Compound of two snub cubes

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Short description: Polyhedral compound
Compound of two snub cubes
UC68-2 snub cubes.png
Type Uniform compound
Index UC68
Schläfli symbol βr{4,3}
Coxeter diagram CDel node h3.pngCDel 4.pngCDel node h3.pngCDel 3.pngCDel node h3.png
Polyhedra 2 snub cubes
Faces 16+48 triangles
12 squares
Edges 120
Vertices 48
Symmetry group octahedral (Oh)
Subgroup restricting to one constituent chiral octahedral (O)

This uniform polyhedron compound is a composition of the 2 enantiomers of the snub cube. As a holosnub, it is represented by Schläfli symbol βr{4,3} and Coxeter diagram CDel node h3.pngCDel 4.pngCDel node h3.pngCDel 3.pngCDel node h3.png.

The vertex arrangement of this compound is shared by a convex nonuniform truncated cuboctahedron, having rectangular faces, alongside irregular hexagons and octagons, each alternating with two edge lengths.

Together with its convex hull, it represents the snub cube-first projection of the nonuniform snub cubic antiprism.

Cartesian coordinates

Cartesian coordinates for the vertices are all the permutations of

(±1, ±ξ, ±1/ξ)

where ξ is the real solution to

[math]\displaystyle{ \xi^3+\xi^2+\xi=1, \, }[/math]

which can be written

[math]\displaystyle{ \xi = \frac{1}{3}\left(\sqrt[3]{17+3\sqrt{33}} - \sqrt[3]{-17+3\sqrt{33}} - 1\right) }[/math]

or approximately 0.543689. ξ is the reciprocal of the tribonacci constant.

Equally, the tribonacci constant, t, just like the snub cube, can compute the coordinates as:

(±1, ±t, ±1/t)

Truncated cuboctahedron

This compound can be seen as the union of the two chiral alternations of a truncated cuboctahedron:

Snubcubes in grCO.svg
A geometric construction of the Tribonacci constant (AC), with compass and marked ruler, according to the method described by Xerardo Neira.

See also

References

  • Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society 79: 447–457, doi:10.1017/S0305004100052440 .