[ \sum_{n=0}^{\infty} {\frac{n^4}{n!}}=15e ]
This is strange enough to provoke wonder, but simple enough to serve as an entry-point to an interesting generalization.
@phonner@johncarlosbaez@paulmasson The “symbolic method” (as in the lovely "Analytic Combinatorics" book by Flajolet and Sedgewick) gives a slick (after you buy into it, i.e. the background) proof of Dobiński's formula. I wrote a quick post about it here a while ago; don't know how understandable it is: https://shreevatsa.net/post/permutations-dobinski/
"A follow-up with 13 of the volunteers three years later revealed that those who had used GPS the most during the intervening period experienced greater declines in their ability to navigate without GPS, strongly suggesting that GPS reliance causes diminished skills, rather than poor skills leading to greater GPS use."
@ColinTheMathmo Personal experience is that my sense of direction, while always poor, became:
• worse when using GPS while driving (navigation in Google Maps) with the default settings,
• better (than ever) when using GPS/Maps while driving, with the orientation fixed to North.
I can strongly recommend the latter; I'm gaining a sense of direction that I never had before.
The main change is that the arrow will e.g. point to the left/right/down depending on whether I'm driving west/east/south, but this way I get instant feedback (e.g. I was driving west and just now I turned left, so I'm now driving south), and get to check my sense of direction against reality, and stay oriented in my head. Something I was never able to do earlier (had no sense of where different places were relative to each other). Strongly recommended.
@ColinTheMathmo I found this one trickier than usual (though I guess it was mostly because of one particular mistake I made early): took me 4:04 and I usually aim to finish in under 90 seconds. Still, fun!
XKCD comic pointing out that that the difference between 91% (or even a 99%) eclipse and a total eclipse is extremely dramatic. An almost total eclipse is barely noticeable, while a total eclipse is a visual phenomenon unlike any other.
@AkaSci How should we understand the lower half of this image? I see the southern hemisphere too in the upper half (South America, Australia etc), and the linked url also has only the upper half, so wondering what the lower half is.
This article has a misleading click-bait title. What's really happening is that some folks are trying to make the proof of Fermat's Last Theorem completely rigorous using a computer system called Lean.
While this is neat, I think it would be even more game-changing to create a bunch of webpages and videos that explain the proof of Fermat's Last Theorem really well. It would take lots of work. But I think the main thing holding math back is how poorly it's explained.
I'm sure that formalizing the proof of Fermat's Last Theorem will help the people who do the formalization understand the proof. But if one's goal were getting lots of people to understand the proof, formalization would not be the most effective approach.
By the way, I don't think Fermat's Last Theorem is especially important in itself! It's mainly just a convenient excuse for getting people interested in cool ideas like modular forms and elliptic curves. And I believe these, in turn, are just a convenient excuse for getting people to think deeply about even cooler ideas about the relation between algebra, geometry and symmetry. These ideas are very beautiful, but very hard to learn at present. You almost have to read them between the lines of existing books and papers.
@johncarlosbaez Terence Tao's recent talk on "Machine Assisted Proof": https://youtu.be/AayZuuDDKP0 (it's a broad overview, but also mentions this Fermat's Last Theorem Lean work in passing)
Also on the subject of bash brace expansion: the editing key ESC { runs the complete-into-braces operation, which generates a brace-expanded list of all the filenames that match what you've typed so far.
But not optimally …
$ touch {0,1}{0,1}{0,1} # typed by hand
$ ls {0{0{0,1},1{0,1}},1{0{0,1},1{0,1}}} # generated by pressing ESC {
… and no wonder, because it's a horrible search problem. I struggle to think of an algorithm to reliably find the shortest answer slowly, let alone fast!
@b0rk As you mentioned "fun" — a lot of times doing things for fun ends up involving algorithms (e.g. lots of people play Sudoku, and anyone who tries to solve Sudoku by computer arrives at backtracking).
(But that's also why they may not fit into the "good" set of topics to explain, because everyone has a different idea of what's fun.)
