# 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.

# 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.

# Origami Dodecahedra

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.

# Grace–Danielsson Inequality

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!