Henry Segerman

Mathematician working mostly in three-dimensional geometry and topology, and mathematical maker/artist working mostly in 3D printing and virtual reality.
#Math #Maths #Art #3DPrinting #Geometry

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npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q

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henryseg@mathstodon-xyz.mostr.pub

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nprofile1qqsg9s6d97384d7fs9g643tk72k0q9v8w2uzrg70d7khpdx56af482gpz3mhxue69uhhyetvv9ujuerpd46hxtnfduqs6amnwvaz7tmwdaejumr0ds2vgh88

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relay.damus.io

nos.lol

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npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q

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Last Notes

2026-04-20 05:39:22 UTC

Here's a second, geared mechanism that transforms between cube and octahedron forms. Full video at https://youtu.be/T9BNMLFHXUw
https://media.mathstodon.xyz/media_attachments/files/116/435/127/285/762/593/original/dda39de7a3c31415.mp4

2026-04-18 16:11:34 UTC

New video, on mechanisms that illustrate the duality between the cube and the octahedron! Here's one of them: a linkage with fabric panels designed by @nprofile…e3ve.
Full video at https://youtu.be/T9BNMLFHXUw
https://media.mathstodon.xyz/media_attachments/files/116/426/549/956/747/975/original/cff67197f29dc424.mp4

2026-01-24 23:12:34 UTC

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@nprofile…7jv6 Ah that’s a problem: a sufficiently tall and narrow triangular prism, if flipped to the dual double tetrahedron, would result in a polyhedron that violates the triangle inequality!

2026-01-24 22:42:32 UTC

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@nprofile…7jv6 Good question! I haven’t thought through the details when there is less symmetry. I imagine that we could allow different edge lengths across a polyhedron, but require that the corresponding edges in the dual polyhedra have the same length. (For example a tall and narrow triangular prism could turn into a short and wide double tetrahedron.)

2026-01-24 21:41:23 UTC

Earlier this week I was challenged to come up with a mechanism that would illustrate the duality of the cube and the octahedron. I got as far as making this animation before discovering that I was scooped three months ago! The paper (https://www.sciencedirect.com/science/article/pii/S0094114X25002769) by Liao, Kiper, and Krishnan has lots of other interesting mechanisms, including one showing duality between the cuboctahedron and the rhombic dodecahedron.
https://media.mathstodon.xyz/media_attachments/files/115/952/226/158/743/301/original/bb18eb63e731218c.mp4

2026-01-24 13:23:31 UTC

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@nprofile…vcny Steve updated the link to a more permanent one: https://shaders.stevejtrettel.site/cirm-2026/lectures/day3/notes

2026-01-24 09:41:54 UTC

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@nprofile…pm95 @nprofile…cz9e Ah you’re right, it isn’t the same as the long line. In the example I gave, there is something weird with the topology around (x,0) for any x: that point doesn’t have a neighbourhood that looks like the neighbourhood of a point on the usual line.

2026-01-23 07:10:43 UTC

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@nprofile…cz9e @nprofile…pm95 if you’re willing to mess with the (point set) topology then you can get a reasonably nice picture. Given a set with a total order, say that the open sets are arbitrary unions of intervals (a,b) = {x | a<x<b}. This defines a topology on the set. Now take the set to be R x [0,1), with the dictionary ordering.

2026-01-22 06:54:52 UTC

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@nprofile…cz9e is it long? The long line is an uncountable number of intervals concatenated together - the picture looks like it could be made with only a countable number of them?

2026-01-21 17:31:48 UTC

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@nprofile…vcny Not the first of course, but these are some wonderful notes by Steve Trettel: https://shaders.stevejtrettel.site/lectures/day3/notes

2026-01-20 08:59:32 UTC

Minesweeper roguelike I’ve been having a lot of fun with out in full release now: https://store.steampowered.com/app/3473250/BroomSweeper/

2026-01-18 09:40:32 UTC

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@nprofile…vcny How many colours of tiles would you need so that you get a valid map colouring at all times?

2026-01-15 05:12:26 UTC

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@nprofile…d4hg @nprofile…vhdw For random doodads, an A1 Mini will work. I have one and an X1C, and the latter is maybe slightly more reliable, but there’s not much difference I’ve noticed. The AMS is very nice though.

2026-01-15 05:12:25 UTC

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2025-11-26 12:42:13 UTC

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@nprofile…u6hy Does the position of your head also change throughout the day? Seems like it could be relevant if so.

