Cube atop small rhombicuboctahedron

Cube atop small rhombicuboctahedron
(OFF file)
Rank	4
Type	Segmentotope
Space	Spherical
Notation
Bowers style acronym	Cubasirco
Coxeter diagram	xx4oo3ox&#x
Elements
Cells	8 tetrahedra, 12 triangular prisms, 1+6 cubes, 1 small rhombicuboctahedron
Faces	8+24 triangles, 6+6+12+24 squares
Edges	12+24+24+24
Vertices	8+24
Vertex figures	8 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	$\sqrt{\frac{3+\sqrt2}{2}} ≈ 1.48563$
Hypervolume	$\frac{19+12\sqrt2}{12} ≈ 2.99755$
Dichoral angles	Tet–3–trip: 150°
	Cube–4–trip: $\arccos\left(-\frac{\sqrt6}{3}\right) ≈ 144.73561°$
	Cube–4–cube: 135°
	Sirco–3–tet: 60°
	Sirco–4–trip: $\arccos\left(\frac{\sqrt3}{3}\right) ≈ 54.73561°$
	Sirco–4–cube: 45°
Height	$\frac{\sqrt2}{2} ≈ 0.70711$
Central density	1
Related polytopes
Army	Cubasirco
Regiment	Cubasirco
Dual	Octahedral-deltoidal icositetrahedral tegmoid
Conjugate	Cube atop quasirhombicuboctahedron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B3×I, order 48
Convex	Yes
Nature	Tame

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.

Segmentochoron display

• 

  Cube atop small rhombicuboctahedron

Vertex coordinates

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

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

along with all permutations of the first three coordinates of:

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

External links

• Klitzing, Richard. "cubasirco".

• Wikipedia Contributors. "Cubic cupola".