Can a convex polyhedron have an odd number of faces, all congruent?

If so the faces would have to all be kites, per comments at the link. Which raises the question: can a convex polyhedron with congruent kite faces avoid being either a trapezohedron ( or formed from deltahedron (a polyhedron with equilateral triangle faces) by subdividing each triangle into three kites? Both automatically have evenly many faces.

The first physical models of the hyperbolic plane, made in 1868 by Beltrami:, blog post by Daina Taimiņa from 2010.

Maybe you could make something like this by wrapping and stretching a disk of wet paper in a roll around a pseudosphere ( The rolled-up photo of Beltrami's model suggests that he did that.

Via which shows this as a tangent to a story about triangulated polygons, frieze patterns, and the Farey tessellation.

Twelve threads:
Vi Hart's latest video mixes up discussions of the nature of social media, the philosophy of mathematical creativity, an exploration of symmetry, and an investigation of the spot patterns of 8-sided dice (which turn out not to all be the same) and how to visualize them.

I'm writing code with Saul Schleimer to (re)implement the 2-3 Pachner move on triangulations of three-manifolds. I had to break out my old ASCII art skills for the comments.

If you go around a face of a spherical dodecahedron, the effect of holonomy rotates you by a sixth of a turn. But which way do you get rotated?

Even after designing and making this holonomy maze, it's not at all intuitive to me!

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If you find yourself in terminal 2 of SFO airport between now and May 2022, you can see two of my sculptures at an exhibit co-curated by Cliff Stoll!

New video: documenting the build of the Mathemalchemy project at Duke University. The installation will tour the US and internationally, starting at the National Academy of Sciences, January 2022.,

A paper almost a decade in the making, now in the Notices of the American Mathematical Society. In 2012, Marco Paoletti asked me a question about rolling acrobatic apparatus. Answering this led to new designs, one of which was fabricated by Lee Brasuell and is now being performed with.

This is a kind of cover of the dodecahedron: locally the graph is exactly the same as the dodecahedron - here the six-colouring of the edges of this dodecahedron lifts to the lattice in RP^3.

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A sculpture I built with Sabetta Matsumoto and Chaim Goodman-Strauss. This is a lattice in RP^3 made from two set of vertices from the 600-cell. Each edge connects a vertex from one 600-cell to a vertex from the other.

Sliding tiles for an upcoming puzzle. This time I (desktop) 3D printed numerals to glue into indentations on the (Shapeways) 3D printed tiles. Thanks to my brother Will Segerman for the idea to do it this way. Much neater than hand painting!

Derek Muller's latest Veritasium video is simply amazing! In half an hour he accurately explains Cantor, Gödel, and Turing, with beautiful animations. All the more amazing given that his original background is physics

Dodecahedral holonomy maze just in from Shapeways, looks like it works! This one should be much harder to solve than the octahedral versions.

New post: The constructive solid geometry of piecewise-linear functions

Or, how the paper that I thought I was writing fell apart and I rescued it by proving a different but related result.

Working on a dodecahedral holonomy maze design. This will be much harder, with 120 nodes in the maze (up from 24 in the original octahedral version).

If you go around a loop on a curved surface, you come back rotated - this effect is called “holonomy”. The green piece cannot rotate in place; you must use holonomy to dodge its arms around the pegs blocking its way and escape the maze!
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Just in time for , a new 3D printed puzzle. Use the 4th (dimension) to solve it!

Mathemalchemy: a collaboration between 24 mathematicians and mathematical artists, to create an installation celebrating the beauty and creativity of mathematics. Video: More details at

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