I'm a mathematical physicist who likes explaining stuff. I'm the Maxwell Fellow of Public Engagement at the School of Mathematics and the School of Physics and Astronomy at the University of Edinburgh. Check out my blog Azimuth! I'm also a member of the n-Category Café, a group blog on math with an emphasis on category theory. I also have a YouTube channel, full of talks about math, physics and the future.
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2026-05-19T13:16:36+02:00 Event JSON
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Last Notes npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez BIOMIMETIC TECHNOLOGIES How can we learn from nature? One of the most obvious ways is to look at natural systems and design technologies based on them. These are called biomimetic technologies. A single example can illustrate some of the issues that arise. Termites maintain nearly constant internal temperatures in their mounds through a system of channels. They don’t need fans that require power. For a time, it was believed that they used a simple convective cooling system, where hot air rises through the central chimney, drawing in cool air at the base. In 1996, a large office and retail building was built based on this idea: the Eastgate Centre in Harare, Zimbabwe, designed by the architect Mick Pearce [TS]. It has chimneys and ventilation channels that draw cool night air through the building’s thermal mass. It uses roughly 90% less energy for climate control than a conventional building of comparable size! That translates directly into far lower carbon emissions from heating and cooling. This success inspired emulation. Pearce himself used similar termite-chimney-inspired designs in a Melbourne office building [HB]. More recently the Startup Lions Campus in Kenya, designed by Kéré Architecture on the banks of Lake Turkana, features three tall terracotta-colored ventilation towers modeled after local termite mounds. (1/n) [TS] Turner, J.S. & Soar, R.C. (2008). Beyond biomimicry: What termites can tell us about realizing the living building, Proc. I3CON, p. 18. [HB] Hes, D. & Bayudi, R. (2005). Council House 2 (CH2), Melbourne CBD: a green building showcase in the making. Proceedings of Conference on Sustainable Building South East Asia, pp. 231-241. https://media.mathstodon.xyz/media_attachments/files/116/483/175/250/666/344/original/7e790bce5fea0853.png npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez Kings Day shows yet again that the least powerful Americans are the most courageous. https://media.mathstodon.xyz/media_attachments/files/116/310/935/159/404/387/original/067696ef1be2cc14.webp npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…j9f5 - the Choctaw don't own most of the land over which they have jurisdiction, and I can't quickly find out how much land they do own. So far they seem to be doing solar on a smaller scale: The Choctaw Nation Solar Farm in Durant, Oklahoma sits on 35 acres of land and contains more than 15,300 solar panels. It's a partnership with Oklahoma Gas & Electric (OG&E), which constructed and operates the facility. The farm first came online in August 2020 producing 5 megawatts, and expanded it to 10 megawatts by the end of 2021. Many Choctaw Nation facilities — including tribal government, culture, and health centers — receive a portion of their power from the farm, with each of the 58 connected facilities using solar for up to half of their total power. The Nation saved $69,000 on energy costs in the first 90 days after the farm went online. The tribe also avoided price spikes during the devastating February 2021 winter storm because half their power consumption came from solar. The utility authority director has mentioned the longer-term possibility of the Choctaw Nation moving into energy production itself, though that was described as still far off. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez As confirmed by a 2020 Supreme Court decision, 15% of Oklahoma is under jurisdiction of the Choctaw Nation. Now the Choctaw have used their power to prevent ICE from getting a big detention center! https://www.projectsaltbox.com/p/choctaw-nation-buys-former-big-lots npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez The proton, 1836 times heavier than the electron, is made of two up quarks and a down, with two of their spins aligned and one pointing the other way. The same quarks with all spins aligned give a new particle, the Δ⁺, that's 2411 times heavier than the electron! But the Δ⁺ is just the first of many 'excited states' of the proton: particles made of two up quarks and a down, but arranged in different ways, with higher energy and thus more mass. They quickly decay, often turning back into a proton. There are two main kinds: • If two quarks have spin pointing the same way and one points the other way, you get a particle with total spin 1/2 + 1/2 - 1/2 = 1/2 It could be a proton, but there are lots of others. Any particle of this kind is called an N*⁺. • If all three quarks have their spins aligned, you get a particle with spin 1/2 + 1/2 + 1/2 = 3/2 Any particle of this kind is called a Δ⁺. When we want to be precise, the Δ⁺ I mentioned before is called Δ(1232)⁺, because its energy at rest is 1232 MeV. That corresponds to its mass being 2411 electron masses. But then come a family of increasingly overweight relatives: the Δ(1600)⁺, Δ(1620)⁺, Δ(1700)⁺, Δ(1750)⁺, and so on, all of spin 3/2. Similarly the proton can be called N(939)⁺, though it'd be like calling water dihydrogen monoxide. Then come the N(1440)⁺, N(1520)⁺, N(1535)⁺, N(1650)⁺, N(1675)⁺, N(1680)⁺, and so on - a seemingly endless series of increasingly heavy relatives, this time all of spin 1/2. Physicists started studying these excited states, or 'resonances', in 1952. By the late 1960s, people were cranking them out. How to understand them??? (1/n) https://media.mathstodon.xyz/media_attachments/files/116/264/701/322/929/059/original/ef37ec6500bcc4ba.png npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez Beato isn't really saying AI will fail: despite the denialists who claim it's never good for anything, it's too useful for too many things to simply "fail". He's claiming something more interesting: for many purposes, big AI can be replaced by LLMs that you can run on your