It is named after the German mathematician Erich Schönhardt, who described it in 1928, although the artist Karlis Johansons had exhibited a related structure in 1921.

One construction for the Schönhardt polyhedron starts with a triangular prism and twists the two equilateral triangle faces of the prism relative to each other, breaking each square face into two triangles separated by a non-convex edge.

Some twist angles produce a jumping polyhedron whose two solid forms share the same face shapes.

Several other polyhedra, including Jessen's icosahedron, share with the Schönhardt polyhedron the properties of having no triangulation, of jumping or being shaky, or of forming a tensegrity structure.

One way of constructing a Schönhardt polyhedron starts with a triangular prism, with two parallel equilateral triangles as its faces.

Thus, the Schönhardt polyhedron can be formed by removing these three tetrahedra from a convex (but irregular) octahedron.

Neither could a model made of a more rigid material like glass: although it could exist in either of the two shapes, it would be unable to deform sufficiently to move between them.

[6] In his original work on this polyhedron, Schönhardt noted a related property: in one special form, when the two equilateral faces are twisted at an angle of 30° with respect to each other, this polyhedron becomes shaky: rigid with respect to continuous motion, but not infinitesimally rigid.

[7] The discovery of this form as a tensegrity structure rather than as a polyhedron has been credited to Latvian-Soviet artist Karlis Johansons in 1921, a few years before the work of Schönhardt.

[12] It was shown by Rambau (2005) that the Schönhardt polyhedron can be generalized to other polyhedra, combinatorially equivalent to antiprisms, that cannot be triangulated.

[14] In a different direction, Bagemihl (1948) constructed a family of polyhedra that share with the Schönhardt polyhedron the property that there are no internal diagonals.

The tetrahedron and the Császár polyhedron have no diagonals at all: every pair of vertices in these polyhedra forms an edge.