Can a convex polyhedron have an odd number of faces, all congruent? https://mathoverflow.net/q/406120/440

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 (https://en.wikipedia.org/wiki/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: http://hyperbolic-crochet.blogspot.com/2010/07/story-about-origins-of-model-of.html, 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 (https://en.wikipedia.org/wiki/Pseudosphere)? The rolled-up photo of Beltrami's model suggests that he did that.

Via http://www.open.ac.uk/blogs/is/?p=731 which shows this as a tangent to a story about triangulated polygons, frieze patterns, and the Farey tessellation.

Twelve threads: https://www.youtube.com/watch?v=-p7C5FrgAzU

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.

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!

Full video: https://youtu.be/wzjUTPCAF4Y

https://www.ams.org/journals/notices/202107/rnoti-p1106.pdf

https://youtu.be/Jqc3nS9Ma3o https://www.mathartfun.com/DiceLabDice.html

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.

Show thread

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 https://youtu.be/HeQX2HjkcNo

New post: The constructive solid geometry of piecewise-linear functions https://11011110.github.io/blog/2021/05/12/constructive-solid-geometry.html

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

Video: youtu.be/LG5HUd0hzzo

Mathematician working mostly in three-dimensional geometry and topology, and mathematical artist working mostly in 3D printing and virtual reality.

Joined Apr 2017