laptop. They already exist and you can download them for free. So the massive investment in data centers, expecting huge profits when we all start paying monthly fees to run LLMs, may soon be undercut by something cheaper. (1/2) https://www.youtube.com/watch?v=YTLnnoZPALI npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez Manet's famous painting Un Bar aux Folies-Bergère never appealed to me. But now I realize its genius, and my spine tingles every time I see it. The perspective looks all wrong. You're staring straight at this barmaid, but her reflection in the mirror is way off to right. Even worse, her reflection is facing a guy who doesn't appear in the main view! But in 2000, a researcher showed this perspective is actually possible!!! To prove it, he did a photographic reconstruction of this scene. Check it out in my next post. This blows my mind. (1/3) https://media.mathstodon.xyz/media_attachments/files/116/188/884/486/749/574/original/54d69d2cfe515f37.jpg npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez Another place I want to visit on my trip to New Mexico is called Aztec Ruins - named that before white folks realized the ancestral Puebloans were a very different civilization. It's huge, with ~450 rooms, some 3 stories high! People lived here from about 1050 to 1200 CE. Aztec Ruins National Monument: https://www.nps.gov/azru/index.htm (1/n) https://media.mathstodon.xyz/media_attachments/files/116/162/185/973/630/938/original/f0f8b280a6618c34.jpg npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez Have you read There Is No Antimemetics Division? Would I like it? A review in The Guardian says: "Memes are ideas that easily spread; antimemes are literally unthinkable, “self-keeping secrets”, impossible to record or to remember. Some feed on memories and pose an existential threat. But how is it possible to win a war when there’s no identifiable enemy, and every attack is immediately forgotten? Against these odds, the Antimemetics Division somehow exists, part of a secret organisation with bases deep underground in the English countryside, as related in this unforgettable, mind-bendingly brilliant novel." https://en.wikipedia.org/wiki/There_Is_No_Antimemetics_Division npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez In the northern hemisphere, the sun moves clockwise in the sky. This is why clocks, which were based on sundials, have hands that move clockwise. In 2014 the Bolivians finally decided to break free of this colonial legacy. They're in the southern hemisphere, after all! So the clock on their parliament now looks like this. I like it. But it must make a tempting target for counter-revolutionaries. https://media.mathstodon.xyz/media_attachments/files/116/128/639/831/911/927/original/1acd7709b64aa2cc.webp npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…gvxt - I don't know what "sort the partials out to see what it looks like without them" means. I'm trying to understand all the rational numbers a for which the Fourier transform spikes at ln(a)/2π. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…gvxt @nprofile…920r - yes, you can call some of them overtones. But I'm not sure that's the full explanation of the pattern we're seeing here! It should be related to the continued fraction expansion of ln(3)/ln(5) - or in other words, which powers of 3 are close to powers of 5. I showed that's part of the story. But I don't know the full story. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez So far I've been trying to understand the complicated waves in the function |1/(1−3^{-(1/2) - ix})(1−5^{-(1/2) - ix})| @nprofile…920r suggested that to do this I should compute the Fourier transform this function. This was indeed very revealing. Check out the graph below! There are the expected big peaks at ln(3)/2π ln(5)/2π ln(15)/2π which we expect from part 3. But there are many more - and many with musical significance! Let me list them - but instead of writing each frequencies ω, which are always of the form ln(a)/2π for rational numbers a, I'll just write the numbers a. Some have fairly simple musical names: 0.0122 27/25 large diatonic semitone 0.0813 5/3 major sixth 0.0935 9/5 minor seventh 0.1626 25/9 two major thirds 0.1748 3 perfect twelfth 0.2561 5 major third + two octaves 0.2684 27/5 0.3375 25/3 0.3497 9 two twelfths 0.4188 125/9 0.4310 15 0.4432 81/5 The musical names are probably less informative than the patterns here. Some of these peaks are barely visible. There are probably more too small to see - infinitely many of them. (9/n) https://media.mathstodon.xyz/media_attachments/files/116/084/515/538/035/299/original/7d78fa68fab8bc77.png npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez For starters, neither 15 × [2π/ln(3)] ≈ 85.8 nor 13 × [2π/ln(3)] ≈ 74.3 are especially close to the true period of the long wave crests in the absolute value of the zeta function. One drifts ahead, while the other drifts behind. But both do a pretty good job of landing on sharp spikes! I suspect that I'm just seeing the beauty of continued fraction expansions playing itself out on this playing field. There's an infinite wealth of structure and substructure, just like in the rings of Saturn - which are also caused by resonance phenomena, governed in part by continued fractions. But this particular function is a lot simpler than the rings of Saturn! (8/n) https://media.mathstodon.xyz/media_attachments/files/116/083/454/154/889/152/original/5cd1f826a1cdb75f.png npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez Aha! 