Small cubicuboctahedron

The small cubicuboctahedron, or socco, is a uniform polyhedron. It consists of 8 triangles, 6 squares, and 6 octagons. One triangle, one square, and two octagons meet at each vertex. It also has 12 square pseudofaces.

Small cubicuboctahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Socco
Coxeter diagram	x4/3o3x4*a ()
Elements
Faces	8 triangles, 6 squares, 6 octagons
Edges	24+24
Vertices	24
Vertex figure	Crossed isosceles trapezoid, edge lengths 1, √2+√2, √2, √2+√2
Measures (edge length 1)
Circumradius	$\frac{\sqrt{5+2\sqrt2}}{2} ≈ 1.39897$
Volume	$2\frac{3+4\sqrt2}{3} ≈ 5.77124$
Dihedral angles	8–4: 90°
	8–3: $\arccos\left(\frac{\sqrt3}{3}\right) ≈ 54.73561^\circ$
Central density	2
Number of external pieces	62
Level of complexity	9
Related polytopes
Army	Sirco
Regiment	Sirco
Dual	Small hexacronic icositetrahedron
Conjugate	Great cubicuboctahedron
Convex core	Cube
Abstract & topological properties
Flag count	192
Euler characteristic	–4
Orientable	Yes
Genus	3
Properties
Symmetry	B3, order 48
Convex	No
Nature	Tame

It is a faceting of the small rhombicuboctahedron, using its 6 squares and 8 triangles, along with 6 new octagons.

Vertex coordinates

Its vertices are the same as those of its regiment colonel, the rhombicuboctahedron.

Representations

A small cubicuboctahedron has the following Coxeter diagrams:

• x4/3o3x4*a
• x4o3ß (, as holosnub)

Related polyhedra

The rhombihyperhombicosicosahedron is a uniform polyhedron compound composed of 5 small cubicuboctahedra.

o4/3o3o4*a truncations

Name	OBSA	CD diagram	Picture
(degenerate, double cover of cube)		x4/3o3o4*a ()
Great cubicuboctahedron	gocco	x4/3x3o4*a ()
(degenerate, oct+6(4))		o4/3x3o4*a ()
(degenerate, double cover of cho)		o4/3x3x4*a ()
(degenerate, oct+6(4))		o4/3o3x4*a ()
Small cubicuboctahedron	socco	x4/3o3x4*a ()
Cuboctatruncated cuboctahedron	cotco	x4/3x3x4*a ()

External links

• Bowers, Jonathan. "Polyhedron Category 4: Trapeziverts" (#37).

• Klitzing, Richard. "socco".

• Wikipedia Contributors. "Small cubicuboctahedron".
• McCooey, David. "Small Cubicuboctahedron"