robinhouston

Maths etc.

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npub1wjc42kn5sa2pek0h5gfygwrs2s6r7cyr7l2mlpna529pkff97jzs97nda9

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robinhouston@mathstodon-xyz.mostr.pub

Profile Code

nprofile1qqs8fv24tf6gw4qum8m6yyjy8pc9gdplvzpl04dlse769zsmy5jlfpgpz3mhxue69uhhyetvv9ujuerpd46hxtnfduqs6amnwvaz7tmwdaejumr0dse6uqr4

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relay.damus.io

nos.lol

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npub1wjc42kn5sa2pek0h5gfygwrs2s6r7cyr7l2mlpna529pkff97jzs97nda9

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Last Notes

2026-10-08 07:48:59 UTC

They’ve just withdrawn three of the papers:
* Algebraicity of Weil classes on split abelian eightfolds
* Algebraicity of Kuga–Satake Correspondences for K3 Surfaces
* The rational Hodge conjecture for products of K3 surfaces
https://github.com/openai/math/blob/main/history.md

2026-10-07 07:23:22 UTC

Looks like something big dropped overnight: https://github.com/openai/math/tree/main/preprints
This is going to take some digestion…

2026-09-28 09:15:44 UTC

For my birthday, my brother-in-law gave me a 4992-digit hex number that, when used as a seed for Python’s random number generator, makes it so that printing a sequence of “random” ASCII characters actually prints a birthday greeting.
I find it rather amazing that this is possible!
This is the program he sent with the seed:
with open("seed.txt", encoding="ascii") as handle:
seed = int(handle.read().strip(), 0)
r = random.Random(seed)
pending = ""
while True:
character = chr(r.randrange(128))
if pending + character == "\x1e\x1f":
break
print(pending, end="")
pending = character
print()

2026-06-29 18:07:02 UTC

I haven't read this paper yet, but I love the abstract! I didn't even know the 2D version of this theorem. Very nice.
https://arxiv.org/abs/2606.02087

2026-05-08 16:28:15 UTC

User cachecrab on Twitter noticed that Google has a rather unusual interpretation of the query “20% of £1k”.
https://media.mathstodon.xyz/media_attachments/files/116/539/889/355/155/788/original/a74666b29ac025ec.png

2026-05-01 13:31:37 UTC

I want to know the relative popularity of the Platonic solids, but a Mathstodon poll can have at most four options.

2026-04-22 15:57:43 UTC

Spotted in The American Mathematical Monthly, Vol. 99, No. 2 (Feb., 1992)
https://media.mathstodon.xyz/media_attachments/files/116/449/176/585/695/946/original/90b6663333261361.png

2026-04-18 18:50:19 UTC

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@nprofile…zl3j The geared mechanism is my favourite.
https://media.mathstodon.xyz/media_attachments/files/116/427/189/841/296/808/original/f25cef914afe766b.mp4

2026-04-15 18:06:03 UTC

I guess we’re all getting a little tired of talking about this sort of thing, but I think this is the most impressive example yet of an LLM-generated solution to a long-open mathematical problem:
https://www.erdosproblems.com/forum/thread/1196
Jared Lichtman, a mathematician who has spent (in his words) “many years” working on the problem himself, describes the proof as “a remarkable artifact […] from The Book”.
(See also https://x.com/jdlichtman/status/2044298382852927894 if you do 𝕏.)

2026-02-28 22:27:35 UTC

Ok, this is kind of cool. Can you find an elegant proof?
#nevent1q…u4m0

2026-02-26 11:57:39 UTC

“In this paper we present work on enumerating all the incomplete open platonic solids, finding 6 tetrahedra, 122 cubes (just like LeWitt), 185 octahedra, 2,423,206 dodecahedra and 16,096,166 icosahedra.”
https://arxiv.org/abs/2602.20425
https://media.mathstodon.xyz/media_attachments/files/116/136/808/021/569/521/original/3159e6e52574a53b.jpeg

2026-02-13 23:31:38 UTC

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@nprofile…k20z I'm so glad it worked out!

2026-02-13 15:44:11 UTC

I know this is short notice, and very geographically specific, but one of our party can't make it so I have a spare ticket for @nprofile…fh5u in Shrewsbury this evening, if anyone wants to come.

