# {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.

# {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.

# {5,3,5} Honeycomb

This is the {5,3,5} honeycomb, drawn by Jos Leys. 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 Euclidean space, we can also have honeycombs in hyperbolic space, which is a 3-dimensional Riemannian manifold with constant negative curvature. The {5,3,5} honeycomb lives in hyperbolic space.

# {6,3,3} Honeycomb in Upper Half Space

A 3-dimensional honeycomb is a way of filling 3d space with polyhedra. It’s the 3-dimensional analogue of a tiiling of the plane. However, not only can we have honeycombs in Euclidean space, we can also have them in hyperbolic space. The {6,3,3} honeycomb lives in hyperbolic space.