2025-09-15 11:49:31 UTC

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@nprofile…w9qq @nprofile…pmvv No practical application as far as I’m aware of, but that’s an interesting question!
Rather than ordering points in a cubic lattice of some dimension (and wanting nearby points to often appear nearby in the sequence), it orders elements of an algebraic number field (associated to some hyperbolic three-manifold). If someone would want to go through those points in a nice sequence then it could be useful.

2025-09-05 19:12:15 UTC

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@nprofile…saev Lets hope that we don’t end up committing atrocities which in the far future would be forgotten dusty history except that people keep being reminded of them because of how often they use Segerman Grabbers.

2025-09-05 19:08:13 UTC

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@nprofile…saev My brother and I came up with it, so you’ve named it after both of us (luckily we have the same last name!).

2025-08-17 01:07:12 UTC

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@nprofile…saev missed an opportunity to do a rickroll there

2025-08-10 14:44:34 UTC

I added a second post on my new Patreon, about my plans for a three-dimensional expanding racks design based on the srs net (also known as the Laves graph). See https://www.patreon.com/posts/135832481
https://media.mathstodon.xyz/media_attachments/files/115/004/979/884/794/547/original/6f55247a10da6757.mp4

2025-08-08 15:37:18 UTC

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@nprofile…4mn6 @nprofile…pmvv This won't make a whole lot of sense without a lot more background information, but: The circular parts correspond to the "cusps" of the universal cover of the triangulation, corresponding to places where the boundary of the manifold (a torus) is. Actually the cusps are dense, and we are just seeing the ones that are closest to us. You can see later in the animation that there are smaller circles that show up - those are further away from our viewpoint.

2025-08-08 15:22:37 UTC

I never got around to posting this animation I made with @nprofile…pmvv. This animates successively better approximations to a space-filling curve. (Or rather, you can tile the plane with the shape, and that gives the approximation to the space-filling curve.) The space-filling curve is the Cannon-Thurston map for a veering triangulation (with veering isosig gLLAQbecdfffhhnkqnc_120012). We prove that veering triangulations have Cannon-Thurston maps in an upcoming paper with Jason Manning.
https://media.mathstodon.xyz/media_attachments/files/114/993/794/188/683/673/original/11de1e963a5fe0dc.mp4

2025-08-04 01:29:13 UTC

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@nprofile…yxkq Gwen Fisher’s partner Paul has a rotary axis on his milling machine - he’s planning to have a go at it in aluminium! One thing that might be tricky is that he can only support a rod at one end - we’ll have to see if vibrations become a problem.

2025-08-02 15:44:08 UTC

Revisiting the geometry of the tetrahedral racks mechanism. A hexagonal cross section looks like it would be easier to make with subtractive techniques. (Original version at https://www.youtube.com/watch?v=sDJk-9QxFro)
https://media.mathstodon.xyz/media_attachments/files/114/959/924/208/418/835/original/f5920cfdb9c4c1c8.png

2025-07-29 16:27:27 UTC

I finally got around to setting up a Patreon! It's at https://patreon.com/HenrySegerman.
At the moment I've got a post up there talking about my plans for taking the expanding racks project (https://youtu.be/iWknov3Xpts) from two-dimensional to three-dimensional networks. In the future I'll plan to post early versions of videos or other updates of things I'm working on.
If you’d like to support my work, please consider joining. Thanks!
https://media.mathstodon.xyz/media_attachments/files/114/937/453/079/367/935/original/e2827537097fc014.png

2025-07-28 19:13:30 UTC

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@nprofile…sj8x @nprofile…dcr8 Here’s a quick slo mo, showing not unusual behaviour (Andrea Hawksley is doing the rolling). The d12 seems to get into a good rolling state, where it is rotating around a five fold axis and not losing much energy. The d20 doesn’t.
https://media.mathstodon.xyz/media_attachments/files/114/932/436/335/991/052/original/48f32ee34702ab2b.mp4

2025-07-18 20:38:55 UTC

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@nprofile…9hsm It's like how the math sometimes forces you into weird bits of design space you wouldn't think to visit otherwise. Also symmetry makes things look way more complicated than they really are!

2025-07-18 20:18:34 UTC

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Turns out that design didn't extend indefinitely - I was trying to shift things around to pack longer racks in but this causes inconsistencies in longer loops. So, shorter racks, but this should still expand by a factor of two or so.
https://media.mathstodon.xyz/media_attachments/files/114/876/079/990/930/523/original/25bee22ea3909e63.png

2025-07-16 21:42:11 UTC

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@nprofile…fqqu @nprofile…saev Can you derive the paths that Oskar needs to carve?