2026-02-12 21:51:44 UTC

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@nprofile…f38c I tried to fix a typo in your post – 6+2+2+1+1 ≠ 13, and presumably ought to be 6+2+2+2+1 – but I am not allowed to make an edit of fewer than six characters, so I cannot.

2026-02-12 13:34:25 UTC

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@nprofile…wc04 Lovely! I was getting a bit worried that loads of people had liked my post but no one had shown any evidence of having successfully printed them.
I'd love to see any models you design.

2026-02-10 13:21:06 UTC

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@nprofile…ft7j There is a cure, but you're not going to like it

2026-02-09 22:47:53 UTC

Achievement unlocked! Feature in an OskarPuzzle video.
https://www.youtube.com/watch?v=Uzc1u2VPNyc
(We actually have a rather more general theory of caltrops than this video describes. We’ve written a paper on it, which we have submitted to a conference. Maybe we’ll publish a preprint soon.)

2026-02-08 17:39:56 UTC

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@nprofile…mhl2 53 is Caol Ila. One of my favourites!

2026-02-07 13:58:48 UTC

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@nprofile…f38c Well, he did make a film in which he appears naked…

2026-02-07 13:33:59 UTC

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(It's pretty much who you'd most expect.)

2026-02-07 12:58:16 UTC

Can you guess which well known mathematician DJs under the name DJ Moonstein?
https://soundcloud.com/moonstein

2026-02-01 09:14:13 UTC

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@nprofile…x7z0 nice!

2026-01-30 12:08:14 UTC

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“For me, creating sculpture is a compulsion. I might try to rationalize it as educational, I might be gratified to receive kind words from an appreciative audience, and I might enjoy the profit of making a sale, but I do it as a compulsive behavior.”

2026-01-30 10:47:52 UTC

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@nprofile…yrk9 @nprofile…cz9e That's a pretty good suggestion! Maybe ‘diminished’ is even better, because ‘truncated’ most commonly means you're truncating _all_ the vertices, and Johnson in his enumeration of Johnson solids uses diminished to mean that a single pyramid or cupola is removed.

2026-01-29 18:59:33 UTC

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@nprofile…7jv6 Well, you know what they say: ask a silly question, get a brilliant and fascinating answer.
Do you happen to know of any especially good sources in which one can read about the work of these geometers?

2026-01-29 18:42:15 UTC

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@nprofile…yrk9 Oh, of course! I was looking for names from mathematics, rather than computer graphics, like a fool.
They aren't quite the same as the polyhedra I'm talking about, but they're closely related. You can get my polyhedra by deleting every alternate vertex of a UV sphere, like forming a demicube from a cube.

2026-01-29 15:15:32 UTC

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@nprofile…4rfv Nor could I, but I googled a picture of it just now.

2026-01-29 15:14:20 UTC

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@nprofile…4rfv Isn’t that an icosahedron with a bit missing?

2026-01-28 09:10:22 UTC

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If you don't know the work of Kagen Sound, which this design is a tribute to, this video is a lovely introduction to his work: https://youtu.be/_NreYBwzRQk

2026-01-28 00:26:41 UTC

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@nprofile…qnl0 Whatever it is that the second link in the message at the head of this thread points to

2026-01-28 00:25:32 UTC

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@nprofile…qnl0 I don't know what, because I don't have permission to view it!

2026-01-28 00:24:28 UTC

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@nprofile…qnl0 😭
https://media.mathstodon.xyz/media_attachments/files/115/969/874/964/861/705/original/82c88a05812dc6e3.png

2026-01-27 21:06:04 UTC

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The best tool I've found for trimming machine screws to precise lengths is apparently only sold in Japan, for some reason.
https://media.mathstodon.xyz/media_attachments/files/115/969/092/229/011/452/original/5c573ce0940c180a.png

2026-01-27 20:53:00 UTC

This is a delightful contraption!
I imagine it's a bore to print if you don't have a dual-nozzle printer, and it's a bit of a journey to assemble. (It requires M3 machine screws of sundry non-standard lengths, so you'll need to get long ones and cut them to size.)
But the end result is lovely to play with.
https://youtu.be/7U9ozXaxz9Y

2026-01-24 20:16:07 UTC

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@nprofile…uy2p That’s why only powers of two can happen. If the order has an odd factor k, we can just take f^k as a duality map with lower order instead.

