (OH)₂D) and CYP24A1 (inactivates 25(OH)D and 1,25(OH)₂D) follow first-order reaction kinetics (Vieth, 2009). This means that doubling the concentration of the precursor doubles the yield of the product, unlike other steroid hormones (e.g., cortisol, oestrogen, testosterone) that follow zero-order kinetics (Vieth, 2020). Intermittent, non-physiologically large vitamin D3 bolus doses may lead to unstable cycling of 25(OH)D and 1,25(OH)₂D levels in blood because the system needs time to adapt to the large doses (Hollis and Wagner, 2013; Keum et al., 2019; Vieth, 2020). In the long run, intermittent bolus regimens at weekly or larger intervals can lead to an up-regulation of countervailing factors (e.g., 24-hydroxylase (CYP24A1), 24,25(OH)2D and fibroblast growth factor 23), all of which ultimately leads to lower synthesis or higher degradation of 1,25(OH)₂D levels (Mazess et al., 2021). Bolus doses, unlike daily doses, failed to reduce C-reactive protein response and actually elevated anti-inflammatory cytokines and doubled the risk of hypercalcemia in previous studies (Krishnan et al., 2012; Martineau et al., 2017; Mazess et al., 2021). Oh no, up-regulation of fibroblast growth factor 23! 23 I don’t feel like I understand this deeply enough to have any opinion beyond the surface level that the body seems to adapt to large doses of vitamin D in ways that could possibly be bad. 24 It seems intuitive that small daily doses would be safer than gigantic monthly doses, but I’m always suspicious of post-hoc mechanistic speculation. Also, if people get enough sun, they can apparently synthesize 10,000-25,000 IU per day, which isn’t that far from the 60,000 IU they got in the D-Health trial. But then again, I think Kunzia et al. are suggesting that the body is designed to adapt to regular exposure to large doses but not intermittent exposure? Well, if you split up the trails by daily vs. bolus dosing, there’s a decent pattern of daily dosing leading to better results: If those bolus dosing trials didn’t exist, I’d think this looked pretty good. So, maybe? Or maybe this is a story made up to hallucinate a positive trend. I would lean towards the latter theory, but there are papers like Mazess et al.’s “Vitamin D: Bolus is Bogus” , that suggested this pattern before D-Health’s dismal results came out. There are even some trials that suggest bolus doses don’t even work for treating rickets. So… I’m still not convinced. But maybe. Aside: There are also many Mendelian randomization studies that look at correlations between health and genes that are related to vitamin D. But I don’t think these provide much information, because the assumptions are shaky and the genes don’t explain much of the variance. 25 Still with me? Here’s a summary of the above 5200 words: So you might be wondering: Isn’t that quite weak? Wasn’t this post supposed to be a defense of vitamin D? Everyone agrees that severe vitamin D deficiency (below ~25 nmol/L) is bad. It leads to rickets , adult rickets , osteoporosis , muscle weakness or even—with profound deficiency—to seizures or cardiac arrhythmia. This makes sense, because below ~25 nmol/L, the kidneys have trouble converting storage vitamin D into active vitamin D, meaning you don’t absorb enough calcium from food. The question is if taking supplement to further raise your levels (say to 50 or 90 nmol/L) is important. We have no mechanistic proof, but it might be true, because many parts of the body use vitamin D as a local signal and because cells are at least somewhat sensitive to circulating storage levels. There’s also this weird thing where parathyroid hormone continues to decline as vitamin D levels rise above ~25 nmol/L even while this seems to make little difference to how much active vitamin D the kidneys make. Nothing in this world comes without trade-offs. Surely, supplementing vitamin D comes with some downsides. But it seems very unlikely that raising vitamin D levels to a “normal” level would cause more harm than benefit. Especially because… According to Luxwolda et al.’s 2012 paper, “Traditionally living populations in East Africa have a mean serum 25-hydroxyvitamin D concentration of 115 nmol/L” , traditionally living populations in East Africa have a mean serum 25-hydroxyvitamin D concentration of 115 nmol/L. Meanwhile, Wahl et al. 2012 try to estimate mean levels around the world today: This map looks weird because of varying lifestyle, diet, supplementation, and needing to combine fragmented studies. But you get the idea. And remember, those are just averages. So there are lots of people with levels far lower than that in our evolutionary history. Of course, just the fact that vitamin D levels have dropped doesn’t mean it’s important. Parasitic worm load, wood smoke inhalation, and cousin marriage have also dropped, but we aren’t rushing to restore those to ancestral levels. But there’s another piece of evidence: After humans migrated out of East Africa, some of them evolved pale skin. Pale skin is bad, because it allows light to destroy folate, which is crucial for pregnancy. 26 Evolution doesn’t typically do things that harm fertility, because evolution wants to increase reproductive fitness. The most common explanation is that pale skin allows more UV light to penetrate, and thus allows people to synthesize more vitamin D. If evolution was willing to pay the high “price” of folate destruction for more vitamin D, that seems like good evidence that vitamin D is important. Some even see contrasts like the Inuits versus Scandinavians as a kind of natural experiment: They lived at similar latitudes, but Inuits ate a diet with vitamin D (fatty fish and whale blubber) and Scandinavians didn’t. The result is that Inuits have darker skin than Scandinavians. 27 This is all speculative, and even if true, might be driven by severe deficiency and rickets. Or perhaps prehistoric benefits don’t translate to your lifestyle. But all the people in Luxwolda’s sample in East Africa had levels above ~60 nmol/L. I just don’t see how you can look at this and not see it as providing some suggestive evidence in favor of the idea that raising levels above severe deficiency is unlikely to be harmful, and could be important. So I think the prior is favorable. A hazard ratio like HR = 0.96 doesn’t look very impressive. But hold on. Suppose that life expectancy is 80 years and that taking vitamin D every day reduces your risk of all-cause mortality by a factor of HR . A reasonable approximation in rich countries is that this would increase your life expectancy by 80 × 0.15 × (1-HR) years = 12 × (1-HR) years, where 0.15 is derived from the entropy of lifespan in rich countries. 28 For example, if all-cause mortality had a true hazard ratio of HR = 0.96, then taking vitamin D every day of your life would increase life expectancy by around 0.48 years. I claim that this would be a lot. Certainly, if I were about to face my destiny, I would pay a lot of money for an extra 0.48 years. Or, you can calculate that this corresponds to an increase of life expectancy per-vitamin-D-pill of 8.6 minutes. 