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Concept
Schönhardt polyhedron
Summary
In geometry, the Schönhardt polyhedron is the simplest non-convex polyhedron that cannot be triangulated into tetrahedra without adding new vertices. It is named after German mathematician Erich Schönhardt, who described it in 1928. The same polyhedra have also been studied in connection with Cauchy's rigidity theorem as an example where polyhedra with two different shapes have faces of the same shapes.
One way of constructing the Schönhardt polyhedron starts with a triangular prism, with two parallel equilateral triangles as its faces. One of the triangles is rotated
around the centerline of the prism, breaking the square faces of the prism into pairs of triangles. If each of these pairs is chosen to be non-convex, the Schönhardt polyhedron is the result.
The Schönhardt polyhedron has six vertices, twelve edges, and eight triangular faces. The six vertices of the Schönhardt polyhedron can be used to form fifteen unordered pairs of vertices. Twelve of these fifteen pairs form edges of the polyhedron: there are six edges in the two equilateral triangle faces, and six edges connecting the two triangles. The remaining three edges form diagonals of the polyhedron, but lie entirely outside the polyhedron.
The convex hull of the Schönhardt polyhedron is another polyhedron with the same six vertices, and a different set of twelve edges and eight triangular faces; like the Schönhardt polyhedron, it is combinatorially equivalent to a regular octahedron. The symmetric difference of the Schönhardt polyhedron consists of three tetrahedra, each lying between one of the concave dihedral edges of the Schönhardt polyhedron and one of the exterior diagonals. Thus, the Schönhardt polyhedron can be formed by removing these three tetrahedra from a convex (but irregular) octahedron.
It is impossible to partition the Schönhardt polyhedron into tetrahedra whose vertices are vertices of the polyhedron. More strongly, there is no tetrahedron that lies entirely inside the Schönhardt polyhedron and has vertices of the polyhedron as its four vertices.
Official source
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