# Octahedron atop cube

Octahedron atop cube Rank4
TypeSegmentotope
SpaceSpherical
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$\sqrt{\frac{4+\sqrt2}{7}} \approx 0.87947$ Hypervolume$\frac{\sqrt{31+22\sqrt2}}{12} \approx 0.65676$ Dichoral anglesTet–3–tet: $\arccos\left(\frac{1-3\sqrt2}{4}\right) \approx 144.16048^\circ$ Tet–3–oct: $\arccos\left(\frac{2-3\sqrt2}{4}\right) \approx 124.10147^\circ$ Tet–3–squippy: $\arccos\left(\frac{2-3\sqrt2}{4}\right) \approx 124.10147^\circ$ Cube–4–squippy: $\arccos\left(\frac{\sqrt2-2}{2}\right) \approx 107.03125^\circ$ Height$\frac{\sqrt{2\sqrt2-1}}{2} \approx 0.67610$ 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:

• $\left(±\frac{\sqrt2}{2},\,0,\,0,\,\frac{\sqrt{2\sqrt2-1}}{2}\right)$ and all permutations of its first 3 coordinates,
• $\left(±\frac12,\,±\frac12,\,±\frac12,\,0\right).$ 