29 A common rule-of-thumb is that smoking a cigarette costs around 11 minutes of life in expectation. If you think HR = 0.96 is trivial, do you also think that smoking one cigarette each day is fine? 30 The correlational studies suggested that vitamin D might drop your risk of all-cause mortality by a third. It’s disappointing that the RCTs refuted this. But those correlational studies were crazy. They imply 31 an increase of life expectancy of around 4 years or around 6.5 cigarettes per day. Could we really believe that you could smoke 6.5 cigarettes, then take a vitamin D pill, and you’re even? Personally, I think hazard ratios just slightly less than one are the best we can reasonably hope for. But I also think that they would be an excellent return on investment. Arguably, modern human life expectancy comes from stacking lots of modest hazard ratios on top of each other. Let’s play a game. Let’s hallucinate some numbers for what vitamin D might do, and then simulate what trials would show. Here are the strongest effects I consider plausible for different baseline levels, along with how common those levels are in the United States. Suppose that were real. Now, say we pick 26,000 people at random, and give half of them vitamin D for give yars. Here are the results of a million simulated trials, assuming a baseline mortality risk of 0.7%: 32 Overall, 9% of trials would find a significant benefit, 63% would find a non-significant benefit, 27% would find a non-significant harm, and 1% would find a significant harm. If you wanted to have an 80% chance of finding a significant decrease, you’d need to run a trial with something like 570,000 people, almost five times more than in all the above trials combined. 33 If you don’t like my numbers, I’ve put up a page where you can run your own simulations with different ones. My point is, the results we see in vitamin D RCTs are what we should expect to see if vitamin D had plausible benefits. That’s not proof, of course—just that if you start with realistic expectations, the trials don’t provide much evidence in either direction. Recent meta-analyses have not consistently found a statistically significant benefit to vitamin D supplementation. But they do suggest a small benefit for cancer mortality and all-cause mortality, and they’re close to being statistically significant. That’s something . And if you buy the argument that bolus dosing is bad, the results get even better. Kunzia et al. did a meta-analysis of cancer mortality using only trials with daily dosing, and found a hazard ratio of 0.88 (confidence interval 0.78 to 0.98). I’d keep this at arm’s length. The bolus dosing trials might have done worse by random chance, meaning this is a kind of p-hacking. But there’s a reasonable chance (maybe 25-50%) that bolus dosing really is bad, in which case those trials would be convincing evidence. I actually think it’s surprising that the meta-analyses look as good as they do, because there just aren’t that many people who started out with low vitamin D levels. Only a handful of trials had mean levels below 60 nmol/L, and they all give semi-promising results: 34 Again, it’s dangerous to dig too deeply looking for these kinds of patterns. If you dig enough, you can always find a way to confirm whatever theory you want. But also again, maybe? You might not personally supplement vitamin D. But for most people reading this, someone else is supplementing it for you. 35 Fortified food is common across the Anglosphere and Scandinavian peninsula. However, it’s rare in the rest of Europe (exceptions: Belgium, Poland) and even-more rare in the rest of the world (exceptions: Chile, Ethiopia, Pakistan). I think this is important for two reasons. First, vitamin D is oddly self-defeating. There are some places in the world where people care about vitamin D. These are the places that run large trials. But these places also fortify their food and tend to be full of people that already supplement vitamin D. These places also tend to believe it’s unethical to tell the control group not to take vitamin D. And here’s another question: If you think vitamin D is worthless, are you comfortable recommending removing vitamin D from food? If not, then why is the particular amount of fortification in food now the right one? Some might argue that the purpose of fortification is to reach the severely deficient, or children, the elderly or pregnant mothers. Maybe! But again, if you could press a button and remove fortification from everyone else, would you feel comfortable pushing that button? Remember, trials don’t test going down from current levels, only going up. This is all very weak, I know! But sometimes weak evidence is all we’ve got. I wish we had at least one large trial done in a population with low starting levels. But as far as I can tell, none are underway. In fact, it’s unlikely that there will be any more large trials anytime soon. So weak evidence is how it’s going to be. Technically, vitamin D itself is a type of steroid although not what people usually mean by “steroid”.  ↩ Here are some of the fancy names for the different forms of vitamin D I’ll talk about: Charmingly named “vitamin D-binding protein” .  ↩ If you eat mushrooms or yeast, it joins the vitamin D from your skin en route to your liver. If you eat animals or animal products, you also get some storage vitamin D, which doesn’t need to be processed by the liver.  ↩ Storage vitamin D is what your doctor measures in your blood test. This is sometimes measured in nmol/L and sometimes in ng/mL. The latter measurement is smaller by a factor of 2.496. So 25 nmol/L ≈ 10 ng/mL.  ↩ Apperly was building on a 1937 paper that observed observed that sailors, exposed to lots of sunlight, had much higher skin cancer rates than the general population, but lower overall cancer rates.  ↩ I theorize that the Garland brothers are alive and writing Slime Mold Time Mold .  ↩ In Biologist, active vitamin D is not just an “endocrine” hormone that sends signals for far away cells through the blood, it’s also a “paracrine” or “autocrine” hormone that sends signals to nearby cells or inside a single cell, through diffusion.  ↩ You might ask, why is vitamin D used by so many different parts of the body for so many different purposes? I think there’s no deep answer here. It’s true for the same reason that dogs sneeze to signal that they’re feeling playful: Evolution re-uses stuff for different purposes all the time. Imagine that DNA already exists coding for the vitamin D receptor and for the enzyme to convert storage vitamin D into active vitamin D. If some cells need to send a local signal, re-using those is easier than inventing something new. There’s nothing unusual or magical about this.  ↩ Don’t try to make sense of this. It doesn’t make sense. You could speculate that this is because the parathyroid glands are trying to make less active vitamin D to compensate for the fact that vitamin-D receptors throughout the body are sensitive to storage vitamin D itself. But I advise against.  ↩ 400 IU is the recommended daily amount  ↩ The WHI trial was a pioneer in salami-slicing results for different outcomes into dozens of different papers, most of which are hard to access. All trials now seem to have adopted this hideous trend which makes it maddening to try to summarize what actually happened in a trial. Also, slightly different numbers for the same quantity appear in different places. I haven’t bothered to chase these down, because the differences are all very small, e.g. a hazard ratio of 0.89 for cancer mortality rather than 0.90.  ↩ Guess what most kidney stones are made of?  ↩ Half of the vitamin D group and the placebo group also got omega 3. These are averaged together in the results. Also, VITAL carefully stratified the assignment to vitamin D or placebo based on baseline vitamin D levels, which should give more statistical power from a given sample size.  ↩ There was also a weird study done on a subset of 1031 people from the VITAL population that looked at telomere length. After starting with around 8700 base pairs, the control group lost around 160 base pairs during the study, while the vitamin D group only lost an average of 20. I’m not sure of what to make of this. For one thing, though the authors claim this is statistically significant, it depends on how you analyze the data. But beyond that, sure, telomere length is a marker of aging, but telomeres get shorter for a reason (likely to fight cancer) and it isn’t obvious that slowing this would always be a good thing.  ↩ This is a little complicated. In VITAL, participants were only eligible if they were taking at most 800 IU per day, and they were restricted to 800 IU per day during the trial. In D-health, participants were only eligible if they were taking at most 500 IU per day, but they were allowed to take up to 2000 IU per day during the trial.  ↩ You might ask: If vitamin D only has a modest effect, then why is it so strongly correlated with health? In principle, I’d like to push back against the idea that we need to explain why these particular correlations don’t imply causation. But the accepted explanation is a combination of (1) reverse causation where being healthy causes people to spend more time outside and thus get more vitamin D; (2) confounding, where obesity is bad for you and leads to lower measured vitamin D levels; (3) confounding, where more healthy lifestyles lead to both more vitamin D and more health; and (4) confounding, where higher socioeconomic status leads to both more vitamin D and more health. You might ask why these correlations would be true at a state level like the Garlands looked at, but then you run into the ecological fallacy and modifiable areal unit problem .  ↩ I took all the trials that got at least 2% weight and were rated as “low risk of bias” in this 2014 Cochrane review of vitamin D and mortality, then manually added all the “major” trials that were published after 2014. I shudder to think of the time it took to make this table. I tried using AI but found it was wildly unreliable. Part of the problem is that each trial’s results are distributed among many papers, in different journals, with different paywalls. And many details aren’t published at all by the original authors but are only scrounged up and put in the depths of the supplementary material of a review years later. In some cases, different sources also give contradictory numbers. The differences were always tiny (e.g. 0.90 rather than 0.89) but it still makes me nervous.  ↩ Here’s a table describing the major contours of the trials: And here’s a table focusing on the change in vitamin D levels: Among the major trials, only VITAL, ViDA, and FIND measured it for more than a tiny number of subjects.  ↩ In VITAL and ViDA, people with baseline levels below 50 nmol/L had a higher hazard ratio for cancer mortality (though with wide confidence intervals), suggesting if anything less benefit. Or, you could use race as a proxy for baseline vitamin D. But in both VITAL and WHI, the hazard ratio for cancer mortality was higher among non-Whites. After looking at many such analyses for many outcomes, the only clear result I could find was for diabetes in the D2d trail, where the hazard ratio was much lower for people below 30 nmol/L (0.38 vs. 0.93).  ↩ The results for VITAL look decent: But in D-Health, excluding the first two years actually increased the hazard ratio for cancer mortality from 1.15 (0.96 to 1.39) to 1.24 (1.01 to 1.54). Most other trials were too short for this kind of analysis to make sense.  ↩ That could downregulate 25-hydroxyvitamin D 1-alpha-hydroxylase, reducing the rate it catalyzes the hydroxylation of hydroxycholecalciferol into 1,25-dihydroxycholecalciferol!  ↩ Dynomight: WTF is this? Dynomight Biologist: Well, C-reactive protein is generally considered inflammatory. Dynomight: So reducing that is good? But then why do they talk like elevating anti -inflammatory cytokines would be bad? Dynomight Biologist: Yeah… That would be good. Unless you have cancer. In which case it’s not good. Dynomight: OK!  ↩ Mendelian randomization studies are based on the idea that certain genes predispose you to have higher levels of circulating vitamin D. If you assume that those genes are randomly distributed in the population and have no effects other than affecting vitamin D, then they serve as a kind of natural experiment. With vitamin D, these studies typically show null results . However, the validity of the assumptions is debatable and the identified genes only explain ~5% of the variance in vitamin D levels, which makes the results very noisy.  ↩ Pale skin also greatly increases the risk of sunburn and skin cancer. In the US, White people get melanoma at around 25 times the rate of Black people, despite (I assume) higher usage of sunscreen and better health outcomes in most other dimensions. But experts generally think folate deficiency created stronger selective pressure, since it’s so closely linked to reproduction.  ↩ It’s a more complicated than this, because you also need to look at the amount of folate in diet, as well as migration patterns and how long populations had to adapt to their environment. But experts seem to consider this the leading explanation for the evolution of pale skin.  ↩ To derive this, suppose that S(t) is the probability that someone survives to age t. Then life expectancy is ∫ S(t) dt , where the integral runs from 0 to ∞. If you change the hazard ratio by a factor of HR , then the new in life expectancy is L(HR) = ∫ S(t)ᴴᴿ dt , so the change under a linear approximation is ΔL ≈ (HR-1) × L’(1) . This is more commonly written as ΔL ≈ (HR-1) × L(1) × H , where H = -L’(1)/L(1) is known as the Keyfitz entropy . This is is chosen because the quantity H is relatively stable, and in rich countries is typically between 0.10 and 0.20. So a decent estimate would be that baseline life expectancy is L(1)=80 years and H = 0.15 in which case the change in life expectancy is around 12 × (1-HR) years.  ↩ Observe that 0.48 years is 252460.8 minutes. Assuming you lived for 80 years and took a pill every day of your life, that would be 80 * 365.25 = 29220 pills. 252460.8 minutes / 29220 pills = 8.64 minutes/pill.  ↩ I expect that a number of you are happy to bite that bullet and say yes, HR=0.96 is trivial and smoking a cigarette each day is also fine. I don’t personally agree, but it’s not my place to question your utility function and I applaud your consistency.  ↩ A hazard ratio of HR=2/3, implies a change in life expectancy of 12 × (1 - 1/3) years = 4 years or 2,103,840 minutes. That corresponds to a per-pill increase of 2,103,840 minutes / 29,220 pills = 72 minutes/pill.  ↩ Technically, this is calculating a relative risk rather than a hazard ratio, but I think the difference isn’t very significant given that we’re assuming a uniform mortality risk. I used AI to create that simulation, though I did test that it replicates a traditional power calculator across a wide range of parameters when the relative risk is constant for all vitamin D levels. So I mostly trust it.  ↩ This simulation is probably a bit pessimistic. Things look a bit better if you use an older population where baseline mortality is higher. (Almost all trials do.) In principle, you could also use a population where more people have low levels, which could help a lot. But, for whatever reason, almost no trials do that. In fact, most trials accidentally under -sample people with low vitamin D, because people who agree to participate tend to be more health-conscious.  ↩ Kunzia et al. made a heroic effort to contact study authors and get data for individual patients. After getting data for 21,558 people (almost all from ViDA + FIND + VITAL + WHI) only 3,663 had levels below 50 nmol/L. That’s not enough to reliably detect a modest effect, meaning their confidence interval for this group is gigantic.  ↩ In this table, I tried to capture foods that are commonly fortified in practice, not just when it’s legally required.  ↩ The kidneys use vitamin D as a boring bone hormone. As long as the blood contains at least ~25 nmol/L of storage vitamin D, the kidneys don’t care. They create the same amount of active vitamin D, in response to calcium levels. But now cells everywhere are using storage vitamin D. To do god-knows-what. With god-knows-what sensitivity to circulating vitamin D levels. The body uses vitamin D in all sorts of weird and complicated ways. It’s biologically plausible that vitamin D could matter beyond bone stuff with severe deficiency, but there’s no convincing mechanistic evidence that it is . Vitamin D levels are strongly correlated with good health outcomes, but RCTs have conclusively shown that most of these correlations are non-causal. RCTs haven’t conclusively shown any benefit for anything beyond beyond bone stuff. At best, they’ve given weak evidence for hazard ratios slightly below one. Biology and evolution suggest a prior that moderate levels of vitamin D (say 80 nmol/L) are quite possibly better than low levels (like 40 nmol/L) and unlikely to be worse. Observational studies say that vitamin D is magical, but those studies are bad and we should ignore them. The RCTs show that vitamin D is non-miraculous. But beyond that they don’t provide much information, because they mostly enrolled people with moderate vitamin D levels, meaning plausible effects would require colossal sample sizes to reliably detect. What evidence the RCTs do provide points weakly towards a modest benefit. If real, that benefit would far exceed the cost of taking vitamin D. Therefore, if you have low vitamin D, it seems wise to supplement. Technically, vitamin D itself is a type of steroid although not what people usually mean by “steroid”.  ↩ Here are some of the fancy names for the different forms of vitamin D I’ll talk about: my name fancy names provitamin D 7-dehydrocholesterol previtamin D previtamin D₃ vitamin D cholecalciferol storage vitamin D calcifediol / ergocalciferol / 25(OH)D / 25-hydroxyvitamin D active vitamin D calcitriol / ercalcitriol / 1,25(OH)₂D / 1,25-dihydroxyvitamin D ↩ Charmingly named “vitamin D-binding protein” .  ↩ If you eat mushrooms or yeast, it joins the vitamin D from your skin en route to your liver. If you eat animals or animal products, you also get some storage vitamin D, which doesn’t need to be processed by the liver.  ↩ Storage vitamin D is what your doctor measures in your blood test. This is sometimes measured in nmol/L and sometimes in ng/mL. The latter measurement is smaller by a factor of 2.496. So 25 nmol/L ≈ 10 ng/mL.  ↩ Apperly was building on a 1937 paper that observed observed that sailors, exposed to lots of sunlight, had much higher skin cancer rates than the general population, but lower overall cancer rates.  ↩ I theorize that the Garland brothers are alive and writing Slime Mold Time Mold .  ↩ In Biologist, active vitamin D is not just an “endocrine” hormone that sends signals for far away cells through the blood, it’s also a “paracrine” or “autocrine” hormone that sends signals to nearby cells or inside a single cell, through diffusion.  ↩ You might ask, why is vitamin D used by so many different parts of the body for so many different purposes? I think there’s no deep answer here. It’s true for the same reason that dogs sneeze to signal that they’re feeling playful: Evolution re-uses stuff for different purposes all the time. Imagine that DNA already exists coding for the vitamin D receptor and for the enzyme to convert storage vitamin D into active vitamin D. If some cells need to send a local signal, re-using those is easier than inventing something new. There’s nothing unusual or magical about this.  ↩ Don’t try to make sense of this. It doesn’t make sense. You could speculate that this is because the parathyroid glands are trying to make less active vitamin D to compensate for the fact that vitamin-D receptors throughout the body are sensitive to storage vitamin D itself. But I advise against.  ↩ 400 IU is the recommended daily amount  ↩ The WHI trial was a pioneer in salami-slicing results for different outcomes into dozens of different papers, most of which are hard to access. All trials now seem to have adopted this hideous trend which makes it maddening to try to summarize what actually happened in a trial. Also, slightly different numbers for the same quantity appear in different places. I haven’t bothered to chase these down, because the differences are all very small, e.g. a hazard ratio of 0.89 for cancer mortality rather than 0.90.  ↩ Guess what most kidney stones are made of?  ↩ Half of the vitamin D group and the placebo group also got omega 3. These are averaged together in the results. Also, VITAL carefully stratified the assignment to vitamin D or placebo based on baseline vitamin D levels, which should give more statistical power from a given sample size.  ↩ There was also a weird study done on a subset of 1031 people from the VITAL population that looked at telomere length. After starting with around 8700 base pairs, the control group lost around 160 base pairs during the study, while the vitamin D group only lost an average of 20. I’m not sure of what to make of this. For one thing, though the authors claim this is statistically significant, it depends on how you analyze the data. But beyond that, sure, telomere length is a marker of aging, but telomeres get shorter for a reason (likely to fight cancer) and it isn’t obvious that slowing this would always be a good thing.  ↩ This is a little complicated. In VITAL, participants were only eligible if they were taking at most 800 IU per day, and they were restricted to 800 IU per day during the trial. In D-health, participants were only eligible if they were taking at most 500 IU per day, but they were allowed to take up to 2000 IU per day during the trial.  ↩ You might ask: If vitamin D only has a modest effect, then why is it so strongly correlated with health? In principle, I’d like to push back against the idea that we need to explain why these particular correlations don’t imply causation. But the accepted explanation is a combination of (1) reverse causation where being healthy causes people to spend more time outside and thus get more vitamin D; (2) confounding, where obesity is bad for you and leads to lower measured vitamin D levels; (3) confounding, where more healthy lifestyles lead to both more vitamin D and more health; and (4) confounding, where higher socioeconomic status leads to both more vitamin D and more health. You might ask why these correlations would be true at a state level like the Garlands looked at, but then you run into the ecological fallacy and modifiable areal unit problem .  ↩ I took all the trials that got at least 2% weight and were rated as “low risk of bias” in this 2014 Cochrane review of vitamin D and mortality, then manually added all the “major” trials that were published after 2014. I shudder to think of the time it took to make this table. I tried using AI but found it was wildly unreliable. Part of the problem is that each trial’s results are distributed among many papers, in different journals, with different paywalls. And many details aren’t published at all by the original authors but are only scrounged up and put in the depths of the supplementary material of a review years later. In some cases, different sources also give contradictory numbers. The differences were always tiny (e.g. 0.90 rather than 0.89) but it still makes me nervous.  ↩ Here’s a table describing the major contours of the trials: Name Country Subjects (n) Age (years) white (%) Duration (years) Lips 1996 Netherlands 2,578 80 ± 6   3.5 Trivedi 2003 UK 2,686 74.7 ± 4.6 74 5 WHI 2006 USA 36,282 (women) 61.8 ± 6.7 84 7 Lyons 2007 Wales 3,440 84 ± 7.5   3 WFPT 2007 UK 9,440 79.1   3 RECORD 2012 UK 5,292 77.5 ± 6 99.2 6.2 Lappe 2017 USA 2303 (women) 65.2 ± 7.0 100 4 VITAL 2018 USA 25,871 67.1 ± 7.1 71.3 5.3 ViDA 2018 New Zealand 5,110 65.9 ± 8.3 83.3 3.3 D2d 2019 USA 2,423 60.0 ± 9.9 67 2.7 DO-HEALTH 2020 Switzerland, Germany, Austria, France, Portugal 2,157 74.9 ± 4.1   3 D-Health 2022 Australia 21,315 69.3 ± 5.5 94.7 5 FIND 2022 Finland 2,495 68.2 ± 4.5 100 5 And here’s a table focusing on the change in vitamin D levels: Name Intervention Allowed personal use (IU/day) Baseline D (nmol/L) Final D (nmol/L) Lips 1996 400 IU daily 0 (screening)     Trivedi 2003 100,000 IU 3× per year (D2) 0 (screening) 200 (trial) 52.5 (in controls) 75 WHI 2006 400 IU daily with Ca 600 (later 1000) 52.0 ± 21.1 (subset) ~67 Lyons 2007 100,000 IU 3× per year <400 (screening) 54.0 (in controls, subset) 80.1 (subset) WFPT 2007 300,000 IU yearly <400 (screening)     RECORD 2012 800 IU daily with Ca 200 ~ 38   Lappe 2017 2000 IU daily with Ca any? 71.8 ± 20.0 96.0 ± 21.4 VITAL 2018 2,000 IU daily 800 77 ± 30 105 ± 25 ViDA 2018 100,000 IU monthly 600 / 800 (younger / holder) 63 ± 24 119 ± 45 D2d 2019 4,000 IU daily 1000 69.9 ± 26.8 98.7 DO-HEALTH 2020 2,000 IU daily 1000 / 800 (screening / trial) 55 ± 22 100 ± 27 D-Health 2022 60,000 IU monthly 500 / 2000 (screening / trial) 77 ± 25 (predicted) 115 ± 30 FIND 2022 1,600 or 3,200 IU daily 800 75 ± 18 100 ± 21 or 120 ± 22 ↩ Among the major trials, only VITAL, ViDA, and FIND measured it for more than a tiny number of subjects.  ↩ In VITAL and ViDA, people with baseline levels below 50 nmol/L had a higher hazard ratio for cancer mortality (though with wide confidence intervals), suggesting if anything less benefit. Or, you could use race as a proxy for baseline vitamin D. But in both VITAL and WHI, the hazard ratio for cancer mortality was higher among non-Whites. After looking at many such analyses for many outcomes, the only clear result I could find was for diabetes in the D2d trail, where the hazard ratio was much lower for people below 30 nmol/L (0.38 vs. 0.93).  ↩ The results for VITAL look decent: outcome (VITAL trial) HR HR excluding first two years Cancer 0.96 (0.88 to 1.06) 0.94 (0.83 to 1.06) Cancer mortality 0.83 (0.67 to 1.02) 0.75 (0.59 to 0.96) Major CVD event 0.97 (0.85 to 1.12) 0.93 (0.79 to 1.09) All-cause mortality 0.99 (0.87 to 1.12) 0.96 (0.84 to 1.11) But in D-Health, excluding the first two years actually increased the hazard ratio for cancer mortality from 1.15 (0.96 to 1.39) to 1.24 (1.01 to 1.54). Most other trials were too short for this kind of analysis to make sense.  ↩ That could downregulate 25-hydroxyvitamin D 1-alpha-hydroxylase, reducing the rate it catalyzes the hydroxylation of hydroxycholecalciferol into 1,25-dihydroxycholecalciferol!  ↩ Dynomight: WTF is this? Dynomight Biologist: Well, C-reactive protein is generally considered inflammatory. Dynomight: So reducing that is good? But then why do they talk like elevating anti -inflammatory cytokines would be bad? Dynomight Biologist: Yeah… That would be good. Unless you have cancer. In which case it’s not good. Dynomight: OK!  ↩ Mendelian randomization studies are based on the idea that certain genes predispose you to have higher levels of circulating vitamin D. If you assume that those genes are randomly distributed in the population and have no effects other than affecting vitamin D, then they serve as a kind of natural experiment. With vitamin D, these studies typically show null results . However, the validity of the assumptions is debatable and the identified genes only explain ~5% of the variance in vitamin D levels, which makes the results very noisy.  ↩ Pale skin also greatly increases the risk of sunburn and skin cancer. In the US, White people get melanoma at around 25 times the rate of Black people, despite (I assume) higher usage of sunscreen and better health outcomes in most other dimensions. But experts generally think folate deficiency created stronger selective pressure, since it’s so closely linked to reproduction.  ↩ It’s a more complicated than this, because you also need to look at the amount of folate in diet, as well as migration patterns and how long populations had to adapt to their environment. But experts seem to consider this the leading explanation for the evolution of pale skin.  ↩ To derive this, suppose that S(t) is the probability that someone survives to age t. Then life expectancy is ∫ S(t) dt , where the integral runs from 0 to ∞. If you change the hazard ratio by a factor of HR , then the new in life expectancy is L(HR) = ∫ S(t)ᴴᴿ dt , so the change under a linear approximation is ΔL ≈ (HR-1) × L’(1) . This is more commonly written as ΔL ≈ (HR-1) × L(1) × H , where H = -L’(1)/L(1) is known as the Keyfitz entropy . This is is chosen because the quantity H is relatively stable, and in rich countries is typically between 0.10 and 0.20. So a decent estimate would be that baseline life expectancy is L(1)=80 years and H = 0.15 in which case the change in life expectancy is around 12 × (1-HR) years.  ↩ Observe that 0.48 years is 252460.8 minutes. Assuming you lived for 80 years and took a pill every day of your life, that would be 80 * 365.25 = 29220 pills. 252460.8 minutes / 29220 pills = 8.64 minutes/pill.  ↩ I expect that a number of you are happy to bite that bullet and say yes, HR=0.96 is trivial and smoking a cigarette each day is also fine. I don’t personally agree, but it’s not my place to question your utility function and I applaud your consistency.  ↩ A hazard ratio of HR=2/3, implies a change in life expectancy of 12 × (1 - 1/3) years = 4 years or 2,103,840 minutes. That corresponds to a per-pill increase of 2,103,840 minutes / 29,220 pills = 72 minutes/pill.  ↩ Technically, this is calculating a relative risk rather than a hazard ratio, but I think the difference isn’t very significant given that we’re assuming a uniform mortality risk. I used AI to create that simulation, though I did test that it replicates a traditional power calculator across a wide range of parameters when the relative risk is constant for all vitamin D levels. So I mostly trust it.  ↩ This simulation is probably a bit pessimistic. Things look a bit better if you use an older population where baseline mortality is higher. (Almost all trials do.) In principle, you could also use a population where more people have low levels, which could help a lot. But, for whatever reason, almost no trials do that. In fact, most trials accidentally under -sample people with low vitamin D, because people who agree to participate tend to be more health-conscious.  ↩ Kunzia et al. made a heroic effort to contact study authors and get data for individual patients. After getting data for 21,558 people (almost all from ViDA + FIND + VITAL + WHI) only 3,663 had levels below 50 nmol/L. That’s not enough to reliably detect a modest effect, meaning their confidence interval for this group is gigantic.  ↩ In this table, I tried to capture foods that are commonly fortified in practice, not just when it’s legally required.  ↩

