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

Website: http://segerman.org
YouTube: https://www.youtube.com/@henryseg

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npub10zcfmh333hf5phqzjwn2avyueg2aly69rauj2c5utdg5y6fggv9svdtds5

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henryseg_at_mathstodon.xyz@momostr.pink

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nprofile1qqs83vyamcccm56qmspf8f4wkzwv59wljdz377f9v2w9k52zdy5yxzcpz3mhxue69uhhyetvv9ujuerpd46hxtnfduqs6amnwvaz7tmwdaejumr0dspn0exg

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Last Notes

2026-05-06 17:01:20 UTC

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Here are some photos of a 3D print @npub1fjz…xz60 and I made, of a toroidal half of the 120-cell. On the torus boundary, you can see the 10x10 grid of vertices that correspond to @npub1mcu…g2al 's 10x10 grid of tetrahedra. I also found a screenshot with solid dodecahedra poking through the torus boundary.
At the core of the solid torus, there is a ring of ten dodecahedra. And there is a dual ring of dodecahedra in the other toroidal half. The two rings face each other, equidistant from the 10x10 grid of vertices.
That core ring of dodecahedra is surrounded by five other rings of dodecahedra, that together make up the toroidal half of the 120-cell. These twist around in a discrete version of the Hopf fibration. (The figure here is from our paper at https://www.ams.org/journals/notices/201511/rnoti-p1309.pdf) Each of the five other rings has a corresponding dual ring in the other half, and a corresponding equidistant 10x10 grid of vertices. Since the pairs of opposite rings of dodecahedra partition the 120-cell, and there is a fixed-point free action moving one pair of rings to another, I think that should be enough to imply that the corresponding 10x10 grids are disjoint.
So, I don't think we ever thought about these sheets of tetrahedra before, but they're not so far away from these patterns of rings in the 120-cell.
https://media.mathstodon.xyz/media_attachments/files/116/528/386/164/665/878/original/da95966a365bddb1.jpg
https://media.mathstodon.xyz/media_attachments/files/116/528/388/969/439/239/original/392f5127d0afa4b3.jpg
https://media.mathstodon.xyz/media_attachments/files/116/528/406/523/085/532/original/30c6cfd7c45e2359.png
https://media.mathstodon.xyz/media_attachments/files/116/528/436/319/893/297/original/518168e4cb4adbb0.png

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 @npub1a8u…8py2.
Full video at https://youtu.be/T9BNMLFHXUw
https://media.mathstodon.xyz/media_attachments/files/116/426/549/956/747/975/original/cff67197f29dc424.mp4

2026-02-06 22:07:00 UTC

“Homunculus”
https://media.mathstodon.xyz/media_attachments/files/116/025/951/500/570/087/original/1f401b34e3dc069b.png

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

2025-12-30 16:10:54 UTC

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
https://media.mathstodon.xyz/media_attachments/files/115/809/354/166/802/018/original/2684a5103f843634.mp4

2025-09-14 20:10:36 UTC

This image (made with @npub1fjz…xz60) shows two different ways to approximate the Cannon-Thurston map associated to the figure-eight knot. Cannon-Thurston maps are space-filling curves, a bit like the Hilbert curve except that these fill a sphere rather than a square.
The black curve is our recreation of Thurston's original algorithm to approximate Cannon-Thurston maps. The boundary between the green and purple regions is an approximation generated by our new algorithm based on veering triangulations.
In this image, our algorithm fills the sphere much more completely than Thurston's does, despite using ten times fewer edges.
https://media.mathstodon.xyz/media_attachments/files/115/204/259/664/394/505/original/e0f318fb588ea70e.png

2025-06-15 20:56:07 UTC

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.@npub1fjz…xz60 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

.@npub1fjz…xz60 and I got our code for generating Cannon-Thurston maps working with @npub1438…0hcg 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

2025-03-19 06:32:48 UTC

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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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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

2025-01-29 15:46:02 UTC

A bigger version of our Geared cube net, for an art exhibition "Intersections: Math, Art, Truth, Humanity", organized by @npub1c8s…lyu0 and Glen Whitney in Seattle. Joint work with @npub1a8u…8py2. (The smaller original version: https://youtu.be/T3CkqXycT9w)
https://media.mathstodon.xyz/media_attachments/files/113/912/360/159/977/380/original/91d09ab711dbe058.jpg
https://media.mathstodon.xyz/media_attachments/files/113/912/360/208/497/698/original/699b8fb94da2194b.jpg
https://media.mathstodon.xyz/media_attachments/files/113/912/360/161/308/305/original/64bd888cfb382903.jpg
https://media.mathstodon.xyz/media_attachments/files/113/912/360/168/740/708/original/32dce2a4c78f51f1.jpg

2025-01-11 18:12:43 UTC

Prototype for a bigger version of Geared cube net I've been working on with @npub1a8u…8py2. The side length is 8cm, almost double that of the previous version (https://youtu.be/T3CkqXycT9w).
https://media.mathstodon.xyz/media_attachments/files/113/811/053/219/189/386/original/bcc2429744a80ede.jpg
https://media.mathstodon.xyz/media_attachments/files/113/811/053/884/503/310/original/6d92221292f6e467.jpg
https://media.mathstodon.xyz/media_attachments/files/113/811/054/448/920/369/original/7f946988f7a252fe.jpg

2025-01-10 04:46:13 UTC

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Hiding the seam around the back!

2025-01-07 23:31:07 UTC

Twisting and untwisting a pair of bevel gears.
The cross-section shape of a bevel gear lives on the two-dimensional surface of a sphere. But real-world gears need some thickness, so we sweep through spheres with a range of radii.
To twist a bevel gear, we twist the cross-section as we change the cross-section. This doesn't make the gear work any better in theory, but in practice real-world gears may transfer torque better or worse at different phases of the rotation. By twisting the gear, we get all phases of the rotation in action at once, just on different radius spheres.
From my video on a geared icosahedron net at https://youtu.be/EP8KJvk_p_g
https://media.mathstodon.xyz/media_attachments/files/113/789/642/106/313/098/original/243db559bb580b20.mp4

2024-12-23 15:48:04 UTC

Geared icosahedron net: Adding gears to the net of an icosahedron turns it into a mechanism with one degree of freedom. Or it would if the real world would cooperate already! The full video explores a deep engineering rabbit hole of trying to get this working better:
https://youtu.be/EP8KJvk_p_g
https://media.mathstodon.xyz/media_attachments/files/113/702/887/321/419/610/original/e27dc608a43d5048.mp4

2024-12-17 01:52:30 UTC

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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.

2024-12-16 22:29:33 UTC

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David Bachman also had some fun making (an approximation to) a flat Möbius strip https://davidbachman.org/mathartblog/?p=483

2024-10-25 03:05:12 UTC

Mathemalchemy (mathemalchemy.org) is a room-sized multimedia art installation that celebrates the creativity and beauty of mathematics.
Next Spring, Mathemalchemy is heading to the Museum of the Navajo Nation; we are raising funds to help make this happen, and to provide stipends to Navajo students and teachers who will act as docents for the 6-month duration of its visit there.
You can help by contributing at https://fundrazr.com/mathemalchemy (and getting a beautiful and/or whimsical object corresponding to your contribution).
https://media.mathstodon.xyz/media_attachments/files/113/365/847/193/011/274/original/81d5e0c6a21d2093.jpg