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.
2-adic Integers
This image created by Christopher Culter shows the compact abelian group of 2-adic integers (black points), with selected elements labeled by the corresponding character on the Pontryagin dual group (colored discs).
Prüfer 2-Group
This is the Prüfer $2$-group, the subgroup of the unit complex numbers consisting of all $2^n$th roots of unity. It is also called $\mathbb{Z}(2^\infty)$.
{3,3,7} Honeycomb Meets Plane at Infinity
The {3,3,7} honeycomb is a honeycomb in 3d hyperbolic space. It is the dual of the {7,3,3} honeycomb shown last time. This image, drawn by Roice Nelson, shows the ‘boundary’ of the {3,3,7} honeycomb: that is, the set of points on the ‘plane at infinity’ that are limits of points in the {3,3,7} honeycomb.
{7,3,3} Honeycomb Meets Plane at Infinity
This picture by Roice Nelson shows the boundary of the {7,3,3} honeycomb. The black circles are holes, not contained in the boundary of the {7,3,3} honeycomb. There are infinitely many holes, and the actual boundary, shown in white, is a fractal with area zero.
{7,3,3} Honeycomb
This is the {7,3,3} honeycomb as drawn by Danny Calegari using his program ‘kleinian’. In this image, hyperbolic space has been compressed down to an open ball using the so-called Poincaré ball model. The {7,3,3} honeycomb is built of regular heptagons in hyperbolic space. These heptagons lie on infinite sheets, each of which is a {7,3} tiling of the hyperbolic plane. The 3-dimensional regions bounded by these sheets are unbounded: they go off to infinity. They show up as holes here.
{7,3} Tiling
This picture, drawn by Anton Sherwood, shows the {7,3} tiling: a tiling of the hyperbolic plane by equal-sized regular heptagons, 3 meeting at each vertex.
Sierpinski Carpet
To build the Sierpinski carpet you take a square, cut it into 9 equal-sized smaller squares, and remove the central smaller square. Then you apply the same procedure to the remaining 8 subsquares, and repeat this ad infinitum. This image by Noon Silk shows the first six stages of the procedure.