Truncated cubic honeycomb

Truncated cubic honeycomb
Rank	4
Type	uniform
Space	Euclidean
Notation
Bowers style acronym	Tich
Coxeter diagram	x4x3o4o ()
Elements
Cells	N octahedra, N truncated cubes
Faces	8N triangles, 3N octagons
Edges	3N+12N
Vertices	6N
Vertex figure	Square pyramid, edge lengths 1 (base) and √2+√2 (sides)
Measures (edge length 1)
Vertex density
Dual cell volume
Related polytopes
Army	Tich
Regiment	Tich
Dual	Square pyramidal honeycomb
Conjugate	Quasitruncated cubic honeycomb
Topological properties
Orientable	Yes
Properties
Symmetry	R4
Convex	Yes

The truncated cubic honeycomb, or tich, is a convex uniform honeycomb. 1 octahedron and 4 truncated cubes join at each vertex of this honeycomb. As the name suggests, it is the truncation of the cubic honeycomb.

Vertex coordinates[edit | edit source]

The vertices of a truncated cubic honeycomb of edge length 1 are given by all permutations of:

$\left(±\frac12+(1+\sqrt2)i,\,±\frac{1+\sqrt2}{2}+(1+\sqrt2)j,\,±\frac{1+\sqrt2}{2}+(1+\sqrt2)k\right),$

Where i, j, and k range over the integers.

A truncated cubic honeycomb has the following Coxeter diagrams:

o4o3o4o truncations
Name	OBSA	Schläfli symbol
Cubic honeycomb	chon	{4,3,4}
Truncated cubic honeycomb	tich	t{4,3,4}
Rectified cubic honeycomb	rich	r{4,3,4}
Bitruncated cubic honeycomb	batch	2t{4,3,4}
Rectified cubic honeycomb	rich	r{4,3,4}
Truncated cubic honeycomb	tich	t{4,3,4}
Cubic honeycomb	chon	{4,3,4}
Small rhombated cubic honeycomb	srich	rr{4,3,4}
Great rhombated cubic honeycomb	grich	tr{4,3,4}
Small rhombated cubic honeycomb	srich	rr{4,3,4}
Great rhombated cubic honeycomb	grich	tr{4,3,4}
Small prismated cubic honeycomb = Cubic honeycomb	chon	t_0,3{4,3,4}
Prismatorhombated cubic honeycomb	prich	t_0,1,3{4,3,4}
Prismatorhombated cubic honeycomb	prich	t_0,1,3{4,3,4}
Great prismated cubic honeycomb	gippich	t_0,1,2,3{4,3,4}