# Hypercubic honeycomb

n -hypercubic honeycomb
Rankn  + 1
TypeRegular
SpaceEuclidean
Notation
Coxeter diagram... (x4o3o3o...o3o4o)
Schläfli symbol{4,3,3,...,3,4}
Elements
Faces${\displaystyle {\binom {n}{2}}}$M
Edgesn M
VerticesM
Vertex figuren -orthoplex, edge length 2
Measures (edge length 1)
Vertex density${\displaystyle 1}$
Related polytopes
DualSelf
ConjugateNone
Abstract & topological properties
OrientableYes
Properties
SymmetryRn +1
ConvexYеs
NatureTame

The hypercubic honeycombs form the only infinite series of regular Euclidean tessellations that exists in all dimensions. As the name suggests, their facets are hypercubes. Four of these facets meet at a ridge, and 2n  of them meet at a vertex.

The hypercubic honeycombs are self-dual, and their vertex figures are orthoplexes.

## Examples

Hypercubic honeycombs by dimension
Rank Name Picture
2 Apeirogon
3 Square tiling
4 Cubic honeycomb
5 Tesseractic tetracomb
6 Penteractic pentacomb
7 Hexeractic hexacomb

## Vertex coordinates

The vertices of an n -hypercubic honeycomb of edge length 1 are given by

• ${\displaystyle \left(p,\,q,\,r,\,...,\,u\right)}$, with n  such variables, all of them ${\displaystyle \in \mathbb {Z} }$.

## Representations

A n -hypercubic honeycomb has the following Coxeter diagrams (with n  nodes), among others:

• x4o3o...3o4o (..., full symmetry)
• x4o3o...3o4x (..., as expanded hypercubic honeycomb)
• o3o3o *b3o...3o4x (..., Sn +1 symmetry, hypercubes of two types)