2025-07-14 22:22:04 UTC

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@nprofile…fqqu Said mathematician was me. I brought in linkage expert Daniel Huczala, who said:
“The linkage [in the astrolabicus puzzle] is Bricard’s orthogonal 6R. You can find its kinematic analysis, that should help you answer your questions, in these papers:
https://doi.org/10.1016/j.mechmachtheory.2017.09.017
https://doi.org/10.1016/j.ijsolstr.2004.09.014”
A precise analysis of the motion that would be needed to get the correct curves seems to be quite technically difficult.

2025-07-13 07:07:31 UTC

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@nprofile…wxl7 A book with that title should have been this one: https://abakcus.com/article/pasta-by-design-the-beautiful-geometry-of-pasta/

2025-06-29 05:46:39 UTC

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@nprofile…r4kn This is just the structure of gears and racks, without any of the axles, rails, and other structures to hold things in place.

2025-06-28 23:53:17 UTC

Working on a few three-dimensional expanding rack mechanisms. This one is based on the NbO lattice.
https://media.mathstodon.xyz/media_attachments/files/114/763/669/345/184/920/original/55c375b23409dc2a.jpg

2025-06-15 20:56:07 UTC

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.@nprofile…pmvv Now playing around with animating the process of generating these approximations to Cannon-Thurston maps.
https://media.mathstodon.xyz/media_attachments/files/114/689/372/953/151/832/original/cb40740a64f98f83.mp4

2025-06-12 17:43:38 UTC

.@nprofile…pmvv and I got our code for generating Cannon-Thurston maps working with @nprofile…hx82 algebraic numbers. The old code used floating point numbers which would eventually lose precision and blow up. With algebraic numbers we can go as far as we want!
https://media.mathstodon.xyz/media_attachments/files/114/671/612/526/008/234/original/3facf16803f7b170.png

2025-06-11 09:19:52 UTC

Here’s another clip from my video on expanding rack and pinion mechanisms - an expanding pythagorean triple triangle. There’s a reason why it’s easier to make this kind of triangle! Full video at https://youtu.be/iWknov3Xpts
https://media.mathstodon.xyz/media_attachments/files/114/663/954/516/993/958/original/2672f5ce7dcb9d48.mp4

2025-06-09 12:49:04 UTC

New video on some expanding 3D printed designs based on rack and pinion mechanisms:
https://youtu.be/iWknov3Xpts
https://media.mathstodon.xyz/media_attachments/files/114/653/471/051/870/527/original/eaa43e0cea825ad5.mp4

2025-06-01 22:50:35 UTC

Saw this unusual logo on the side of a truck in the suburbs of Melbourne, Australia. Strange mix of 2D and 3D.
https://media.mathstodon.xyz/media_attachments/files/114/610/544/411/465/489/original/10850a5c915c26f6.png

2025-05-22 20:43:22 UTC

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@nprofile…saev @nprofile…uetd @nprofile…qqx9 Very cool stuff, and it's the first method (with a physical justification) I know of for making dice with differently shaped faces and probabilities.
However, they don't pay attention to momentum, friction, or bouncing, and it's still true that changes in the physical setup like friction etc. can change the probability distribution when you don't have symmetry.
They don't claim that they can perfectly simulate 3d6 with a dragon - but now the question is how much those other physical parameters matter. (If you want to roll dragons then maybe it doesn't matter too much to you how accurate the probabilities are?)

2025-05-13 13:53:15 UTC

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@nprofile…phcs Hilbert curve, gyroid, platonic solids, Klein quartic, 27 lines on a cubic, Klein bottle (or Möbius strip), figure eight knot (or trefoil). There are lots of less visual things that deserve to be featured - a Turing machine, a field, a group, a set, etc.

2025-04-22 18:03:39 UTC

Some pretty pictures of mathematical things in this Scientific American article - I contributed a discussion of the complement of the figure eight knot, and the image from that appears on the cover of the May 2025 issue! https://www.scientificamerican.com/article/these-mysterious-shapes-are-at-the-heart-of-maths-biggest-puzzles/

2025-04-12 12:41:25 UTC

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@nprofile…y75c Was that me? Could have been @nprofile…pmvv?

2025-03-19 06:32:48 UTC

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@nprofile…fa4k Perhaps the “broad interest” question only makes sense to answer for mathematicians who have very broad interests, such as yourself!
I am much more of a frog (in Freeman Dyson’s terminology https://www.ams.org/notices/200902/rtx090200212p.pdf).

