Compound of two cubes

Compound of two cubes
(OFF file)
Rank	3
Type	Uniform
Notation
Coxeter diagram	xo4ox xx
Elements
Components	2 cubes
Faces	8 squares, 4 squares as 2 stellated octagons
Edges	8+16
Vertices	16
Vertex figure	Equilateral triangle, edge length √2
Measures (edge length 1)
Circumradius	${\frac {\sqrt {3}}{2}}\approx 0.86603$
Volume	2
Dihedral angle	90°
Height	1
Central density	2
Number of external pieces	18
Level of complexity	6
Related polytopes
Army	Semi-uniform Op, edge lengths ${\sqrt {\frac {2-{\sqrt {2}}}{2}}}$ (base), 1 (sides)
Regiment	*
Dual	Compound of two octahedra
Conjugate	Compound of two cubes
Abstract & topological properties
Flag count	96
Orientable	Yes
Properties
Symmetry	I2(8)×A1, order 32
Convex	No
Nature	Tame

The stellated octagonal prism or compound of two cubes is a prismatic uniform polyhedron compound. It consists of 2 stellated octagons and 8 squares. Each vertex joins one stellated octagon and two squares. As the name suggests, it is a prism based on a stellated octagon.

Its quotient prismatic equivalent is the square antiprismatic prism, which is four-dimensional.

Vertex coordinates

Coordinates for the vertices of a stellated octagonal prism of edge length 1 centered at the origin are given by:

• $\left(\pm {\frac {\sqrt {2}}{2}},\,0,\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right).$

Variations

This compound has variants where the bases are non-regular compounds of two squares. In these cases the compound has only square prismatic symmetry and the convex hull is a ditetragonal prism.