Compound of two snub cubes

Compound of two snub cubes
(3D model) (OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Disco
Coxeter diagram	ß4ß3ß ()
Elements
Components	2 snub cubes
Faces	48 triangles, 16 triangles as 8 hexagrams, 12 squares as 6 stellated octagons
Edges	24+48+48
Vertices	48
Vertex figure	Floret pentagon, edge lengths 1, 1, 1, 1, √2
Measures (edge length 1)
Circumradius	≈ 1.34371
Volume	≈ 15.77896
Dihedral angles	3–3: ≈ 153.23459°
	4–3: ≈ 142.98343°
Central density	2
Number of external pieces	160
Level of complexity	26
Related polytopes
Army	Semi-uniform Girco
Regiment	Disco
Dual	Compound of two pentagonal icositetrahedra
Conjugate	Compound of two snub cubes
Convex core	Cuboctatruncated disdyakis dodecahedron
Abstract & topological properties
Flag count	480
Orientable	Yes
Properties
Symmetry	B3, order 48
Convex	No
Nature	Tame

The disnub cuboctahedron, disco, or compound of two snub cubes is a uniform polyhedron compound. It consists of 48 snub triangles, 16 further triangles, and 12 squares (the latter two can combine in pairs due to faces in the same plane). Four triangles and one square join at each vertex.

Its quotient prismatic equivalent is the snub cubic antiprism, which is four-dimensional.

Measures

The circumradius R ≈ 1.34371 of the disnub cuboctahedron with unit edge length is the largest real root of

$32x^{6}-80x^{4}+44x^{2}-7.$

Its volume V ≈ 15.77896 is given by the largest real root of

$729x^{6}-182736x^{4}+310176x^{2}-798848$.

Its dihedral angles can be given as acos(α) for the angle between two triangular faces, and acos(β) for the angle between a square face and a triangular face, where α ≈ –0.89286 equals the unique real root of

$27x^{3}-9x^{2}-15x+13,$

and β ≈ –0.79846 equals the unique negative real root of

$27x^{6}-99x^{4}+129x^{2}-49.$

Vertex coordinates

A disnub cuboctahedron of edge length 1 has coordinates given by all permutations of:

• (±c₁, ±c₂, ±c₃),

where

• $c_{1}={\sqrt {{\frac {1}{12}}\left(4-{\sqrt[{3}]{17+3{\sqrt {33}}}}-{\sqrt[{3}]{17-3{\sqrt {33}}}}\right)}},$
• $c_{2}={\sqrt {{\frac {1}{12}}\left(2+{\sqrt[{3}]{17+3{\sqrt {33}}}}+{\sqrt[{3}]{17-3{\sqrt {33}}}}\right)}},$
• $c_{3}={\sqrt {{\frac {1}{12}}\left(4+{\sqrt[{3}]{199+3{\sqrt {33}}}}+{\sqrt[{3}]{199-3{\sqrt {33}}}}\right)}}.$

External links

• Bowers, Jonathan. "Polyhedron Category C10: Disnubs" (#68).

• Klitzing, Richard. "disco".

• Wikipedia Contributors. "Compound of two snub cubes".