| 5-simplex Hexateron (hix) | ||
|---|---|---|
| Type | uniform 5-polytope | |
| Schläfli symbol | {34} | |
| Coxeter diagram |   
| |
| 4-faces | 6 | 6 {3,3,3}
|
| Cells | 15 | 15 {3,3}
|
| Faces | 20 | 20 {3}
|
| Edges | 15 | |
| Vertices | 6 | |
| Vertex figure |  5-cell | |
| Coxeter group | A5, [34], order 720 | |
| Dual | self-dual | |
| Base point | (0,0,0,0,0,1) | |
| Circumradius | 0.645497 | |
| Properties | convex, isogonal regular, self-dual | |
In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(1/5), or approximately 78.46°.
The 5-simplex is a solution to the problem: Make 20 equilateral triangles using 15 matchsticks, where each side of every triangle is exactly one matchstick.