2026-01-24 20:14:14 UTC

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@nprofile…uy2p These examples are specifically demonstrating cases where the lowest possible order of self-duality map has a surprisingly large order.

2026-01-24 20:13:28 UTC

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@nprofile…uy2p That may be possible, but if so we could just take f^3 as our duality map instead, and it would have order 2.

2026-01-24 19:51:23 UTC

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@nprofile…uy2p Figure 1(b) of https://dml.cz/bitstream/handle/10338.dmlcz/132561/MathSlov_47-1997-2_4.pdf shows the edge-graph of a polyhedron that is claimed to have a rank-8 self-duality map. Perhaps I will look at that one next.

2026-01-24 19:46:44 UTC

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@nprofile…uy2p I have read that there exist examples in which the order is any power of two, but I haven’t yet studied them.

2026-01-24 19:46:01 UTC

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@nprofile…uy2p In this particular example, the dual map has order 4. You can more or less see that from the two-color model: the dual map represents a rotation by 90°.

2026-01-24 17:36:38 UTC

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You can pretty much see just from that photograph that passing from the polyhedron to its dual corresponds to a 90° rotation; so of course doing it twice must correspond to a 180° rotation, rather than to the identity transformation.

2026-01-24 17:29:25 UTC

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I printed a two-colour model of the polyhedron and its dual. I'm pretty happy with how it came out.
https://media.mathstodon.xyz/media_attachments/files/115/951/248/799/385/531/original/d30b55de4b9dfb18.png

2026-01-23 16:50:39 UTC

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It would have been a delightfully neat coincidence if the other one had turned out to have a different type of self-duality, but no, that one is also a 180° rotation.
https://media.mathstodon.xyz/media_attachments/files/115/945/430/457/519/769/original/983b96216bdca0f5.png

2026-01-23 09:39:49 UTC

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I’m trying to understand the precise nature of the self-duality of these polyhedra. Since the polyhedra have no symmetry, it must be involutive, so there are three possibilities. (Either the two polyhedra are related by central inversion, by a 180° rotation, or by a reflection in an equatorial plane.)
Here is one of them with its dual. It is apparently related to its dual by a 180° about some arbitrary-looking axis.
https://media.mathstodon.xyz/media_attachments/files/115/943/730/395/015/091/original/ae04424b7f270e71.png

2026-01-22 20:28:06 UTC

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@nprofile…v0c7 If it’s not too late, this model is probably a better one to print, because the duality is easier to see from the canonical form of the polyhedron. https://mathstodon.xyz/@robinhouston/115940630890307894

2026-01-22 20:27:10 UTC

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And a standalone model of just the canonical polyhedron, in case anyone wants that for some reason. https://sketchfab.com/3d-models/jendrol-canonical-e6eaf1530d7245be9c7cc78d63f335df

2026-01-22 20:15:07 UTC

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Here is a 3D model that illustrates the duality. https://sketchfab.com/3d-models/jendrol-non-involutary-selfduality-complex-0f153bfb8b844768951a9d1043d556af

2026-01-21 13:19:21 UTC

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@nprofile…v0c7 That's a lovely idea! I hope you can.
3d printing has become very cheap and easy. The material cost of such a model would be less than $1, and even the printers themselves are astonishingly affordable.
I'm hoping to design a two-colour model that will illustrate the duality, but that will take more work and I'm not sure when I will finish it. It would also require a more sophisticated machine to print.

2026-01-20 20:52:00 UTC

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@nprofile…sjc5 Yes, absolutely. It's interesting how the other 7-vertex 7-face polyhedra are very obviously symmetrical from their Schlegel diagrams, so these two stand out clearly.

2026-01-20 20:12:39 UTC

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A puzzle: can you find the self-duality just by inspecting the polyhedron, without looking at Jendrol’s paper?

2026-01-20 20:11:12 UTC

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Okay! It is printing now, so no promises about how it will come out if you try to print it, but here is the 3D model: https://sketchfab.com/3d-models/jendrols-example-of-non-involutary-selfduality-25129ef1283643e6a304df6cd14236af

2026-01-20 20:07:19 UTC

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@nprofile…f38c Thank you! I have added the missing word, which you correctly guessed.