💡 Now I see what the number 13/19 being close to ln(3)/ln(5) does for us. I said it predicts peaks in our zeta function spaced at a distance of roughly 74.3. And indeed, there are big peaks at x = 0 and x ≈ 74.3! There's a bigger peak at x ≈ 85.84, since the number 15/22 is even closer to ln(3)/ln(5). But I was wrong in suggesting that the period of the really big waves is 85.84! In fact the multiples of 85.84 drift away from crests of those waves. But you'll notice they do lie on sharp spikes. And this is only possible because 85.84 - 74.3 = 11.54 which is very close to the distance between the sharp spikes!!! Remember, I computed distance in part 4, that using the fact that 2/3 is another rational approximation to ln(3)/ln(5). But where does the above equation come from? Is it a coincidence? No, earlier in this thread we approximately got the numbers 85.84, 74.3 and 11.54 in two different ways. If we use one of these ways, 85.84 - 74.3 = 11.54 is telling us 15 × [2π/ln(3)] - 13 × [2π/ln(3)] = 2 × [2π/ln(3)] If we use the other, it's telling us 22 × [2π/ln(5)] - 19 × [2π/ln(5)] = 3 × [2π/ln(5)] Both of these are of course true. So, something very nice is going on here. But I'm still confused about what. (7/n) https://media.mathstodon.xyz/media_attachments/files/116/083/327/896/010/609/original/85c4a086ad93e119.png npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez Now I think I see why the distance between crests of the big long waves in this graph is about 85.84! We saw ln(3)/ln(5) ≈ 2/3 gives spikes in this graph spaced at a distance of about 2 × [2π/ln(3)] ≈ 3 × [2π/ln(5)] These numbers are 11.44 and 11.71, respectively. The actual spike spacing is about halfway between these two. Similarly, the better approximation ln(3)/ln(5) ≈ 13/19 should give peaks spaced at a distance of about 13 × [2π/ln(3)] ≈ 19 × [2π/ln(5)] These numbers are 74.35 and 74.17, respectively. Alas, that's not explaining our number 85.84. 😢 But there's an even better approximation ln(3)/ln(5) ≈ 15/22 which should give peaks spaced at a distance of about 15 × [2π/ln(3)] ≈ 22 × [2π/ln(5)] and these numbers are 85.79 and 85.89. The number 85.84 is about halfway between these two!!! 🎉 Of course this is still a bit mysterious. Why does 15/22 do something for us, but apparently not 13/19? Actually I believe 13/19 *does* do something for us. I'm just not sure what. I also haven'st studied what even better rational approximations to ln(3)/ln(5) do for us. But I imagine they create subtler waves in the zeta function, with even longer periods. (6/n) https://media.mathstodon.xyz/media_attachments/files/116/083/076/496/793/756/original/578d1c03af47a7d1.png npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez More interestingly, the fact that 2/3 is a good rational approximation to ln(3)/ln(5) says that 3 ln(3) ≈ 2 ln(5) or exponentiating both sides, 3³ ≈ 2⁵ i.e. 27 ≈ 25. The ratio 27/25 is so important in music that it has a name! It's called the 'large diatonic semitone'. It's one of four semitones that naturally show up in just intonation. Just intonation is ruled by the primes 2, 3, and 5, but today I'm just looking at the primes 3 and 5. That's why I'm looking at the zeta function of the commutative ring ℤ/3 × ℤ/5, and that's why the number 27/25 showed up! (5/n) https://media.mathstodon.xyz/media_attachments/files/116/082/712/140/724/441/original/d11a95c69a961edf.jpg npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez The slow 'beats' in the absolute value of the zeta function 1/(1 - 3⁻ˢ)(1 - 5⁻ˢ) arise because it's the product of two functions: 1/1 - 3⁻ˢ with period 2π/ln(3) ≈ 5.719 and 1/1 - 3⁻ˢ with period 2π/ln(5) ≈ 3.904 These functions both become big simultaneously when there are integers m,n with m 2π/ln(3) ≈ n 2π/ln(5) or in other words m/n ≈ ln(3)/ln(5) So finding the tallest peaks in the absolute value of the zeta function amounts to looking for good rational approximations of this number: ln(3)/ln(5) ≈ 0.682606194 Here are the first few: 2/3 ≈ 0.6667 13/19 ≈ 0.6842 15/22 ≈ 0.6818 The first says we expect tall peaks spaced apart by roughly 2 × [2π/ln(3)] ≈ 3 × [2π/ln(5)] These numbers are close but not equal! Look at them: 4π/ln(3) ≈ 11.44 6π/ln(5) ≈ 11.71 This explains why I empirically found that the very tall spikes in the graph below are separated by a distance of about 11.58. However, my guess that this distance is really 5 × [2π/ln(15)] ≈ 11.6009 may be completely wrong. (4/n) https://media.mathstodon.xyz/media_attachments/files/116/082/638/110/150/050/original/156231969416dcca.png npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez In the graph here, the sharp peaks seem to be spaced by a distance of 11.58 ≈ 5 × [2π/ln(15)] The slow beating has a period of roughly 62.69 ≈ 27 × [2π/ln(15)] I don't know if these approximate formulas are good - based on some deeper math - or just coincidences. With luck I can figure this out pretty soon, but I thought I'd throw it out here for y'all to play around with. We can take the reciprocal of the zeta function and notice that (1 - 3⁻ˢ)(1 - 5⁻ˢ) = 1 - √3 e^(ix ln 3) - √5 e^(ix ln 5) + √15 e^(ix ln 15) So for *this* function we expect oscillations with periods 2π/ln(3) ≈ 5.719 2π/ln(5) ≈ 3.904 2π/ln(15) ≈ 2.320 but this does not instantly explain the longer periods that stand out so dramatically in the graph here. (3/n) https://media.mathstodon.xyz/media_attachments/files/116/082/180/348/029/195/original/1497f0dc712c8afb.png npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez To get a better picture of the *slow* oscillations in |1/(1 - 3⁻ˢ)(1 - 5⁻ˢ)| where s = ½ + ix, let's plot it from x = 0 to x = 300. https://media.mathstodon.xyz/media_attachments/files/116/082/100/436/960/223/original/1875827997fb8e9d.png npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez There are fascinating connections between the Riemann zeta function and music theory. I'll probably write a paper about this, but I can't resist talking about a little piece of the story. I will *not* explain what this has to do with music, since I want to tell that exciting story later on, and do a really good job of it. Any commutative ring has a zeta function! The Riemann zeta function is the zeta function of ℤ, but the zeta function of ℤ/3 × ℤ/5 is simpler: it's just 1/(1 - 3⁻ˢ)(1 - 5⁻ˢ) Let's graph this along the 'critical line' where the famous zeros of the Riemann zeta function live. So, let's take s = ½ + ix and plot |1/(1 - 3⁻ˢ)(1 - 5⁻ˢ)| as a function of x from x = 0 to x = 100. We get this picture here: (1/n) https://media.mathstodon.xyz/media_attachments/files/116/082/076/373/675/815/original/798686c7fe217f56.png npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…h6a3 - thanks, I'll check out this show in preparation for my own pilgrimage! npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…8qh7 @nprofile…xdkd - I used to read Gene Ward Smith on sci.math and such, but I never paid much attention to his tuning theory stuff. I feel terribly sorry that I didn't, since he died of COVID in 2021: https://en.xen.wiki/w/Gene_Ward_Smith His work seems