# Packing Regular Pentagons

This picture by Toby Hudson shows the densest known packing of the regular pentagon.

This picture by Toby Hudson shows the densest known packing of the regular pentagon.

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.

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.

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.

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).

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 $\mathbb{Z}(2^\infty)$.

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.

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.

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.

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.

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.

There is a nice photograph of some interlocking origami dodecahedra created by Dirk Eisner on the website Mathematical Origami. But it’s hard to be sure how many dodecahedra the whole model contains, since some are hidden from view. This raises a puzzle: assuming the configuration is as symmetrical as possible, how many dodecahedra are there? Here you see Greg Egan’s answer to this puzzle—and to a much more challenging puzzle.

When can you fit a tetrahedron between two nested spheres? Suppose the radius of the large sphere is $R$ and the radius of the small one is $r$. Suppose the distance between their centers is $d$. Then you can fit a tetrahedron between these spheres if and only if the **Grace–Danielsson inequality** $ d^2 \le (R + r)(R – 3r) $ holds. This was independently proved by Grace in 1917 and Danielsson in 1949. But Antony Milne has found a new proof of this inequality using quantum information theory!

The Penrose kite and dart are a pair of tiles that can be used to create aperiodic tilings of the plane. This image illustrates a ‘pattern-equivariant 1-chain’, a tool used by James J. Walton to study the topology of the kite and dart tiling, and other aperiodic tilings.

This is the {6,3,6} honeycomb, drawn by Roice Nelson. A 3-dimensional honeycomb is a way of filling 3d space with polyhedra or infinite sheets of polygons. Besides honeycombs in 3d Euclidean space, we can also have honeycombs in 3d hyperbolic space, a non-Euclidean geometry with constant negative curvature. The {6,3,6} honeycomb lives in hyperbolic space… and it’s special, because it’s self-dual!