Tesseractihemioctachoron

Tesseractihemioctachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Tho
Coxeter diagram	/2
Elements
Cells	8 tetrahedra, 4 octahedra
Faces	32 triangles
Edges	24
Vertices	8
Vertex figure	Tetrahemihexahedron, edge length 1
Edge figure	tet 3 oct 3 tet 3 oct 3
Measures (edge length 1)
Circumradius	$\frac{\sqrt2}{2} ≈ 0.70711$
Dichoral angle	60°
Number of external pieces	40
Level of complexity	5
Related polytopes
Army	Hex
Regiment	Hex
Company	Hex
Dual	Tesseractihemioctacron
Conjugate	None
Abstract & topological properties
b0	1
b1	0
b2	3
b3	0
Euler characteristic	4
Orientable	No
Properties
Symmetry	D4, order 192
Convex	No
Nature	Tame

The tesseractihemioctachoron, or tho, is a uniform hemipolychoron. It has 8 regular tetrahedra and 4 central octahedra as cells, with 4 tetrahedra and 3 octahedra at each vertex in the form of a tetrahemihexahedron. It is the 4D demicross.

It is a faceting of the hexadecachoron, analogous to how the tetrahemihexahedron is a faceting of the octahedron. It shares the hexadecachoron's vertices, edges, and faces. The 8 tetrahedra are half of those of the hexadecachoron, while the octahedra are its original vertex figures.

Notably, as well as being isogonal, the tesseractihemioctachoron is also edge- and face-transitive. It is one of only four non-regular uniform polychora to be vertex, edge, and face transitive.

Cross-sections

Vertex coordinates

Its vertices are the same as those of its regiment colonel, the hexadecachoron.

External links

• Bowers, Jonathan. "Category 1: Regular Polychora" (#17).

• Klitzing, Richard. "tho".