to be scattered in various newsgroups and chat rooms, articles on the Xenharmonic Wiki, etc. Besides the material on the Riemann zeta function another exciting thing is his study of "Don Page commas": https://en.xen.wiki/w/Don_Page_comma npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez My god! What an amazing blog article! For the last few weeks I've been studying the work of Gene Ward Smith, who discovered a connection between muic theory and the Riemann zeta function. But it turns out @nprofile…xdkd has been thinking abou this for years... and what she has discovered is much richer and more beautiful than I had imagined. This changes everything! npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…uk82 - oh! I'm starting to read about this, and so far it seems like there were extensive trading networks in the Americas. I haven't read about anything particularly linking Puebloans to Mayans genetically. But I'll keep reading. The Aztecs started up around 1300, while the Chaco Canyon civilization ended around 1126, probably due to a drought. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…ygne - was it easy to find everything when you got there? npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…uk82 - scarlet macaws in New Mexico? Or cacao? npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…f5pz - oh, interesting! I didn't know that about the thick-billed parrots. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…ju9m - this is amazing! Or rather: many of us don't learn enough about this in school, so we tend to underestimate these people. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…a638 - sounds cool. I went to White Sands once, but have no clear picture of what's between that and Chaco Canyon. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…8qh7 - the use of honey suggests they weren't insane. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez For more on parrots and scarlet macaws at Chaco Canyon: https://archaeologymag.com/2025/12/ancient-puebloans-macaws-ceremonial-use/ "Chaco Canyon saw occupation from the mid-9th to the mid-12th centuries, which also coincided with the growth of what eventually became monumental masonry pueblos, known as Great Houses. While macaw and parrot remains have intrigued researchers for decades, the last analysis of them was published more than half a century ago. This study reexamines that old material using modern zooarchaeological methods and contextual reconstruction. The reanalysis identified the remains of 45 birds from five different sites within the canyon. Most of them were scarlet macaws, with a small number of thick-billed parrots, a species that is not native to the region, and provide evidence of long-distance acquisition. Most of the birds were found in the Great Houses, particularly Pueblo Bonito, the largest and most studied Chacoan building. There, archaeologists found dozens of macaws in large plastered rooms, which often included thermal features, indicating a deliberate effort was put into keeping the birds warm in a harsh environment. Many of the rooms showed clear signs that live birds had been held inside for long periods. Researchers observed thick layers of droppings, food debris, and what looked like perches, which provides proof that macaws lived in these spaces rather than just being put there for a short time or processed. Individuals ranged widely in age from juveniles to those over the age of twenty, which points to long-term care rather than short-lived use." (3/3) npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez For more on chocolate at Chaco Canyon: https://www.nps.gov/chcu/learn/historyculture/pre-columbian-chocolate-discovered-at-chaco.htm "From 2004-2007 a University of New Mexico (UNM) research project re-excavated the trenches first dug in Pueblo Bonito’s middens under Neil Judd in the 1920s. Of the hundreds of thousands of pot sherds that were recovered, archaeologist Patricia Crown selected five for her research. She is a ceramics specialist at UNM’s Department of Anthropology. She designed the project, and W. Jeffrey Hurst from The Hershey Center for Health and Nutrition performed the research. They chose five pot sherds for organic residue analysis, three of which were likely from cylinder jars. The pieces date to between 1000 and 1125 AD based on their decorative styles. Only the three sherds most likely from cylinder jars exhibited trace theobromine, a conclusive indicator of cacao or chocolate. The implications of this find are extraordinary. The cacao plant grows only in certain tropical climates, and the nearest possibility for Chaco is Central Mexico. We already know the Chacoan people traded with Mesoamerican cultures for exotics like copper bells and scarlet macaws, but cacao suggests a more ritual connection than other Mesoamerican goods. In some Maya ceremonies a cacao beverage was frothed by pouring the liquid from one vessel to another. Likewise, the cacao found at Chaco was probably in liquid form because the residue had absorbed into the clay itself. Further, the limited distribution of the cylinder jars could be evidence that only an elite or small segment of the population consumed the beverage." (2/3) npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez Chaco Canyon should be called Choco Canyon, because researchers have found traces of *chocolate* in cups found at this site dating back to 1000 - 1125 AD. This is amazing: Chaco Canyon is in a dry part of New Mexico, 1900 kilometers north of where cacao grows. But the cups look like those that Mayans used for chocolate-drinking rituals! And archeologists have also found remains of parrots and macaws in Chaco Canyon. This suggests enormous trading routes. (1/3) https://media.mathstodon.xyz/media_attachments/files/116/037/400/946/848/573/original/2a0a486577f4affe.jpg npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…l7vu - thanks, that was a great listen. I'm planning to do a lot of study before visiting various sites this spring. This video was a good springboard toward digging deeper into these matters, because it situated Chaco in an interacting network of different cultures. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…t5a9 - I said it would be fun, but I don't know if I'll ever do it: I have too many project half-finished. Right now my main passion is the mathematics of tuning systems. