Tesseractic honeycomb

Tesseractic honeycomb

Perspective projection of a 3x3x3x3 red-blue chessboard.
Type Regular 4-space 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}x{∞}
{4,4}x{∞}2
{∞}4
Coxeter-Dynkin diagrams




4-face type {4,3,3}
Cell type {4,3}
Face type {4}
Edge figure 8 {4,3}
(octahedron)
Vertex figure 16 {4,3,3}
(16-cell)
Coxeter groups {\tilde{C}}_4, [4,3,3,4]
{\tilde{B}}_4, [4,3,31,1]
Dual self-dual
Properties vertex-transitive, edge-transitive, face-transitive, cell-transitive

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.

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.

Contents

Coordinates

Vertices of this honeycomb can be positioned in 4-space in all integer coordinates (i,j,k,l).

Constructions

There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,3,3,4}. Another form has two alternating tesseract facets (like a checkerboard) with Schläfli symbol {4,3,31,1}. The lowest symmetry Wythoff construction has 16 types of facets around each vertex and a prismatic product Schläfli symbol {∞}4. One can be made by stericating another.

Related polytopes and tessellations

The tesseract can make a regular tessellation of the 4-sphere, with three tesseracts per edge, with Schläfli symbol {4,3,3,3}, called a order-3 tesseractic honeycomb. It is topologically equivalent to the regular polytope penteract in 5-space.

The tesseract can make a regular tessellation of 4-dimensional hyperbolic space, with 5 tesseracts around each edge, with Schläfli symbol {4,3,3,5}, called an order-5 tesseractic honeycomb.

See also

References