Hypercubic honeycomb

n -hypercubic honeycomb
Rank	n  + 1
Type	Regular
Space	Euclidean
Notation
Coxeter diagram	... (x4o3o3o...o3o4o)
Schläfli symbol	{4,3,3,...,3,4}
Elements
Faces	${\binom {n}{2}}$M
Edges	n M
Vertices	M
Vertex figure	n -orthoplex, edge length √2
Measures (edge length 1)
Vertex density	$1$
Related polytopes
Dual	Self
Conjugate	None
Abstract & topological properties
Orientable	Yes
Properties
Symmetry	Rn +1
Convex	Yеs
Nature	Tame

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.

Vertex coordinates

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

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

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 (..., S_n +1 symmetry, hypercubes of two types)

External links

• Klitzing, Richard. The Honeycomb Product.
• Wikipedia contributors. "Hypercubic honeycomb".