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…t5a9 - thanks! I try to b careful, but I'm also indebted to a referee who found more typos. It would be fun to take that paper on motives, expand it, and add an introduction to the Tate conjecture, Hodge conjecture and standard conjectures - for people who don't know much algebraic geometry! For now these expository papers are my favorites. They touch on some of these conjectures and why they matter: James Milne, Motives: Grothendieck’s dream, in Open Problems and Surveys of Contemporary Mathematics, eds. Lizhen Ji, Yat-Sun Poon and Shing-Tung Yau, International Press, Somerville, Massachusetts, 2013, pp. 325–342. Available at https://www.jmilne.org/math/xnotes/mot.html James Milne, The Riemann Hypothesis over finite fields: from Weil to the present day, in The Legacy of Bernhard Riemann after One Hundred and Fifty Years, vol. II, eds. Lizhen Ji, Frans Oort and Shing-Tung Yau, International Press, Somerville, Massachusetts, 2015, pp. 487–565. Available at https://www.jmilne.org/math/xnotes/pRH.html James Milne, Motives over finite fields, Proc. Sympos. Pure Math. 55, Part 1, AMS, Providence, 1994, pp. 401–459. Available at https://www.jmilne. org/math/articles/1994a.html npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…t5a9 - I'm not sure I learn very fast, but if I do it's probably by ignoring details until they seem to become necessary. If you want to learn about motives, you might try my talk "Motivating motives", or the slides for that, or the paper I wrote based on those: https://math.ucr.edu/home/baez/motives/ They have an intimidating reputation, but I blundered in and tried to understand them. The idea is that any variety can be chopped into basic 'pieces' in a very abstract sense, which are not usually subsets, since they can have a negative number of points. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…t5a9 - Does a typical good, meaty first course on algebraic geometry give some intuition for the bigrading on the cohomology of a smooth complex variety? I don't know. I never took such a course, but I think there are lots of ways such a course could go. This stuff is in Griffiths and Harris, but that book says a lot more about complex manifolds than some intros to algebraic geometry. (That's why I like it: I'm really more of an analyst.) I feel I understand why the Hodge conjecture says "the world is nice and simple". You take the two most obvious properties of a cohomology class that comes from a rational combination of algebraic cycles and say "that's it - that's all we need!" So in other words, if it's true a mysterious gap between algebraic topology and algebraic geometry is gone. Looking at Deligne's intro, I see he makes this vague feeling much more precise: he says the Hodge conjecture implies that the category of motives over ℂ is a full subcategory of the category of Hodge structures. I hadn't known that, but this is a big deal. It turns this category of motives, which is important but very elusive, into something far more concrete that can be described using linear algebra. So, thanks for pressing me on this point! I've spent a bunch of time trying to understand the bare basics of motives, and this is a step forward. I don't think a good meaty first course in algebraic would explain motives or Hodge structures, though! npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…xuq7 - I think that first section was really just a rapid attempt to convince people that there's more to algebraic geometry than a pile of slick abstract definitions. It may have scared as many students as it intrigued. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…rgsr - I didn't actually ever sleep through a class. I just never took a course on commutative algebra. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…xuq7 - nice! It looks like a rocket-fuel-propelled approach to algebraic geometry where he talks about zeta functions and cohomology theories on page 3 and then gets serious. 😆 npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…xuq7 - Oh great! I haven't ever looked at a book by Raskin, in fact I don't even know the name Raskin. But I've been thinking there should be an approach to algebraic geometry that goes like this. I only heard about this approach long after I suffered through the Hartshorne approach (listening to people talk about it, not actually studying it very hard). npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…xuq7 - Neat! I was just joking: I actually skipped commutative algebra entirely myself, too - I just zoned out whenever anyone tried to explain this stuff to me. I just posted a comment sketching in a very sketchy way the "modern" approach to defining schemes, via the Zariski site. Maybe this is what you're studying now. Hartshorne takes a more traditional approach - or maybe we should say the traditional approach is to follow Hartshorne. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez If like me you slept through part of your algebra class and spent years later trying to catch up on algebraic geometry, this is the thread for you! npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…a3ey - it's a great read, though not all at once. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…jrcx - it looks like an error-ridden text. Throw it out and get a better book on number theory! npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…sz8l @nprofile…u3x7 - It's pretty obvious that having Terry Tao use their software is better than any advertisement that money could buy. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…az9a - I was explaining the situation to my followers around the world, many of whom don't know the intricacies of the US federal system. I don't think Trump cares about laws and the constitution, but there's a whole network of people who do care, so when he breaks the laws they get energized to push back, and this does have an effect. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…ezv6 - there's no link there. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…az9a - it would be illegal, indeed against the US Constitution, for Trump to attempt to take over the election process. In fact I recently added a remark about this to my original post, for people unfamiliar with the US system. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…5ahq - I'll check it out. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…5ahq - I'd never say radically improved democracy is impossible. We need not only an appealing new system, but a way to get there from here. I hope you've written, or will write, a detailed tactical manual on how to get there from here. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…5ahq - Lots of people are discussing those questions. But it's also important to avoid falling off the cliff right now. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez Trump now says the quiet part out loud: "The Republicans should say: 'We want to take over. We should take over the voting in at least 15 places.' The Republicans ought to nationalise the voting." For folks outside the US: Elections in the US are run, not nationally, but by the individual states - until the final stage. Thus, nationalizing the elections would be a way to end fair elections in the US. We can guess that the "15 places" include states where ICE thugs currently roam in large numbers: states with Democrat-dominated cities such as Minneapolis, Portland, Chicago, New York City, Los Angeles and Atlanta. Trump can use ICE, the National Guard and - if he invokes the Insurrection Act - the military to suppress demonstrations against this takeover. Funding DHS, the department ICE belongs to, is a very bad idea. https://www.bbc.com/news/articles/c0mke841zj0o npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…9fju - sure thing! I've never been to the Calumet region. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez Ecologist Alison Anastasio visited a former US Steel South Works site in Chicago. She expected to find “all crap plants” — common invasive weeds. To her surprise she spotted little bluestem and three species of native milkweed. She had already decided she didn't want a career as an academic scientist. But she came up with the idea of forming a group to study this ecosystem: “a dream team of people I wanted to work with.” She knew Laura Merwin from the University of Chicago, and later she met Lauren Umek, a project manager for the Chicago Park District. She invited them to brunch to pitch her idea to research plants growing on slag. Not for any obvious career goal. Just from sheer curiosity. Merwin and Umek were excited to join her project - a “reverse side hustle,” since it involved a lot of work, but cost money. Their first paper, “Urban post-industrial landscapes have unrealized ecological potential,” was published in Restoration Ecology in 2022. It argues that slag fields don't need to be fixed. They have ecological value in and of themselves. And land managers should forget whatever ecosystem was there before. Instead, they should look to more exotic ecosystems as a guide, like the dolomite prairies of Illinois, where magnesium-rich rock near the surface makes it hard for ordinary plants to thrive. Slag too is rich in magnesium. The Slag Queens are continuing their revolutionary work even now! For more, start here: • Carrie Gous, The beauty of slag, https://mag.uchicago.edu/science-medicine/beauty-slag Some of what I just wrote is a paraphrase of this article. (2/2) https://media.mathstodon.xyz/media_attachments/files/116/002/537/476/696/335/original/67600c88728578df.jpg npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez Here's a tale of how nature triumphs in the end. Steel mills dumped molten slag in parts of Chicago and nearby areas. The slag hardened in layers up to 15 feet deep. These places became barren wastelands. Some were also dumping grounds for hot ash and cinders. Eventually the steel mills closed. The deep layers of hard, toxic material were not friendly to plants. Cottonwoods are usually 30 meters tall or more. In the slag fields, stunted cottonwoods grow to just 2 meters. But rare species that could handle these conditions began to thrive. The lakeside daisy, a federally threatened species lost to Illinois for decades, turns out to grow taller on slag than on topsoil! The capitate spike-rush, last recorded in Illinois in 1894 and considered locally extinct, was rediscovered growing on slag. And more! Native prairie grasses like little bluestem. Native milkweeds. Even tiny white orchids called sphinx ladies' tresses. A team of women ecologists began studying these unusual landscapes. They call themselves the Slag Queens. (1/n) https://media.mathstodon.xyz/media_attachments/files/116/001/935/711/383/568/original/f6921ce7e25c2330.jpg npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…3vpz - interesting enough. A bit more color variation on short distance scales might make the landscapes more realistic, especially the one at lower right which has large regions of dark orange (cloud shadows?). The clouds are simple blobs. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…lfhd @nprofile…t6k2 - I read the vaguely similar one Trading Places, about a British and a California professor who trade places. Fun study of stereotypes, probably dated now. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…t6k2 - My wife is reading Katabasis. She says it's too long (450 pages), and could have used editing to remove repetition. But she likes some of the ideas, e.g. how people in the Department of Analytic Magick at Cambridge need to compare Western and Chinese accounts of the geography of Hell to get a good understanding of the place. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez I thought they wouldn't release the Epstein files until hell freezes over... but in fact it only took frost quakes. Yes, FROST QUAKES, also called cryoseisms! It got so cold so fast in the US that freezing soil created earthquakes. They even cracked the wall in someone's house. https://www.npr.org/2026/01/30/nx-s1-5693186/winter-storm-causes-weather-phenomenon-known-as-frost-quakes-in-parts-of-the-south npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…fh5j - oh well! npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…t6k2 @nprofile…z2n3 @nprofile…s7d2 - just to be clear, where @nprofile…z2n3 quoted me as writing ω2 and ωω, I actually wrote ω^2 (ω squared) and ω^ω (ω to the ω). I was trying to say that PRA is powerful enough to prove that the Ackermann function is total because it can handle recurious up to ω^ω, but proving the Ackermann function is total only requires recursion up to ω^2. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez Fractions built from the numbers 2, 3 and 5 play a big role in music theory! Here I list those as close to 1 as possible for a given complexity. First we get a bunch of famous ones like the major third: 5/4 = 1.25 the minor third: 6/5 = 1.2 the diatonic semitone: 16/15 = 1.06666... and the syntonic comma: 81/80 = 1.0125 Eventually we get absurdly small ones like the quark of Baez: 2⁻⁵⁷³ 3²³⁷ 5⁸⁵ = 1.0000005104.... Yes, it's bad to name things after yourself. In fact that's item 25 on the crackpot index! So if you do it, you should do it for something silly. But the math here is not silly. I think it's really cool how musicians systematically explored fractions close to 1. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…g0zc @nprofile…gu9p @nprofile…ck2y - If Gödel really did get angry and threw out Chaitin, this could be one of the first examples of scientists getting disgusted by people trying to connect Gödel's theorem and Heisenberg's uncertainty principle. By now it's a routine occurrence. Speaking of "adding axioms", Grothendieck added an extra axiom to ZFC to carry out his work on algebraic geometry: the axiom of universes. This is a trick to get around the problem with "the set of all sets". The idea is that for any cardinal, there's a larger cardinal so big that the collection of sets with cardinality smaller than that is almost indistinguishable from the set of all sets. Since Grothendieck's work relying on the axiom of universe was later used to prove Fermat's Last Theorem, this raises the question of whether we only know FLT conditional on the axiom of universes! My friend Colin McLarty has been trying to sort this out. He believes that in all cases the axiom of universes can be sidestepped - at the expense of making various arguments more technical. Here's a surprisingly readable introduction to this business: https://www.jstor.org/stable/20749620 npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…r9vg - I guess they'll have to see by talking to each other! npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez We need all the humor we can get, these days: The Guardian reports that UK ticket sales for the new movie Melania are 'soft'. This is a great example of the British gift for understatement. "UK ticket sales for Melania are so far “soft”, according to Tim Richards, the chief executive of Vue, one of the country’s biggest cinema operators. Just one ticket has been sold for the first 3.10pm screening on Friday at its flagship Islington branch in London, while two have been booked for 6pm. At the time of publication, all seats remained available for the 28 screenings of Melania at the Blackburn, Castleford and Hamilton branches. The picture was slightly rosier at the Cineworld in Wandsworth, which had sold four tickets, while five backrow seats were also booked at the Cineworld in Broughton." https://www.theguardian.com/us-news/2026/jan/26/melania-trump-documentary-uk-cinemas-vue-soft-sales npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…zyzh - I even heard that some people listed their MathOverflow (or was it StackOverflow) reputation scores on their CVs. Blech. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…zyzh - Interesting. For some kinds of math questions there's a right answer, e.g. true-or-false questions (which can be very powerful in mathematics). For these it can be really helpful for beginners to know what's the right answer. For other questions, more open-ended, there's nothing at all like a unique right answer. I personally never even paid much attention to this check mark business. Personally I think the competitiveness comes in when we see charts like this: https://mathoverflow.net/users/2893/john-c-baez I'm the 100th best person on MathOverflow this month, etc. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…zyzh - nothing in MathOverflow says a question has a single answer. Many questions there have lots. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…wncd - great! I have been very busy and distracted but I ***do*** want to return to this project and finish it off. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…fh5j - I find MathOverflow extremely useful myself, because it attracts lots of experts who can help me with problems I'm stuck on. However, I suspect you're right that "gamifying" the process of asking and answering questions by giving people points and publicly displaying their "reputation" scores could be off-putting to lots of women (and also certain kinds of men). When I suggested that women might not enjoy this competitive approach, someone reacted by pointing out the existence of highly competitive women tennis players, and then the conversation digressed to the meta-question of whether this analogy was useful - and the meta-meta-question of whether analogies of this sort were ever useful. I found it quite unsatisfying to see my point shunted off in this way. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez Not very urgent compared to some things, but okay: MathOverflow is a website for asking and answering math questions. Very few women participate. After hearing some women discuss the reasons why, I asked what has been done to make MathOverflow more welcoming to women: https://meta.mathoverflow.net/questions/6367/what-is-being-done-to-make-mathoverflow-more-welcoming-to-women Answer: something was done once, but nothing is being done now. But some preferred to argue against the very question. And now I see that someone has posted a followup, proposing that "questions that invite general discussion about how to address the gender imbalance on MO are off-topic and should be closed/locked/deleted" https://meta.mathoverflow.net/questions/6390/proposal-free-floating-discussions-of-things-like-women-and-mo-should-be-bann The reason, supposedly, is that my question started a conversation that "did not reflect well on MathOverflow". Personally I think the solution is not to shut people up, but to start doing things to make MathOverflow more welcoming to women. Some women say this is hopeless - they'll just ignore the place and/or hope it dies. I hope people who think that start up a new site. But I hope people who like MathOverflow try to improve it. (For nonexperts I should clarify that I asked my question on MathOverflow Meta. This is precisely the place to talk, not about math, but MathOverflow policies. So it's not as if I were interrupting math conversations with "off-topic" discussion of policy. The objection was solely that I started a conversation that became unpleasant in some ways. By now the moderators have deleted the most obnoxious comments.) npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…g0zc - Interesting! So if you say a young man is luckless and feckless, that's a nice euphemism for something that rhymes. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…g0zc - opposite of feckless? npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…g0zc - "This shepherd's pie is f***ing good!" npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…g0zc - the shepherd's pie is f***ing good! npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…4qpt - the symmetry brings out the beauty. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…jrcx - @nprofile…t6k2 would have opinions on this, though he may be tired of talking about it. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…a3ey - I guess this is early enough that he's not saying it with the somewhat exasperated tone of a modern mathematician? npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…j4wh - really??? Can't AI solve that problem? npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez "There's a danger that our technology could fail if it fails!" Maybe this man's impressively large dome makes such remarks sound profound. https://www.irishtimes.com/business/2026/01/20/ai-boom-could-falter-without-wider-adoption-microsoft-chief-satya-nadella-warns/ npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…a3ey - noble of you! npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…a3ey - I too promise to no longer publish more than 7 papers per year. 😆 npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…2sx2 - they've got lots of nice posts here. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…2sx2 - there are some very nice experts on constructive mathematics here on Mathstodon, e.g. @nprofile…t6k2 and @nprofile…ldyp. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…2sx2 - there are mathematicians who question all the things you're questioning, and they write papers about that. However, it pays to learn the standard rules. This doesn't mean "believing" in them, indeed one great thing about math is that checking the validity of a proof never requires belief. It pays to learn the standard rules because if you don't, you're cut off from an enormous body of centuries of work. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…e2fl - now I get to check out #SolarPunkSunday! npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…6fwm - what a great story! I could enjoy a whole novel about beavers on the San Pedro, written in a similar style, with maps. But I'll be frequenting your blog now, for other tales. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…nekg - that would definitely be disturbing if I were out camping. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…80ng - will do! npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…ccq6 - indeed, that's how I often feel these days. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…x8fk - we see animals called bobcats in our back yard in Southern California, but these are actually lynxes... so I'm a big fan of the lynx. Hurrah for the Iberian lynx! npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez Some good news in a time of darkness: the European wildcat, Felis sylvestris, is making a comeback! This thoughtful-looking example was photographed in a mountainous region of the Czech republic. The European wildcat's extreme elusiveness may have helped it avoid hunters in places where a larger native cat, the lynx, has been killed off. There may be about 140,000 European wildcats spread across more than two dozen countries. But they are very hard to find! Wildlife photographer Andrea Giovanni, who made a video of one, writes: "I'd never even thought of taking photos of wildcats, for a simple reason: I thought it was impossible, or at least, extremely difficult. It's considered 'the ghost of the forests' because it's very, very elusive, and it's hard to predict where it can be spotted. Other animals tend to follow the same trails through the forest. The wildcat goes wherever she wants to." One reason the European wildcat is coming back is increased legal protections. But another is that villages in Italy and other regions are becoming depopulated! Some are very worried about declining human populations. But it does make room for other species. I got this picture, taken by Vladimír Čech Jr in the Doupov mountains, from a very nice article on the European wildcat: https://www.bbc.com/future/article/20260112-rare-images-of-europes-ghost-cat For more on this species: https://en.wikipedia.org/wiki/European_wildcat https://media.mathstodon.xyz/media_attachments/files/115/911/651/790/519/381/original/c65d70c11d981b0c.jpg npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…659g - is that on the California coast south of Big Sur? That's where I've seen such rocks. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…r0xq - I'm afraid so! npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…r0xq - I hope they had a big party at the Department of Humanities at MIT. Yes, they have just one department for all "humanities". https://catalog.mit.edu/schools/humanities-arts-social-sciences/humanities/ npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…r0xq - Okay, I can't figure out how to see after the 4th on my laptop. Let's just say it's not optimally designed. But it turns out they really *do* claim MIT is the top university in the world for arts and humanities, and that's a complete joke. I did my PhD there, it's a great place, but it has very few people working on arts and humanities. npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6m John Carlos Baez @nprofile…qlgd - Was Lior Pachter a grad student at UCR for a while?