Schönhardt polyhedron
Non-convex polyhedron with no triangulation / From Wikipedia, the free encyclopedia
Dear Wikiwand AI, let's keep it short by simply answering these key questions:
Can you list the top facts and stats about Schönhardt polyhedron?
Summarize this article for a 10 year old
In geometry, a Schönhardt polyhedron is a polyhedron with the same combinatorial structure as a regular octahedron, but with dihedral angles that are non-convex along three disjoint edges. Because it has no interior diagonals, it cannot be triangulated into tetrahedra without adding new vertices. It has the fewest vertices of any polyhedron that cannot be triangulated. It is named after German mathematician Erich Schönhardt, who described it in 1928, although artist Karlis Johansons exhibited a related structure in 1921.
Schönhardt polyhedron | |
---|---|
Faces | 8 |
Edges | 12 |
Vertices | 6 |
Properties | non-convex no interior diagonals cannot be triangulated |
Net | |
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 produces a jumping polyhedron whose two solid forms share the same face shapes. A 30° twist instead produces a shaky polyhedron, rigid but not infinitesimally rigid, whose edges form a tensegrity prism.
Schönhardt polyhedra have been used as gadgets in a proof that testing whether a polyhedron has a triangulation is NP-complete. 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.