@b0rk I guess another thing that makes algorithms/theory a poor fit for teaching from any real-world application is that we don't know beforehand what theory we'll need — in your example (https://social.jvns.ca/@b0rk/111734271606635587) you needed both graph algorithms and mixed integer linear programs, but one wouldn't guess beforehand to put those two topics together — so yeah, hard to say "here, this is useful" :)
There are only about 80,000 old giant sequoias left in California. After years of drought, roughly 10% of these enormous trees died in a massive fire in 2020. The future for them does not look good.
But I just learned that there are about 500,000 younger giant sequoias and closely related coastal redwoods in the UK!
They were first introduced in 1853 by the Scottish grain merchant Patrick Matthew. Later that year, the famous plant collector William Lobb brought over many more. Because of their rarity and novelty, these trees were very expensive. But that worked in their favor: they became a symbol of wealth in Victorian Britain. People planted them at the entrances of grand houses and estates, along avenues, and in churchyards and parks.
The map shows just 4949 of the giant sequoias in the UK. Surprisingly, they thrive there, despite the climate being very different from that in their native range - the Sierra Nevada mountains, dry in the summer and snowy in the winter.
They are now the largest trees in the UK! A new study shows each tree sequesters carbon at the rate of 85 kilograms per year, on average. This is very high for trees, and sequoias keep growing for centuries. People are wanting to plant more.
Invasive species - or wonderful rescue of a species that might otherwise go extinct? Evolution does its thing regardless of our value judgements, but what we do affects it. Long ago, giant sequoias were common in North American and Eurasian coniferous forests. By the last ice age, their range had shrunk to a small area in California. Now, thanks to humans, they are spreading again.
@johncarlosbaez The 80000 vs 500000 number was quite surprising, but this paragraph on the Wikipedia clarified it:
> In 2024, there were an estimated 5,000 mature sequoias in the UK, and up to 500,000 sequoias and sequoiadendrons that are not mature. Growing conditions are generally more conducive than in their native range in the US though they do not propagate naturally.
@ColinTheMathmo I forgot to do this one until I saw this post -- looked at the time and was disappointed it was already past 4pm here (the time here when a new one used to come out every day), then realized DST has just started here so I still had time until 5pm :-)
A group I'm in is reviewing and revising a document. It's a word document, because some of them are seriously, seriously technically challenged, and that seems to be the only thing they can cope with.
But making comments on and passing copies around of a Word document is just a completely nightmare, and it ends up ... as you will all know ... a mess of formatting, wrong versions, just ... urgh.
If there is a website with the document visible and little boxes spread throughout into which people can type comments to be collected and collated, that would be much easier.
@ColinTheMathmo I'm surprised to hear you found Google Docs confusing to use. At least for the use case you're mentioning (of having people type comments on a document), it seems to be designed for it... In case it helps, I set up an example Google Docs document for you (or anyone) to play with : https://docs.google.com/document/d/1PeyrfppS0pZNoWvibF6BP_X2B-gZasqpkssLfT1dSjI and recorded a 40-second video of adding comments to it (in two ways: clicking on the comment icon in the margin, and right-click and comment). It can be used without logging in to a Google account too (just tried from an incognito window). Hope it helps!
Galileo put a lot of work into explaining why we don't feel the motion of the Earth. I just learned that similarly, early scientists had to explain why we don't hear the music of the spheres!
Aristotle writes:
"Some thinkers suppose that the motion of bodies of that size must produce a noise, since on our earth the motion of bodies far inferior in size and in speed of movement has that effect. Also, when the sun and the moon, they say, and all the stars, so great in number and in size, are moving with so rapid a motion, how should they not produce a sound immensely great? Starting from this argument and from the observation that their speeds, as measured by their distances, are in the same ratios as musical concordances, they assert that the sound given forth by the circular movement of the stars is a harmony. Since, however, it appears unaccountable that we should not hear this music, they explain this by saying that the sound is in our ears from the very moment of birth and is thus indistinguishable from its contrary silence, since sound and silence are discriminated by mutual contrast. What happens to men, then, is just what happens to coppersmiths, who are so accustomed to the noise of the smithy that it makes no difference to them."
Aristotle didn't believe in the music of the spheres. But much later, Kepler was fascinated by it. Here's a picture from his book 𝘏𝘢𝘳𝘮𝘰𝘯𝘺 𝘰𝘧 𝘵𝘩𝘦 𝘞𝘰𝘳𝘭𝘥𝘴.
