# {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!

# Pattern-Equivariant Homology of a Penrose Tiling

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.

# {6,3,6} Honeycomb

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!

# {6,3,5} Honeycomb

This is the {6,3,5} 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 {6,3,5} honeycomb lives in hyperbolic space, and every vertex has 12 edges coming out, just as if you drew edges from the middle of an icosahedron to its corners.

# {6,3,4} Honeycomb

This is the {6,3,4} 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 {6,3,4} honeycomb lives in hyperbolic space, and each vertex has 6 edges coming out of it, just as if you drew edges from the middle of an octahedron to its corners.

# {6,3,3} Honeycomb

This is the {6,3,3} honeycomb, drawn by Roice Nelson. A 3-dimensional honeycomb is a way of filling 3d space with polyhedra. It’s the 3-dimensional analogue of a tiling of the plane. Besides honeycombs in 3d Euclidean space, we can also have honeycombs in 3d hyperbolic space. The hexagonal tiling honeycomb lives in hyperbolic space, and each vertex has 4 edges coming out, just as if we drew edges from the middle of a tetrahedron to its 4 corners.