2026-01-20 19:10:00 UTC

A tangent from my main thread: here is a surprising fact about polyhedral self-duality.
It’s possible for a polyhedron to be self-dual, but only in such a way that applying the duality map twice DOESN’T return the polyhedron to its starting position.
Here’s the paper (from 1989), but because it’s 2026 I’m going to make a 3d-printable version of this polyhedron. Hopefully that won’t take tooooo long – bear with me.
https://www.sciencedirect.com/science/article/pii/0012365X89901441

2026-01-20 19:00:37 UTC

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You can think of the quad graph as being a combination of the original planar graph with its planar dual.
Now, a (spherical) polyhedron is self-dual if and only if it’s isomorphic to its dual, or equivalently iff there is a part-reversing automorphism of its quad graph.
So that’s what I mean when I say that self-duality is akin to symmetry. Precisely, it is a symmetry of the quad graph. Since our original polyhedra(l graphs) have no symmetry, the only symmetries of the quad graph there could possibly be are part-reversing ones, i.e. dualities.

2026-01-20 18:51:55 UTC

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That’s interesting, because self-duality is almost a sort of symmetry.
Let me explain. Given a polyhedron P, which for these purposes we can think of as a graph of vertices and edges drawn on the surface of a sphere, we can construct another graph, a bipartite one, that has two types of vertex: a vertex for every of P, and also a vertex for every face of P. Whenever a vertex is on the boundary of a face, we connect the corresponding vertices of our new graph.
It would be nice if there were a standard name for this derived graph, but every paper I’ve read that uses it gives it a different name. Archdeacon and Richter call it the radial graph; Felsner and Rote call it the angle graph; Ziegler calls it the quad graph.
Of those names, I think I like ‘quad graph’ the best, because it describes an important feature of this graph, which is that all of its faces are quadrilaterals. So for now let’s call it that.

2026-01-20 18:43:04 UTC

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Since they have the same number of faces as vertices, and their defining property of asymmetry is preserved by duality, they must either be dual to each other or both be self-dual.
It’s easy to see they are not dual to each other. For example, one has a pentagonal face, and the other has at most 4 edges meeting at each vertex.
So they must both be self-dual.

2026-01-20 13:03:28 UTC

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Here are some pictures, in case you don’t want to look at the 3D models or you can’t make them work.
https://media.mathstodon.xyz/media_attachments/files/115/927/552/906/609/915/original/92ca4e6ff42cacf0.png
https://media.mathstodon.xyz/media_attachments/files/115/927/553/651/380/014/original/10104f531b39ad20.png

2026-01-20 13:00:12 UTC

We hear so much about symmetrical polyhedra. What about their asymmetrical cousins?
Here are the two simplest – I think – convex polyhedra that have no symmetry at all, even combinatorially. In other words, they have no non-trivial automorphisms as graphs.
Both of them have 7 vertices and 7 faces.
https://sketchfab.com/3d-models/asymmetric-heptahedron-1-bc580cf801444bbea24e8f7ac42fb88c
https://sketchfab.com/3d-models/asymmetric-heptahedron-2-0b6db5620b4b486186edbca686dd5bff

2026-01-18 23:02:11 UTC

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@nprofile…u9m4 if they are connected in the following regular fashion, they can be formed into a rhombic dodecahedron in at least one way
https://media.mathstodon.xyz/media_attachments/files/115/918/571/185/646/277/original/0c86901954287e4b.png
https://media.mathstodon.xyz/media_attachments/files/115/918/589/532/101/329/original/038ab0977d5fbd39.png

2026-01-18 11:46:01 UTC

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@nprofile…7md8 I assume it's already solved and this is the intended solution?

2026-01-16 17:15:37 UTC

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@nprofile…u9m4 That's a very interesting idea, thanks. Yes, I have several Shashibo cubes and I've played with them a lot.
I've just tried connecting 24 of these tetrahedra with tape to make a hinged loop that can fold into a rhombic dodecahedron. There appears to be considerable creative freedom in how to do that! I don't yet know if there a particularly regular or symmetrical arrangement that works.
It's more difficult than I expected to fold the hinged loop back up into the rhombic dodecahedron. This has potential as a puzzle.

2026-01-15 10:19:20 UTC

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@nprofile…hmdu @nprofile…udvu in fact the only sense in which an event of probability 0 can possibly happen is if you ‘draw the target around the arrow’, i.e. you choose the event after it has already happened.

