# Octahedron atop cube

Octahedron atop cube
Rank4
TypeSegmentotope
Notation
Bowers style acronymOctacube
Coxeter diagramxo4oo3ox&#x
Elements
Cells8+12 tetrahedra, 6 square pyramids, 1 octahedron, 1 cube
Faces8+24+24 triangles, 6 squares
Edges12+12+24
Vertices6+8
Vertex figures6 square antiprisms, edge length 1
8 triangular antipodiums, edge lengths 2 (large base) and 1 (small base and sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {4+{\sqrt {2}}}{7}}}\approx 0.87947}$
Hypervolume${\displaystyle {\frac {\sqrt {31+22{\sqrt {2}}}}{12}}\approx 0.65676}$
Dichoral anglesTet–3–tet: ${\displaystyle \arccos \left({\frac {1-3{\sqrt {2}}}{4}}\right)\approx 144.16048^{\circ }}$
Tet–3–oct: ${\displaystyle \arccos \left({\frac {2-3{\sqrt {2}}}{4}}\right)\approx 124.10147^{\circ }}$
Tet–3–squippy: ${\displaystyle \arccos \left({\frac {2-3{\sqrt {2}}}{4}}\right)\approx 124.10147^{\circ }}$
Cube–4–squippy: ${\displaystyle \arccos \left({\frac {{\sqrt {2}}-2}{2}}\right)\approx 107.03125^{\circ }}$
Height${\displaystyle {\frac {\sqrt {2{\sqrt {2}}-1}}{2}}\approx 0.67610}$
Central density1
Related polytopes
ArmyOctacube
RegimentOctacube
DualCubic-octahedral tegmoid
ConjugateOctahedron atop cube
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryB3×I, order 48
ConvexYes
NatureTame

The octahedron atop cube, or octacube, is a CRF segmentochoron (designated K-4.15 on Richard Klitzing's list). As the name suggests, it consists of a cube and an octahedron as bases, connected by 6 square pyramids and 8+12 tetrahedra.

It is also commonly referred to as a cubic or octahedral antiprism, as the two bases are a pair of dual polyhedra.

## Vertex coordinates

The vertices of an octahedron atop cube segmentochoron of edge length 1 are given by:

• ${\displaystyle \left(\pm {\frac {\sqrt {2}}{2}},\,0,\,0,\,{\frac {\sqrt {2{\sqrt {2}}-1}}{2}}\right)}$ and all permutations of its first 3 coordinates,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0\right).}$