Now we know each planet makes gravitational waves with a frequency equal to its year - very faint, and a very deep bass. But as neutron stars spiral into each other, in the last second their gravitational waves make a powerful 'chirp' with frequencies that soar over an octave above middle C!
> Galileo put a lot of work into explaining why we don't feel the motion of the Earth.
In India too, after Aryabhata (5th century) declared that day and night (and the motion of the stars) are because the earth rotates — and gave the example of how when on a boat the trees on the shore seem to be moving — some later astronomers (like Brahmagupta, 7th century) were sceptical, and raised questions (which took Aryabhata's students some effort to answer) like how birds returned to their nests in the evening if the earth had moved substantially during the day.
@ColinTheMathmo thanks for posting about this puzzle every day, it keeps me playing and I'm enjoying it :) Yes I too found today's puzzle one of the easier ones in recent days -- finished in 1:45 or so even including the time to handle (reject) a phone call interruption -- though the path I took was amusing:
Hindustani music has lots of ragas, but they're often organized into the 32 kinds shown here, of which only 10 are very commonly used. These kinds are called "thaats".
Each thaat is a 7-note scale. I'll explain them in a western way, not an Indian way. This will make it clear how 𝑢𝑛𝑠𝑢𝑟𝑝𝑟𝑖𝑠𝑖𝑛𝑔 the ideas are to anyone who has studied western music theory.
Let's imagine our thaat starts with the note C. Then:
• it must contain C
• it must contain D or D♭, but not both
• it must contain E or E♭, but not both
• it must contain F or F♯, but not both
• it must contain G
• it must contain A or A♭, but not both
• it must contain B or B♭, but not both.
So, we get 2⁵ = 32 thaats. It's really unsurprising that C and G are locked in place, while the other 5 notes are flexible - after all, C and G are the 'tonic' and 'dominant', also called the 1 and 5, the most important notes in a scale.
A lot of thaats match familiar western modes, but later I'll show you some that may not. Maybe some expert on jazz can say if they've seen them.
Thaats are just the start of the story, since we can get extra ragas by leaving out some notes in a thaat... and as a result, different ragas can come from the same thaat.
But there's a more urgent issue: what are all the letters in this chart? The notes in the thaat are called
sa re ga ma pa dha ni
or for short
S R G M P D N
These are a lot like the western "do re mi fa so la ti".
But as I've said, we get a binary choice only for 5 of these notes, namely R G M D N, since the other 2 are locked in place. That's what the chart shows. It's a binary tree with 32 leaves, namely the thaats.
@johncarlosbaez I have nothing to add, but it's nice to see your further interest in Indian music 🙂
I don't know much about Hindustani music, but looking at the Wikipedia article, I was surprised to find that although Hindustani music and its ragas are centuries old, the system of classifying them into thaats is only about a century old!
> The modern thaat system was created by Vishnu Narayan Bhatkhande (1860–1936) […] Bhatkhande modelled his system after the Carnatic melakarta classification, devised around 1640 by the musicologist Vidwan Venkatamakhin.
@johncarlosbaez Without going into the history and Hindustani vs Carnatic, to me what this shows is that the ragas are “real”/primary, while the classification/theory is secondary: the primary purpose of music (at least Indian classical music) is to give joy to the listeners by evoking specific emotions in them, and for this in practice there were several ragas refined over the centuries—they could exist independently and be passed down from teacher to student individually, even before someone came up with an overarching theory classifying them and bringing/noticing structure in these disparate ragas.
[As for the difference between the two traditions, I think they have more in common than they have differences! But one joke I've heard (by a late comedian from North Karnataka, a "border" region where both styles of music have popularity) was comparing them to local toddy vs an imported drink (of some sort, I don't drink so I don't remember): his analogy was that with Carnatic music you get the "kick" sooner (relatively), while with Hindustani music it takes a while to take effect but the "high" lasts longer after it's over.]
It's sort of ridiculous that I've never studied even the very basics of Indian music, like what are the ragas. Thanks to my friend Todd Trimble I'm dipping my toe into it. And it turns out that in Carnatic music, the 72 Mēḷakarta ragas are just subsets of the usual 12-tone scale!