2025-03-18 17:27:33 UTC

I was recently asked to provide a quick opinion on a paper for a journal. One of the criteria the journal weighed heavily was whether or not the paper was "of broad interest to mathematicians in diverse fields".
Mathematics is so specialized at this point, that in my opinion, essentially zero papers meet this criterion. In recent years, I might list Yitang Zhang's work, or the discovery of aperiodic monotiles. Perhaps I am overly specialized myself, but I look at the titles of recent papers in non-field-specific mathematics journals, and I find no papers that look relevant to my work. Which is fine! Publishing papers that are big advances in their respective fields is great. But I would almost certainly be lying if I claimed that (in my opinion) any given modern paper in mathematics is of broad interest.
Is this a me problem?

2025-03-16 05:44:44 UTC

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@nprofile…kcjk Very cool project! I had to figure out a similar problem to get bits of Oklahoma for my students to work with to make this:
https://media.mathstodon.xyz/media_attachments/files/114/170/401/851/989/328/original/49e5bad1fa7013ba.png

2025-03-12 23:12:42 UTC

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@nprofile…0lhr They’re probably not important.

2025-03-12 20:15:32 UTC

If you’re in Sydney, Australia tomorrow, I’ll be giving a public lecture for Pi Day, details at https://events.humanitix.com/artistic-mathematics-truth-and-beauty-by-henry-segerman-international-day-of-mathematics-public-lecture
@nprofile…9mcm and I will also be doing some events at the Chau Chak Wing Museum over the weekend, details at https://www.sydney.edu.au/museum/whats-on/talks-and-events/ccwm-x-smri-maths-at-the-museum.html

2025-03-11 20:57:54 UTC

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@nprofile…47vr No worries! Did she end up solving it?

2025-03-10 13:58:49 UTC

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@nprofile…ehd3 Looks cool, but no way to submit a paper in LaTeX?

2025-03-07 21:15:14 UTC

The next big Illustrating Mathematics event will be a trimester program at Institut Henri Poincaré in Paris from January 5 to April 3, 2026.
If you are interested in learning or using mathematical illustration, particularly in research, check out the program webpage at https://indico.math.cnrs.fr/event/13123/.
There will also be three week-long conferences throughout the trimester if you can only visit for a shorter time.
https://media.mathstodon.xyz/media_attachments/files/114/123/212/277/369/571/original/5568e1eea414f0d4.jpg

2025-03-02 18:05:33 UTC

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@nprofile…47vr https://divisbyzero.com/2010/07/26/algebra-the-faustian-bargain/

2025-02-20 14:31:03 UTC

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@nprofile…q8c7 Too lazy to find out whose law that really is.

2023-07-18 14:07:27 UTC

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@npub1am3…njaj @npub1n4h…74jl Yes, and with 60 sides, the numbers have to be quite small… By the way, that d7 doesn’t look like it’s isohedral (https://en.m.wikipedia.org/wiki/Isohedral_figure), so I doubt that there’s a justification for it being fair in principle.

2023-07-18 04:20:33 UTC

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@npub1n4h…74jl @npub1am3…njaj The problem with them is that they don’t have parallel faces on opposite sides. So when the die is lying on a face, it isn’t clear which face is on top. So how do we write the numbers?

2023-07-18 01:22:47 UTC

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@npub1n4h…74jl @npub1am3…njaj We did a 120-sided one! https://youtu.be/516U4whg4GU
To add to the Catalan dice list, we also have a disdyakis dodecahedron d48 and our "dLX" dice are 60-sided alphabet dice are pentakis dodecahedra.

2023-07-18 01:12:16 UTC

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@npub1am3…njaj The d30 is a countdown (also called spindown) die which we don't make, and the numbering on the rhombic d12 is different from ours. But the d24 and d60 look like they were made by us (or were made by someone who copied our numbering): http://thedicelab.com

2023-07-16 14:05:07 UTC

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@npub1svd…5cuv Yesterday we were on our feet talking to people pretty much the whole day. Exhausting but great meeting so many people! Looking forward to the same thing today.

2023-07-16 13:18:55 UTC

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@npub1g86…6pkm The classic 22/7 approximation!

2023-07-15 03:40:47 UTC

Booth all set up at http://opensauce.live with @npub1svd…5cuv
https://media.mathstodon.xyz/media_attachments/files/110/716/019/705/764/495/original/a556d2290527078b.jpeg