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Manuel Moreale 1 months ago

Edoardo Baldi

This week on the People and Blogs series we have an interview with Edoardo Baldi, whose blog can be found at edoardob.blog . Tired of RSS? Read this in your browser or sign up for the newsletter . People and Blogs is supported by the "One a Month" club members. If you enjoy P&B, consider becoming one for as little as 1 dollar a month. Hello! I’m Edoardo, in my thirties, born near Milan (Italy) and raised in the Alps of the same region, to escape the boredom of too flat a horizon. I studied physics, first in Milan, then abroad in Switzerland, where I spent a little over four years on a PhD that convinced me academic research wasn’t for me – or so I thought, since I didn’t stray too far. In the following years I became a “research software engineer”, meaning a software developer who works closely with research. It took me a while to realize that, despite the many benefits, that work had become a routine I was taking too much for granted. Or better: I had lost sight of why I was staying there; why I kept choosing that configuration for my life. Now I’m trying to figure out if teaching the two subjects I’m most passionate about – math and physics – is what I want to do in the next chapter of my career. I can never get enough of hiking in the mountains, especially over multiple days – as long as my body agrees. And sharing an experience with other people who love the same thing is my ideal vacation. Books, writing – I don’t know how many experiments with novels and short stories I’ve done over the years – and puzzles of all kinds (including programming challenges, even though I’m a particularly slow coder) are some of the activities that can easily fill my free time. Having always loved tinkering with computers, I think I started writing random things online quite early. If I remember correctly, it was on LiveJournal or MySpace, prehistoric stuff now. I discovered WordPress during high school, following a guy from my same school who wrote ironic essays on philosophy topics. I tried to emulate that model, but I didn’t get very far as it wasn’t my thing. Years later, with some friends fond of cinema, again on WordPress, I started a collective blog where we wrote our opinions on the movies we watched, often together. The name of the blog – Sweet Sue and Her Society Syncopators – was a tribute to a classic 50s American comedy. (I’ll let you work that one out.) During my PhD, I collaborated on and managed the university cinema club’s blog. At the time, however, I also started publishing my very personal ideas on books and movies on another blog, whose name or domain I honestly don’t even remember now. I think I tried to recover something from that blog via the Wayback Machine, with no success. Fast-forward several years, I realized why none of those blogs had survived: I was writing on commission – I loved the perk of press screenings, but writing something afterwards was non-negotiable. Or I was performing for some imagined audience by covering whatever was trending, not what I actually cared about. I could say that my personal blog was born when I decided that my online space would be only a public personal journal: the only rule was to write about what interested me the most, in the way that felt most natural. This is still the reason behind my current blog. How long is it going to survive? I don’t know. It did well, so far, with ups and downs. Beyond my hiking recaps, almost everything I write starts from curiosity – a science-based question (“if I ate an apple a day for a year, how many kg of peel could I accumulate?”), something I want to understand well enough to explain, a brain teaser that sometimes keeps me awake. Since it’s often something I don’t know, a research phase almost always follows – and I admit that, sometimes, it derails my intention to write. I keep a dedicated note for each idea, where I track its evolution. When I feel like I’ve reached a conclusion of sorts, I then sketch out a structure and use it as a guide for the first draft. Curiously, all my notes are in English, but the first draft of anything I write is always in Italian. Then I translate into English, and very often rewrite some parts that don’t flow very well in the other language. And yes, I often use Claude for a final proofread: I’ve given it strict instructions on what it can and can’t touch, and how. The content is always mine, and I’m careful to keep it that way: I don’t want to end up with a voice I no longer recognize as my own. As for the tools, my personal notes live in an Obsidian vault – because they must be plain text files – and I write all my drafts almost exclusively in iA Writer. It’s been my first choice for many writing projects, at least in their early stages. One feature I particularly love is its support for authorship , without violating the plain text pact. When I sit down to write the first draft, I have only one need: to be alone in a fairly quiet environment. Honestly, I’ve never tried writing in a public place, like a café – and the few times I did write on a train, it was surely due to a deadline I couldn’t avoid. As far as I’m concerned, it’s more the act of moving through space that stimulates what I might call creative thinking – which I take to mean authentic rather than original , as in “totally new”. And I’m also convinced that the environment influences my creativity, but I couldn’t say how or why. Often I’ve only realized much later that I had visited an environment from which I returned with ideas I considered creative – whether these didn’t go very far is another, unresolved story. I think I’ve tried dozens of frameworks to create a blog, starting with the large family of static-site generators. After several attempts, intrigued by some input from Manu, I gave Kirby a chance and discovered that it met all my needs. One above all: my blog’s content must be in plain text, as I don’t want to deal with any kind of problem taking it with me, wherever it might be in the future. So, for the moment: Kirby CMS, hosted on a fairly basic server managed by Hetzner. The domain is registered on Porkbun, and the DNS is managed by Cloudflare. I’ve also written a dozen custom plugins to tweak many aspects of my website because, for me, tinkering with the mechanics of a personal blog is part of the joy of having one. I just can’t resist – and I keep telling myself “tinker less, write more”. I would probably study web design and web technologies properly from the start – I mostly stumbled into this stuff through my day job. I say this to avoid having to settle for some preconfigured service that isn’t right for me. I would love to have a domain like , but the problem isn’t availability so much as the popularity of my name. And, honestly, I’m not ready to pay $200 a year for a personal website. The maintenance costs for my blog are quite low: 4€ and something a month for the server, plus the annual cost of the domain – about 20€. Kirby CMS requires a one-time license (100€, renewed every four years), and this is the only expense I periodically re-evaluate: the moment it no longer aligns with my needs, I will have no problem planning a migration elsewhere. In fact, I’ve already done it several times as a stress test , but for now I don’t feel the need to. My website generates no revenue, nor have I ever tried to make it do so. Personally, I have nothing against monetising a personal website, provided it’s done honestly. If I were to do it, I probably wouldn’t rely on platforms like Substack – only because I like building things myself. Even today I financially support some blogs because I believe in the work of the people behind them – or to give a friend a small nudge to keep going. A good part of the blogs I follow, or like to return to from time to time, I discovered thanks to “People & Blogs” – or through “Ye Olde Blogroll” . I think it’s unlikely that anyone reading this page doesn’t know either of them; but if that’s the case, I invite you to take a look, exploring even the older, less obvious stuff. I want to mention a friend’s project, halfway between a personal blog and a photography portfolio, that I had the pleasure of contributing to . I’m very fond of it: partly for my friendship with the author, and partly because it circles a theme that has quietly followed me for years: the sense of belonging to a place, or to multiple places; the idea, the concept, the experience of what we call home . The project is “Stay Stay Stay” by Elettra Pistoni: if you’re not into reading about this topic, her pictures are well worth a look. I also think she would more than gladly welcome the opportunity for this interview, but I’ll leave the decision to those in charge. I’ve lost count of how many newsletters or feeds I’ve subscribed to over the years, and it doesn’t really matter. I’ve reached the point where the list of online content I follow consistently has no more than ten items. Among these, two blogs and a newsletter (in Italian) that I return to quite regularly, even to reread older things: I’ll take this as a cue to share a bit of what’s going through my head – two thoughts and a side project that will maybe see the light someday. Finally, a heartfelt thanks to Manu for offering me the opportunity to share a bit of myself with this community! Now that you're done reading the interview, go check the blog and subscribe to the RSS feed . If you're looking for more content, go read one of the previous 146 interviews . People and Blogs is possible because kind people support it. “Useful Fictions” by Cate Hall Julia Evans ’s blog, a trove for tech enthusiasts The newsletter “It’s Friday I’m (not) in love” , partly inspired by “Modern Love”, the New York Times’ well-known column. Whenever I feel like telling someone “I don’t have time”, I stop and remind myself that it’s almost never true. In fact, never. It’s just my fear of making a commitment, or a lack of courage to admit what I really care about. I try never to hide behind this excuse with the people I really care about, because they don’t deserve it. I’ve also written a short post about it . This could be one of my guiding tenets , because I haven’t been able to refute it yet: “Actions, not words, reveal our real values” . It’s not mine , and I often struggle to accept it myself. But I’m convinced that if we actually lived by it, we would have far more genuine and satisfying relationships with other people – in whatever sense you want to take that. Being a hiker obsessed with traveling light, I started working on an app (web only to begin with) that lets me keep track of my gear and which items I decide to bring on each trip. Dozens of these tools already exist, but this is my vision of what I’d want such an app to do. I called it “Baseweight”, and I hope to have an alpha version out in the near future. If someone is curious, the app’s future home will be at baseweight.my . And if you’d like to share your thoughts on it, don’t hesitate to reach out ! Opinions and suggestions are especially welcome at this early stage.

