Small snubahedron

Small snubahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Sis
Elements
Components	6 tetrahedra
Faces	24 triangles
Edges	12+24
Vertices	24
Vertex figure	Equilateral triangle, edge length 1
Measures (edge length 1)
Circumradius	$\frac{\sqrt6}{4} ≈ 0.61237$
Inradius	$\frac{\sqrt6}{12} ≈ 0.20412$
Volume	$\frac{\sqrt2}{2} ≈ 0.70711$
Dihedral angle	$\arccos\left(\frac13\right) ≈ 70.52878^\circ$
Central density	6
Related polytopes
Army	Semi-uniform Toe
Regiment	Sis
Dual	Small snubahedron
Conjugate	Small snubahedron
Abstract & topological properties
Schläfli type	{3,3}
Orientable	Yes
Properties
Symmetry	A3, order 24
Convex	No
Nature	Tame

The small snubahedron, sis, or compound of six tetrahedra with rotational freedom is a uniform polyhedron compound. It consists of 24 triangles, with three faces joining at a vertex.

This compound has rotational freedom, represented by an angle θ. At θ = 0°, all six tetrahedra coincide. We rotate these tetrahedra around their 2-fold axes of symmetry (2 each), seeing them as digonal antiprisms. At θ = 45° the compound has double symmetry resulting in the snubahedron.

This compound can be formed by taking one of the tetrahedra inscribed in each cube of the rhombisnub dishexahedron.

Vertex coordinates

The vertices of a small snubahedron of edge length 1 and rotation angle θ are given by all even permutations of:

• $\left(±\frac{\cos(\theta)+\sin(\theta)}{2\sqrt2},\,±\frac{\cos(\theta)-\sin(\theta)}{2\sqrt2},\,±\frac{\sqrt2}{4}\right).$

External links

• Bowers, Jonathan. "Polyhedron Category C5: Tets and Cubes" (#32).

• Klitzing, Richard. "sis".

• Wikipedia Contributors. "Compound of six tetrahedra with rotational freedom".