Mathematician working mostly in three-dimensional geometry and topology, and mathematical maker/artist working mostly in 3D printing and virtual reality.
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2026-08-25T12:08:49Z Event JSON
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Last Notes npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @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! npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @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.) npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman 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. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…vcny Steve updated the link to a more permanent one: https://shaders.stevejtrettel.site/cirm-2026/lectures/day3/notes npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @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. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @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. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @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? npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…vcny Not the first of course, but these are some wonderful notes by Steve Trettel: https://shaders.stevejtrettel.site/lectures/day3/notes npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman Minesweeper roguelike I’ve been having a lot of fun with out in full release now: https://store.steampowered.com/app/3473250/BroomSweeper/ npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…vcny How many colours of tiles would you need so that you get a valid map colouring at all times? npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @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. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @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. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman New video out just now, on a mechanism that uses racks and gears to expand and contract, this time in all three dimensions! https://youtu.be/NEJZlGuWGV8 npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…u6hy Does the position of your head also change throughout the day? Seems like it could be relevant if so. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman Stillwater has a very small airport - only one gate. They fly twice a day to Dallas Fort Worth. This is the first time I’ve ever seen three passenger planes here at the same time! (Exciting, I know.) npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman The OSU Museum of Art is in a beautiful old building that used to be a post office. They have a small but heavily-fortified room that used to be a safe. Not such a great location to view paintings, but will work nicely for light-based sculpture, like "(5,3,2) triangle tiling", by @nprofile…3pws and me. Museum preparator Michael King was testing the mounting system today and invited me along to check it out! npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…w9qq @nprofile…pmvv No practical application as far as I’m aware of, but that’s an interesting question! npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @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. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @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!). npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…saev missed an opportunity to do a rickroll there npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman 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 npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @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. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman 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. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @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. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman 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) npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman I finally got around to setting up a Patreon! It's at https://patreon.com/HenrySegerman. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…saev @nprofile…wxl7 That’s a Nobbly Wobbly, designed by Dick Esterle. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @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. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @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! npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman 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. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…fqqu @nprofile…saev Can you derive the paths that Oskar needs to carve? npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…fqqu Said mathematician was me. I brought in linkage expert Daniel Huczala, who said: npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…hr6z You could go back before the current colonizers, at least past the Norman conquest, but ideally as far back as we know. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…wxl7 A book with that title should have been this one: https://abakcus.com/article/pasta-by-design-the-beautiful-geometry-of-pasta/ npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @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. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman Working on a few three-dimensional expanding rack mechanisms. This one is based on the NbO lattice. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman .@nprofile…pmvv Now playing around with animating the process of generating these approximations to Cannon-Thurston maps. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman .@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! npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman 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 npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman New video on some expanding 3D printed designs based on rack and pinion mechanisms: npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman Saw this unusual logo on the side of a truck in the suburbs of Melbourne, Australia. Strange mix of 2D and 3D. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @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. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @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. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman 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/ npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…y75c Was that me? Could have been @nprofile…pmvv? npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…fa4k Perhaps the “broad interest” question only makes sense to answer for mathematicians who have very broad interests, such as yourself! npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman 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". npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @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: npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…xqed Oh no! Sorry you couldn’t make it - but maybe you can get to the museum on the weekend? Also, not as good as being there in person, but they recorded the talk so that’ll be up sometime soon. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…0lhr They’re probably not important. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman 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 npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…47vr No worries! Did she end up solving it? npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…ehd3 Looks cool, but no way to submit a paper in LaTeX? npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman The next big Illustrating Mathematics event will be a trimester program at Institut Henri Poincaré in Paris from January 5 to April 3, 2026. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…47vr https://divisbyzero.com/2010/07/26/algebra-the-faustian-bargain/ npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…q8c7 Too lazy to find out whose law that really is. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…elud @nprofile…9q2p I might make it even more wordy with “The rules of every logically consistent universe.” npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman Prototype for a bigger version of Geared cube net I've been working on with @nprofile…9mcm. The side length is 8cm, almost double that of the previous version (https://youtu.be/T3CkqXycT9w). npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…yu7x Hiding the seam around the back! npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…hr6z If it’s already frozen and taking up space in your freezer, then it would be replaced by air that you’d be keeping cold instead? If so, it might be saving you electricity, if the cold air escapes each time you open the door and then must be cooled again. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…yu7x I’m not sure exactly how he made his, but I imagine that with Rhino/Grasshopper he was using numerical methods rather than directly parametrising the surface. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @nprofile…yu7x David Bachman also had some fun making (an approximation to) a flat Möbius strip https://davidbachman.org/mathartblog/?p=483 npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman https://media.mathstodon.xyz/media_attachments/files/113/397/153/969/537/392/original/a632a281eded34ac.png npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman I finally got a couple of my 3D printed designs back up and available again at i.materialise.com: npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @npub1yxx…p7am Would be interesting to see this applied to ambigrams. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @npub1w6g…5ddw It is a reduced alternating diagram, so the number of crossings is minimal. There is only one such knot with three crossings: so yes it is the trefoil. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman Spherical conference photos from the Bridges 2024 #Mathart conference at Virginia Commonwealth University in Richmond, Virginia. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman Surprised to find out that our Cannon-Thurston carving won a prize in the Mathematical Art Exhibition at the Joint Mathematics Meetings! With @npub16gf…scgm and Will Segerman. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman A "recursive" mechanism, where the position of each part is determined by the positions of the previous two parts in the chain. Full video at npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @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. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @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? npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @npub1n4h…74jl @npub1am3…njaj We did a 120-sided one! https://youtu.be/516U4whg4GU npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @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 npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @npub1svd…5cuv Not much time to look at other booths, but this was very cool, from https://www.planarmotor.com npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @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. npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman @npub1g86…6pkm The classic 22/7 approximation! npub1stp56taz02munq234tzhdu4v7q2cwu4cyx3u7madwz6df46n2w5sx22d3q Henry Segerman Booth all set up at http://opensauce.live with @npub1svd…5cuv