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Unsung 1 months ago

“Every pigment in this catalogue has a paper trail.”

Today at Unsung, three efforts with perhaps thought-provoking depth. The first one: a new website called Storied Colors . = 2x) and (width >= 700px)" srcset="https://unsung.aresluna.org/_media/every-pigment-in-this-catalogue-has-a-paper-trail/1.2096w.avif" type="image/avif"> = 3x) or (width >= 700px)" srcset="https://unsung.aresluna.org/_media/every-pigment-in-this-catalogue-has-a-paper-trail/1.1600w.avif" type="image/avif"> It’s a… catalogue of [two hundred and fifty] named colors — pigments, dyes, lakes, glazes, and a small number of digital hues — each accompanied by the documentary evidence required to call it by its name. I wouldn’t normally link to this, as this feels closer to graphic design than UX design, even if the typography of the site is pretty exquisite, and the history of some colors truly fascinating. What particularly stood out to me about the site that felt worth celebrating was its rigor ; there are a lot of cheap color tools and some great ones , but this approaches the subject matter with a different kind of energy: There are good color dictionaries. There are good histories of paint. There are excellent technical references on conservation chemistry. What is harder to find is one place where the chemical formula, the workshop floor, the trade route, the patent dispute, and the eventual ban all sit on the same page — sourced, dated, and free of the gloss that surrounds color writing online. Most of what you can read about historical color on the web has been rewritten three or four times from the same Wikipedia paragraph, with the citations dropped along the way. What you are reading here is an attempt to put the citations back. There’s also this… The corpus is curated, not comprehensive. There are perhaps a thousand pigments worth knowing about; the launch corpus selects two hundred and fifty whose stories are best documented, most consequential, or most strange. The catalogue is actively expanded; new entries land regularly. Editorial discipline is what keeps the standard honest. …buyoed by clean information architecture with tags and deep search. I’d love to see more of that applied to UX. (You might remember that I missed it in the review of the Laws of UX book , which feels on the surface like roughly a similar idea.) Equally importantly, however, for Color Stories, none of this stands in the way of some beautiful writing : This is not a forgotten oddity. This is mid-twentieth-century American consumer culture casually serving food on uranium-glazed plates for thirty-five years across two production runs, marketed as everyday tableware to ordinary households, and discontinued only when the second uranium supply ran out. The plates sit in display cabinets across the country, in good condition, still glowing faintly under a Geiger counter. Kudos to the (anonymous) creator. #above and beyond #colors #web