2026-01-15 10:15:22 UTC

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@nprofile…hmdu @nprofile…udvu Yes, I think that's the underlying effect. I've talked to several people who have been told “events of probability 0 are not necessarily impossible” and mentally appended “…just very unlikely” which is not at all the right way to think about it. I've seen people worry that a randomised algorithm terminates with probability 1, and think that means there's a possibility it will run for ever.

2026-01-15 10:03:06 UTC

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@nprofile…udvu I understand the sense in which this is true, but without further nuance I think it's a highly misleading claim that has done a lot to damage people's intuition about events that have probability 0.

2026-01-14 22:59:57 UTC

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@nprofile…d4hg Depending on exactly what you need and why, there may be an argument for the P1S over the P2S. Not only is the P1S considerably cheaper for a fairly similar machine, I've heard horror stories of P2Ses being unreliable, whereas the P1S is very solid and reliable. (I have been running six of them continuously for weeks, with no problems at all.)

2026-01-13 22:59:44 UTC

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@nprofile…4cdk if you have 50 then you should be able to make the first stellation of the rhombic dodecahedron, and have two left over.

2026-01-13 21:00:49 UTC

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@nprofile…4cdk Oh, I'm sorry! They're definitely imperfect in various ways, which is why I'm still iterating on the design. I just printed 400 of them (this older version) and they're definitely tough to connect and disconnect, and sometimes require more tools than just a pair of human hands.

2026-01-12 14:42:28 UTC

An inverted tensegrity structure with magnets. I've never seen one of those before!
https://www.grand-illusions.com/products/tensegrity-floating-loop

2026-01-11 11:34:53 UTC

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@nprofile…a9a8 Cool! Can you make its corridor too? https://mathstodon.xyz/@robinhouston/115504027747246443

2026-01-11 09:31:52 UTC

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@nprofile…u9m4 I wasn't sure, since there are so many LaTeX symbol packages and I'm not familiar with all of them, but from this I would guess probably not: https://mathstodon.xyz/@svat/115875283593216261

2026-01-11 09:19:37 UTC

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@nprofile…tfwq @nprofile…g03k It may be this would cause more controversy that you are willing to take on, but you could always post a sci-hub link in those cases where you can't find a legitimate open copy of a paper…

2026-01-10 21:45:26 UTC

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@nprofile…5mqn And you, Francis.

2026-01-10 17:52:33 UTC

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@nprofile…em2h It is staggeringly unfair in so many ways

2026-01-09 19:47:24 UTC

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@nprofile…f38c https://en.wikisource.org/wiki/Captain_Hook_at_Eton

2026-01-09 17:59:19 UTC

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@nprofile…f38c @nprofile…9d4g @nprofile…tfwq Wow! There was a time when I knew a LOT about perl, but I don’t think I ever knew it supported uppercase versions of the string comparison operators.

2026-01-09 17:40:02 UTC

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@nprofile…9d4g @nprofile…tfwq I still harbour an increasingly unrealistic fantasy that not all programming languages will have to use C expression notation for ever. You would think people would be tired of it! But in fact they become angry and upset if anyone ever tries anything different.

2026-01-09 10:09:10 UTC

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@nprofile…tfwq
https://media.mathstodon.xyz/media_attachments/files/115/864/586/065/875/869/original/cda0efb18a7c8cd3.jpeg

2026-01-09 10:07:17 UTC

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@nprofile…tfwq The constraint modelling language MiniZinc allows you to use the correct symbols for certain operators, including ¬ for not, which I take full advantage of.

2026-01-06 23:06:31 UTC

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Thanks again for this observation, @nprofile…u9m4. I’ve been thinking about it a lot.
I find it particularly interesting that this honeycomb is, in that sense, the three-dimensional analogue of the equilateral-triangular tiling of the plane.

2026-01-04 13:22:19 UTC

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@nprofile…f5cp Isn't this just the eternal tension between geometry and algebra? In this case, that affine transformations are a more natural class geometrically, whereas linear transformations are more natural algebraically.

2026-01-04 13:18:47 UTC

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@nprofile…4hp5 Thanks. I do like videos. I've just finished watching the first one, which was great.

