In geometry, the snub disphenoid is a convex polyhedron with 12 equilateral triangles as its faces.

Its vertices may be placed in a sphere and can also be used as a minimum possible Lennard-Jones potential among all eight-sphere clusters.

The dual polyhedron of the snub disphenoid is the elongated gyrobifastigium.

As suggested by the name, the snub disphenoid is constructed from a tetragonal disphenoid by cutting all the edges from its faces, and adding equilateral triangles (the light blue colors in the following image) that are twisted in a certain angle between them.

[1] The snub disphenoid may also be constructed from a triangular bipyramid, by cutting its two edges along the apices.

[2] Alternatively, the snub disphenoid can be constructed from pentagonal bipyramid by cutting the two edges along that connecting the base of the bipyramid and then inserting two equilateral triangles between them.

It has 4 vertices in a square on a center plane as two anticupolae attached with rotational symmetry.

An alternative net suggested by John Montroll has fewer concave vertices on its boundary, making it more convenient for origami construction.

[2] As a consequence of such constructions, the snub disphenoid has 12 equilateral triangles.

[7] The dual polyhedron of the snub disphenoid is the elongated gyrobifastigium.

Up to symmetries and parallel translation, the snub disphenoid has five types of simple (non-self-crossing) closed geodesics.

These are paths on the surface of the polyhedron that avoid the vertices and locally look like the shortest path: they follow straight line segments across each face of the polyhedron that they intersect, and when they cross an edge of the polyhedron they make complementary angles on the two incident faces to the edge.

Intuitively, one could stretch a rubber band around the polyhedron along this path and it would stay in place: there is no way to locally change the path and make it shorter.

For example, one type of geodesic crosses the two opposite edges of the snub disphenoid at their midpoints (where the symmetry axis exits the polytope) at an angle of

A second type of geodesic passes near the intersection of the snub disphenoid with the plane that perpendicularly bisects the symmetry axis (the equator of the polyhedron), crossing the edges of eight triangles at angles that alternate between

Shifting a geodesic on the surface of the polyhedron by a small amount (small enough that the shift does not cause it to cross any vertices) preserves the property of being a geodesic and preserves its length, so both of these examples have shifted versions of the same type that are less symmetrically placed.

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.

[10] In the study of computational chemistry and molecular physics, spheres centered at the vertices of the snub disphenoid form a cluster that, according to numerical experiments, has the minimum possible Lennard-Jones potential among all eight-sphere clusters.

The dodecahedral molecular geometry describes the cluster for which it is a snub disphenoid.

[13][14] It was studied again in the paper by Freudenthal & van d. Waerden (1947), which first described the set of eight convex deltahedra, and named it the Siamese dodecahedron.

[15][14] The dodecadeltahedron name was given to the same shape by Bernal (1964), referring to the fact that it is a 12-sided deltahedron.

There are other simplicial dodecahedra, such as the hexagonal bipyramid, but this is the only one that can be realized with equilateral faces.

Bernal was interested in the shapes of holes left in irregular close-packed arrangements of spheres, so he used a restrictive definition of deltahedra, in which a deltahedron is a convex polyhedron with triangular faces that can be formed by the centers of a collection of congruent spheres, whose tangencies represent polyhedron edges, and such that there is no room to pack another sphere inside the cage created by this system of spheres.

This restrictive definition disallows the triangular bipyramid (as forming two tetrahedral holes rather than a single hole), pentagonal bipyramid (because the spheres for its apexes interpenetrate, so it cannot occur in sphere packings), and regular icosahedron (because it has interior room for another sphere).

Bernal writes that the snub disphenoid is "a very common coordination for the calcium ion in crystallography".