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ava's blog 1 months ago

the girly wellness aesthetic as a white supremacist dog whistle

Since reading Naomi Klein’s Doppelganger and its parts about Covid and fitness influencer culture a while ago (especially the chapter "The Far Right Meets the Far Out"), I cannot help but see that “ Pinterest clean girl fitness and fruit bowl gua sha yoga mat pilates in the forest ” content as covert white supremacy and eugenicist ideals; dog whistles, shared far and wide by people who probably don’t know better and just think it looks good and want to be like that. I cannot quote the entire book and how it adds it together and builds this narrative up, but I especially liked these parts: " There are deep and healthy pleasures to be found in exercise, as there are in other aspects of wellness. For many of the evangelists in these worlds, however, both fitness and diet are intensely value-laden endeavors. Achieving goals means setting rigorous targets and displaying relentless discipline to meet them (a.k.a. "putting in the work"). That's how you reach your idealized body double. Which is all fine if it stays there. But the trouble is, it often doesn't. As Carmen Maria Machado draws out in her doppelganger short story, once the slim, perfect body has been achieved, the less controlled body that once was can persist as an ever-present shadow-self - and this discarded double is deeply loathed. [...] And that is the trouble with this more private kind of doppelganger; when body mania sets in, the fit self may well not be satisfied with crushing its own unfit self; it may look for other targets, its self-hatred seeping out and projecting itself onto other people's less fit, less conventionally able bodies. These kinds of moralistic physical judgments deepened during the pandemic, especially when it became clear that obesity, diabetes, and some forms of addiction increased the risks posed by Covid-19, along-side other factors, including age. Much of the pressure to wear a mask and get vaccinated, meanwhile, was framed as a duty to care for those with greater vulnerabilities. It was then that wellness culture, and its barely submerged hostility toward less conventionally perfect bodies and less "clean" lifestyles, began to bare its teeth. [...] The core Covid-era public health message - that we all needed to undergo some individual inconveniences for the sake of our collective health - enjoyed majority support. Yet it simply could not be reconciled with the wellness industry's own overarching message: that individuals must take charge over their own bodies as their primary sites of influence, control, and competitive edge. And that those who don't exercise that control deserve what they get. Neoliberalism of the body, in distilled form. [...] On the contrary, the lesson they seem to have extracted from the race and class disparities of Covid's early death toll was "This virus is going to kill people who do not look like me.". [...] This willingness to write off huge swaths of humanity that are cast as lesser within supremacist narratives is the strongest glue that binds together the pastel-hued, self-loving world of women's wellness with the fire-breathing, immigrant-bashing world of the Bannon right. [...] These are the histories currently being conjured up in mainstream wellness culture, which has adopted Silicon Valley's notion of self-optimization, itself a by-product of the personal-branding culture that torments so many young people today. Every step counted. Every sleep measured. Every meal "clean". And it is in this context that has prepared the ground for a redux of the 1930s fascist/New Age alliance. The very idea that a human can and should be "optimized" lends itself to a fascistic worldview - because if your food is extra-clean, it can easily mean other people's food is extra-dirty. If you are safe because your immune system is strong, it can flip to man others are unsafe because they are weak. If you are optimized, others are, by definition, suboptimal. Defective. Next door to disposable. " Together with a lot of quotes of fitness trainers, and the fact that the Lululemon founder donated his money to right-wing causes. I used to enjoy looking at this stuff. Since reading, I notice how monotonous the entire aesthetic is, all these social media profiles and suggestions; it’s always white or racially ambiguous people, always women with European beauty standards and highly genderconforming bodies and style. Always the minimalist white beige pastel pink outfits and surroundings, always huge living spaces that look basically unused, always so clean and perfectly styled that it insinuates either lots of available time or paid household help. It goes directly against much of the color celebrated in other cultures, something I already read about in Chromophobia by David Batchelor, in which the author makes compelling arguments that certain groups are obsessed with pure white design because color is seen as corrupting, as racialized and as queer. It’s always with messages about working on yourself that are laid over bodies and food, subtle hints about how you can cure almost anything if you just eat extra clean, avoid evil chemicals, filter everything, drink herbal tea, take supplements and do the sort of exercise regimen that gives you a body like the images. The message is clear: this is what the happy, healthy, perfect body looks like, and everything else is gross, impure, sick, and in need of fixing. This is also presented as almost effortless, and you as the one being out of tune, your body derailed, that you have to get back into its natural equilibrium by detoxification and debloating (rapid weightloss). How it got so out of balance? The poison they now put in your food, the water, the packaging, the air, whatever. There is no space for visibly disabled and chronically ill bodies in this narrative that only permits good health as the default. Acknowledging them would mean admitting that your health is somewhat out of your control beyond the basics, and that it isn’t your juicing and Pilates regimen or your 300$ supplements keeping you together, but luck, genes, not having had an accident, and maybe handwashing. It would be admitting that you could end up sick and hurt despite all the money and time you pour into this, or that your body won’t look like this (for)ever. The other bodies are considered ugly, weak, lazy, a victim of their lifestyle, their greed. It’s simply cooler and seemingly “natural” to throw herbs and greens into a smoothie and pretend that this is your medicine, than the sterile, branded packaging of a syringe or pill, which doesn’t look natural at all. I think especially in America, these content creators love the juxtaposition of the fat Black woman in a food desert with some KFC and burgers, and their white skinny selves in Erewhon. What this content is after is somewhat an image of the Übermensch - the one basically never sick, always strong, beautiful, fertile, white or white-passing, disciplined, hardworking. There’s a reason why so many fitness influencers are conservative or are even MAGA, why so many of them shifted to tradwife content, and how much tradwife content is just like the above but focused on very palatable and stereotypical household chores instead of gym fits, while still featuring almost the same foods and regimens. They post “farmers market haul!” and it shows three impressively tasty looking leafy greens and other vegetables, and you just know that those three items cost what others need for 3 days of food, and can be used for just one meal, or more of you severely undereat. This can’t feed a family, and they couldn’t frolick through the park with their chives and kale in a bag if they really had to transport several cans of food and tetrapaks, too. Wedged in-between are pictures from far-away, expensive travels: impressive beaches, forests, parks, mountains. People, posting in the tone of being just smol little beans !! 🥺, saying: taking a walk through my parents’ backyard! And it’s a whole forest. Generational wealth, but wholesome, ecological, wellness-focused, back-to-the-roots. It’s where cottagecore aesthetic and eco-fascism are able to meet. It’s where criticism about cities, pollution, ecological collapse, loving nature aesthetics can be combined with “retvrn” and “reject modernity, embrace tradition”. All that is why when seeing this type of stuff now, it looks dystopian, it looks like propaganda, highly exclusionary, eugenicist. And I have these feelings despite being the target group and easily passing for one of them as someone who’s white, going to the gym, with a fitting skincare routine, lots of supplements and eating lots of whole foods; the only thing not making me fit in is my chronic illness, and not fully adhering to heterosexual standards of beauty. I know others will think it's "not that deep", but this stuff doesn't exist in a vacuum and frequently gets co-opted in meme warfare and the normalization of a certain standard. It makes me deeply uncomfortable, and thinking of when this really popped off, it paved the way into our current skinny/Ozempic culture and the rise of fascism in many areas. Reply via email Published 17 Jun, 2026

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Kev Quirk 1 months ago

📝 2026-06-17 07:04: We have the incubator setup incubating a dozen eggs, including the last 2 our hen...

We have the incubator setup incubating a dozen eggs, including the last 2 our hen that was caught by the fox layed. Expect regular updates. 🐣 Thanks for reading this post via RSS. RSS is ace, and so are you. ❤️ You can reply to this post by email , or leave a comment .

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Chris Coyier 1 months ago

How You Get the Mentos into the Diet Coke

Did you think this was gonna be some grand metaphor blog post? You probably know: if you put Mentos in Diet Coke, it makes the carbonation go crazy and fizz shoots out of the bottle. It’s a “science” experiment akin to baking soda & vinegar volcano, freezing a banana in liquid hydrogen and shattering it, or playing with a bit of mercury in your hands. No? Just me? Anyway. What do you picture in your mind when you think of getting that tube of Mentos into the Diet Coke bottle? I bet you don’t even really think of anything at all. You just do it or whatever. You put them in there. Allow me to tell you, it’s not going to go good. In my experience children just cup them in their hands and try to funnel them in. Kid or adult, you’ll get half of them in a best. Then you freak out when the fizz starts come up and drop the rest, probably knocking over the bottle. Maybe you’re like, bro, it’s a tube already, you just open one end and squoosh them out. I don’t blame you for trying, but it’s no dice. Big Mint™ has that paper wrapped on there too tight. I’ve seen adults try to use basic kitchen gadgetry to try to luge them down in there, which is also just too failure prone. I’ve also seen YouTube videos with extraordinarily complex inventions to do this, crafted from wood and levers and tubes and whatnot. Too much. Listen, this activity is already wasteful enough, we might as well do it right. My idea is to drill a hole into each Mento and string them from a bit of wire. Probably just best to watch my sick video proving that my idea rules. I used this online video editor thingy Kapwing to blur the faces (obviously, from the watermark). It’s not like ultra mission critical, I just prefer that. Looks like it missed a few frames. So it’s both impressive and not terribly reliable for one-shots.

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