Tesseractic honeycomb

In four-dimensional euclidean geometry, the tesseractic honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol {4,3,3,4}, and constructed by a 4-dimensional packing of tesseract facets.

Tesseractic honeycomb
Perspective projection of a 3x3x3x3 red-blue chessboard.
Type	Regular 4-space honeycomb Uniform 4-honeycomb
Family	Hypercubic honeycomb
Schläfli symbols	{4,3,3,4} t0,4{4,3,3,4} {4,3,31,1} {4,4}(2) {4,3,4}×{∞} {4,4}×{∞}(2) {∞}(4)
Coxeter-Dynkin diagrams
4-face type	{4,3,3}
Cell type	{4,3}
Face type	{4}
Edge figure	{3,4} (octahedron)
Vertex figure	{3,3,4} (16-cell)
Coxeter groups	${\displaystyle {\tilde {C}}_{4}}$, [4,3,3,4] ${\displaystyle {\tilde {B}}_{4}}$, [4,3,31,1]
Dual	self-dual
Properties	vertex-transitive, edge-transitive, face-transitive, cell-transitive, 4-face-transitive

Its vertex figure is a 16-cell. Two tesseracts meet at each cubic cell, four meet at each square face, eight meet on each edge, and sixteen meet at each vertex.

It is an analog of the square tiling, {4,4}, of the plane and the cubic honeycomb, {4,3,4}, of 3-space. These are all part of the hypercubic honeycomb family of tessellations of the form {4,3,...,3,4}. Tessellations in this family are Self-dual.