# Snub disphenoid

Snub disphenoid
Type Johnson
J83 - J84 - J85
Faces 4+8 triangles
Edges 18
Vertices 8
Vertex configuration 4(34)
4(35)
Symmetry group D2d
Dual polyhedron -
Properties convex, deltahedron
Net

In geometry, the snub disphenoid or dodecadeltahedron is a dodecahedron and one of the Johnson solids (J84). It is a three-dimensional solid that has only equilateral triangles as faces, and is therefore a deltahedron. It is not a regular polyhedron because some vertices have four faces and others have five. It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids.

A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform (that is, they are not Platonic solids, Archimedean solids, prisms or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]

The snub disphenoid is constructed, as its name suggests, as a snub disphenoid, and represented as ss{2,4}, with s{2,4} as a digonal antiprism, being the first of an infinite set of snub antiprisms. This construction also produces two degenerate digonal faces from the digonal antiprism.[2]

s{2,4} ss{2,4}
Digonal antiprism
(disphenoid)
Snub disphenoid

It can also be seen as the 8 triangular faces of the square antiprism with the two squares replaced by pairs of triangles.

It was called a Siamese dodecahedron in the paper by Freudenthal and van der Waerden which first described it in 1947 in the set of eight convex deltahedra.

The snub disphenoid is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size. The other three polyhedra with this property are the regular octahedron, the pentagonal bipyramid, and an irregular polyhedron with 12 vertices and 20 triangular faces (Finbow et al. 2010).

The snub disphenoid has three dihedral angles, approximately 121.7°, 96.2°, 166.4°.

## References

1. ^ Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
2. ^ Snub Anti-Prisms