# Snub disphenoid

Snub disphenoid
Rank3
TypeCRF
Notation
Elements
Faces4+8 triangles
Edges2+4+4+8
Vertices4+4
Vertex figures4 rhombi, edge length 1
4 pentagons, edge length 1
Measures (edge length 1)
Volume≈ 0.85949
Central density1
Number of external pieces12
Level of complexity9
Related polytopes
DualElongated gyrobifastigium
ConjugateSnub disphenoid
Abstract & topological properties
Flag count72
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry(B2×A1)/2, order 8
Flag orbits9
ConvexYes
NatureTame

The snub disphenoid (OBSA: snadow), also known as the Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron or dodecadeltahedron is one of the 92 Johnson solids (J84). It consists of 4+8 triangles as faces.

It can be constructed from a tetrahedron, seen as a digonal antiprism or disphenoid, by expanding the two halves outward and inserting a set of 8 triangles in between the halves.

The snub disphenoid (with theoretical edge length 1) is the vertex figure of the 13-5 step prism, which cannot be made uniform, because the Johnson solid variant is not circumscribable.

## Vertex coordinates

Coordinates for a snub disphenoid of unit edge length are given by

• ${\displaystyle \left(\pm t,\,0,\,-r\right)}$,
• ${\displaystyle \left(0,\,\pm t,\,r\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,0,\,s\right)}$,
• ${\displaystyle \left(0,\,\pm {\frac {1}{2}},\,-s\right)}$,

where r, s and t are given in terms of the unique positive root q ≈ 0.16902 of

${\displaystyle 2x^{3}+11x^{2}+4x-1}$

as

${\displaystyle r={\frac {\sqrt {q}}{2}},\quad s={\sqrt {\frac {1-q}{8q}}},\quad t={\sqrt {\frac {1-q}{2}}}}$.

With these coordinates, it's possible to calculate the volume of a snub disphenoid with unit edge length as ξ ≈ 0.85949, where ξ is the unique positive root of the polynomial

${\displaystyle 5832x^{6}-1377x^{4}-2160x^{2}-4.}$[1]

## Related polyhedra

The snub disphenoid can be considered to be the digonal case in the family of snub antiprisms. The snub triangular antiprism is the regular icosahedron, and the snub square antiprism is another Johnson solid. No other convex members of this family can be made to have all regular faces, though nonconvex regular-faced cases do exist.

If each square of a square antiprism are turned into two triangles, the result is a snub disphenoid. If only one square is turned into triangles, the result is a biaugmented triangular prism.

## In vertex figures

The snub disphenoid appears as the vertex figure of the nonuniform 13-5 step prism. This vertex figure has an edge length of 1, and has no corealmic realization, because the Johnson snub disphenoid has no circumscribed sphere.