I just learned why there are 72. Let's use western notation and call the 12-tone scale
C Db D Eb E F F# G Ab A Bb B
The subset needs to contain C and G. (So, it needs to have the tonic and perfect fifth - a very reasonable constraint by western standards.)
It has to have 2 of the 4 notes Db D Eb E - so that's 6 options. (In western terminology we'd say it needs to have a second and a third, but the second can be minor, major or augmented and the third can be diminished, minor or major.)
It has to have either F or F# - so that's 2 options. (So, it needs to have a fourth, which can be perfect or augmented.)
It also has to have 2 of the 4 notes Ab A Bb B - so that's 6 options. (So, it needs to have a sixth and seventh, but the sixth can be minor, major or augmented, and the seventh can be diminished, minor or major.)
That gives a total of 2 × 6 × 6 = 72 options!
All this seems very reasonable by western standards: it's just very systematic and thus includes more cases than the most common western modes.
But beware! There are many more 'janya ragas' derived from the basic 72 by leaving out notes and other tricks. This calculation claims there are 28,864 possible janya ragas:
@johncarlosbaez I'm mostly just a casual listener, but re "what fraction of these are commonly played", AFAIK a typical performing musician has something like 40–50 rāgas in their repertoire (ones they've truly mastered the "nature" of), and across musicians maybe some 200–300 hundred rāgas are commonly played. I'm sure someone will correct me if this is wrong :-)
@johncarlosbaez Oh also, you may also like this, as more closely related to notes and intervals: https://www.carnaticcorner.com/articles/22_srutis.htm (I've been meaning to re-read it and rewrite in terms I can understand more easily, and will let you know if I do, but sharing in the meantime…)
@mattmcirvin - as you probably know the Babylonian method for computing square roots converges quadratically, roughly doubling the number of correct digits each time. On this tablet, made around 1750 BC, they got an answer equivalent to roughly
The same value "577/408" is also found in India in the Baudhāyana Śulbasūtra, "possibly compiled around 800 BCE to 500 BCE", but arrived at (probably) by very different methods. The expression is of the form
> "The measure is to be increased by its third and this [third] again by its own fourth less the thirty-fourth part [of that fourth]; this is [the value of] the diagonal of a square [whose side is the measure]."
There's a 1972 paper by Donald Knuth called “Ancient Babylonian Algorithms” where he discusses some of the Babylonian mathematical tablets from a computational perspective. I've just posted it here for now: https://shreevatsa.net/tmp/2023-12/DEK-CS-11-P53-Ancient.Babylonian.Algorithms.pdf
He makes comments like "the Babylonians actually worked with floating-point sexagesimal numbers", and “This procedure reads surprisingly like a program for a "stack machine" like the Burroughs B5500”.
It's the number of cards in a deck, not counting jokers. This may not be a coincidence: the 4 suits correspond to the 4 seasons, there are 13 weeks per season, and the total value of all the cards is
with the joker giving 365. (The second joker helps out on leap years.)
There are 52 partitions of a 5-element set into disjoint nonempty subsets. So, we say 52 is the fifth 'Bell number'. They're named after Eric Temple Bell, who wrote the famous book Males of Mathematics. He did not discover these numbers.
In Japan, 52 chapters of the Tale of Genji have traditional symbols on the top called 'genji-mon', which correspond visibly to the 52 partitions of a 5-element set. Unfortunately the Tale of Genji has 54 chapters, so they needed to make up two extra symbols.
@johncarlosbaez BTW, re. the implied criticism in E. T. Bell writing “Men of Mathematics” — we learn from Constance Reid's biography that he submitted it to the publisher under the title “The Lives of Mathematicians”; they changed it to the title he hated, to tie in with their “Men of Art” book. Also, I vividly remember reading about Sophie Germain in that book, plus one of the chapters is about (Weierstrass and) Sofya Kovalevskaya. https://mathstodon.xyz/@svat/109701209533195575
has anyone seen a really good analysis of the problems with git's command line UI? Would love to read it. for example:
git checkout is dangerous and has too many different jobs (though git switch is trying to fix that!)
for a tool that's supposed to make changes easy to undo, you actually need to learn a LOT of ways to undo
(not looking for git tutorials, explanations of git’s underlying model, or explanations of why you think git's UI is actually good, just an analysis of the problems)
