Schönhardt polyhedron
Jump to navigation
Jump to search
| Schönhardt polyhedron | |
|---|---|
 | |
| Rank | 3 |
| Type | Isogonal |
| Elements | |
| Faces | 6 scalene triangles, 2 triangles |
| Edges | 3+3+6 |
| Vertices | 6 |
| Vertex figure | Irregular tetragon |
| Measures | |
| Central density | 1 |
| Related polytopes | |
| Army | Trigyp |
| Dual | Nonconvex triangular gyrotegum |
| Abstract & topological properties | |
| Euler characteristic | 2 |
| Orientable | Yes |
| Genus | 0 |
| Properties | |
| Symmetry | (A2×A1)+, order 6 |
| Convex | No |
| Nature | Tame |
The Schönhardt polyhedron is the simplest polyhedron that cannot be triangulated into tetrahedra. It is a concave variant of the triangular gyroprism.
It cannot be divided into tetrahedra without introducing new vertices. Rambau showed that this is also true of every "nonconvex twisted prism", or concave gyroprism, where the base triangle is replaced by another non-degenerate regular polygon.[1]
The Schönhardt polyhedron also appears in the study of flexible polyhedra as the jumping octahedron.
References[edit | edit source]
- ↑ Rambau, Jörg. "On the Generalization of Schönhardt's Polyhedron".