Schmidt Arrangement of the Eisenstein Integers - Katherine Stange

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.

Hammersley Sofa - Claudio Rocchini

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.

Densest Packings of the Smoothed Octagon - Greg Egan

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.

Densest Packing of Regular Octagons - Graeme McRae

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.