2026-01-04 11:45:12 UTC

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@nprofile…4hp5 @nprofile…g0zc I'm ok with a “bring your own motivation” text in this case, because I already know why I want to understand this theory, and I just need someone who already understands it to lay it out carefully and correctly. But I will check out your lectures too!

2026-01-04 07:34:07 UTC

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@nprofile…f5cp Nice, thanks. It reminds me a little of the commutativity of multiplication, specifically multiplication of natural numbers defined as repeated addition, which is not obvious at all.

2026-01-04 01:27:08 UTC

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@nprofile…g0zc It’s a subject I know less about than I feel I ought to, which means I always feel vaguely guilty when it comes up. I am hoping a careful study of the book will correct that.

2026-01-04 01:24:48 UTC

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@nprofile…f5cp I’m struggling to access your intuition that it isn’t obvious! Is there any way you can make it seem less obvious to me?

2026-01-04 01:06:55 UTC

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@nprofile…cz9e I agree it's not helpful writing, but I don't think it's trying to be helpful in that way. It's aimed at a reader who already knows perfectly well what a reflection is, and its purpose is to fix terminology and notation.

2026-01-04 00:43:12 UTC

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@nprofile…s4qa Reflection Groups and Coxeter Groups by James E. Humphreys.

2026-01-03 23:14:14 UTC

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@nprofile…cz9e Thanks for letting me know. Interesting. I’m not seeing any z-fighting there.

2026-01-03 22:13:11 UTC

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I couldn’t find one online, so I sketched a very rough approximation of the Toblehedrone here: https://sketchfab.com/3d-models/enneahedron-06563f7881d34b23b7708e2c3419a135

2026-01-03 15:23:00 UTC

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@nprofile…f38c @nprofile…eljp What a good thing they made sure C didn’t have any obvious footguns! :-)

2026-01-03 15:19:50 UTC

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@nprofile…f38c @nprofile…eljp Thanks, that must indeed be the reason. (Though I suspect it would usually have worked ok in practice, as long as the the comparison operators returned 0 for false and ~0 for true.)

2026-01-03 15:12:30 UTC

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@nprofile…eljp @nprofile…f38c Considering that C was initially extremely minimalist, and did not have a boolean type separate from int, it’s curious that it had a logical negation operator distinct from bitwise negation. I wonder why.

2026-01-03 14:43:36 UTC

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@nprofile…f38c I don’t know which programming language was the first to use ! for negation, but it can’t have been C because C’s immediate predecessor B also used it: https://web.archive.org/web/20150611114427/https://www.bell-labs.com/usr/dmr/www/kbman.pdf

2026-01-03 05:49:10 UTC

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@nprofile…a9a8 This one seems to work well; or at least if there is a false solution, it isn't significantly easier to discover than the true one. I'm afraid I didn't take a note of how long it took me to solve, but I do remember that I failed at my first attempt and returned to it the following day, so it wasn't trivial.

2026-01-03 05:44:27 UTC

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@nprofile…a9a8 I messed about a few years ago with trying to design puzzles based on geometric packings, mainly of circles into rectangles (which I implemented as bearing balls in a wooden tray). I ran into the problem that there very often exist alternative packings that are only micro-fractionally worse: for a good puzzle one of course needs a decent gap between the intended solution and the next best packing.
(I was rather encouraged to see that Stewart Coffin's Five Fit puzzle has the same flaw.)

2025-12-31 00:22:55 UTC

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I've just spotted a peak @nprofile…6j80 video comment: ‘I am mid way through working out how to turn a stellated cuboctahedron on a lathe…’

2025-12-31 00:05:08 UTC

∀x . what does the x say?
https://en.wikipedia.org/wiki/List_of_animal_sounds

2025-12-30 23:06:24 UTC

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@nprofile…cyh9 (https://girlymicrobiologist.com/wp-content/uploads/2025/05/a-trip-to-the-intensive-care-unit-an-intravenous-injection-of-psilocybin.pdf)

2025-12-30 23:04:07 UTC

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@nprofile…cyh9 omg this is going to take over from the one about the chap who injected himself with mushroom tea and the mushrooms started growing in his bloodstream

2025-12-30 15:53:49 UTC

A lovely new video from @nprofile…6j80: https://youtu.be/NEJZlGuWGV8
Stunning photography, as well as a fascinating mechanism.