Small Cubicuboctahedron
This is the small cubicuboctahedron, as drawn by Robert Webb’s Great Stella software. It looks simple enough, but it conceals some interesting mathematics.
Schmidt Arrangement
This picture drawn by Katherine Stange shows what happens when we apply fractional linear transformations $ z \mapsto \frac{a z + b}{c z + d} $ to the real line sitting in the complex plane, where $a,b,c,d$ are Eisenstein integers that is, complex numbers of the form $ m + n \exp(2 \pi i/3) $ with $m,n$ being integers.
Pentagon-Decagon Branched Covering
Two regular pentagons and a regular decagon fit snugly at a point: their interior angles sum to 360°. Despite this, you cannot tile the plane with regular pentagons and decagons. However, there is a branched covering of the plane tiled with pentagons and decagons, which map to regular pentagons and decagons on the plane. Here Greg Egan has drawn a portion of this branched covering.
Pentagon-Decagon Packing
Two regular pentagons and a regular decagon meet snugly at a vertex: their interior angles sum to 360°. However, they can’t tile the plane. However, they come fairly close, as shown in this picture by Greg Egan.
Hammersley Sofa
The moving sofa problem asks: what is the shape of largest area that can be maneuvered through an L-shaped hallway of width 1? This animated image made by Claudio Rocchini shows one attempt to solve this problem.
Icosidodecahedron from D6
The icosidodecahedron can be built by truncating either an icosahedron or a dodecahedron. It has 30 vertices. It is a beautiful, highly symmetrical shape. But it’s just a shadow of an even more symmetrical shape with twice as many vertices in twice as many dimensions!
Packing Regular Pentagons
This picture by Toby Hudson shows the densest known packing of the regular pentagon.
Packing Regular Heptagons
This picture by Toby Hudson shows the densest known packing of the regular heptagon. Of all convex shapes, the regular heptagon is believed to have the lowest maximal packing density.
Packing Smoothed Octagons
Which shape is worst of all for packing the plane? That is, which has the lowest maximal packing density? Suppose we demand that our shape be convex and also centrally symmetric: that is, a subset $S \subseteq \mathbb{R}^2$ such that $x \in S$ implies $-x \in S$. Then a certain ‘smoothed octagon’ is conjectured to be the worst. Amazingly, this shape has a 1-parameter family of maximally dense packings, shown in this image created by Greg Egan.
Packing Regular Octagons
This is the densest packing of regular octagons in the plane, drawn by Graeme McRae. It is interesting because it is a counterexample to the 2-dimensional analogue of a conjecture made in 3 dimensions by Stanislaw Ulam.