# Cube atop small rhombicuboctahedron

Cube atop small rhombicuboctahedron
Rank4
TypeSegmentotope
SpaceSpherical
Notation
Bowers style acronymCubasirco
Coxeter diagramxx4oo3ox&#x
Elements
Cells8 tetrahedra, 12 triangular prisms, 1+6 cubes, 1 small rhombicuboctahedron
Faces8+24 triangles, 6+6+12+24 squares
Edges12+24+24+24
Vertices8+24
Vertex figures8 triangular antipodiums, edge lengths 1 (base 1) and 2 (base 2 and sides)
24 isosceles trapezoidal pyramids, base edge lengths 1, 2, 2, 2, side edge lengths 1, 1, 2. 2
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{3+\sqrt2}{2}} ≈ 1.48563}$
Hypervolume${\displaystyle \frac{19+12\sqrt2}{12} ≈ 2.99755}$
Dichoral anglesTet–3–trip: 150°
Cube–4–trip: ${\displaystyle \arccos\left(-\frac{\sqrt6}{3}\right) ≈ 144.73561°}$
Cube–4–cube: 135°
Sirco–3–tet: 60°
Sirco–4–trip: ${\displaystyle \arccos\left(\frac{\sqrt3}{3}\right) ≈ 54.73561°}$
Sirco–4–cube: 45°
Height${\displaystyle \frac{\sqrt2}{2} ≈ 0.70711}$
Central density1
Related polytopes
ArmyCubasirco
RegimentCubasirco
DualOctahedral-deltoidal icositetrahedral tegmoid
ConjugateCube atop quasirhombicuboctahedron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryB3×I, order 48
ConvexYes
NatureTame

The cube atop small rhombicuboctahedron, or cubasirco, is a CRF segmentochoron (designated K-4.71 on Richard Klitzing's list). As the name suggests, it consists of a cube and a small rhombicuboctahedron as bases, connected by 8 tetrahedra, 12 triangular prisms, and 6 further cubes.

It is also sometimes referred to as a cubic cupola, as one generalization of the definition of a cupola is to have a polytope atop an expanded version.

A small disprismatotesseractihexadecachoron can be formed by attaching cube atop small rhombicuboctahedron segmentochora to the bases of the small rhombicuboctahedral prism.

## Vertex coordinates

Coordinates for the vertices of a cube atop small rhombicuboctahedron of edge length 1 are given by:

• ${\displaystyle \left(±\frac12,\,±\frac12,\,±\frac12,\,±\frac{\sqrt2}{2}\right),}$

along with all permutations of the first three coordinates of:

• ${\displaystyle \left(±\frac{1+\sqrt2}{2},\,±\frac12,\,±\frac12,\,0\right).}$