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| 5-cell (4-simplex) | |
|---|---|
 A 3D projection of a 5-cell performing a simple rotation | |
| Type | Convex regular 4-polytope |
| Schläfli symbol | {3,3,3} |
| Coxeter diagram |    |
| Cells | 5 {3,3}  |
| Faces | 10 {3}  |
| Edges | 10 |
| Vertices | 5 |
| Vertex figure |  (tetrahedron) |
| Petrie polygon | pentagon |
| Coxeter group | A4, [3,3,3] |
| Dual | Self-dual |
| Properties | convex, isogonal, isotoxal, isohedral |
| Uniform index | 1 |
In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol {3,3,3}. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, hypertetrahedron, 'pentachoron,[1] pentatope, pentahedroid,[2] tetrahedral pyramid, or 4-simplex (Coxeter's polytope),[3] the simplest possible convex 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The 5-cell is a 4-dimensional pyramid with a tetrahedral base and four tetrahedral sides.
The regular 5-cell is bounded by five regular tetrahedra, and is one of the six regular convex 4-polytopes (the four-dimensional analogues of the Platonic solids). A regular 5-cell can be constructed from a regular tetrahedron by adding a fifth vertex one edge length distant from all the vertices of the tetrahedron. This cannot be done in 3-dimensional space. The regular 5-cell is a solution to the problem: Make 10 equilateral triangles, all of the same size, using 10 matchsticks, where each side of every triangle is exactly one matchstick, and none of the triangles and matchsticks intersect one another. No solution exists in three dimensions.
- ^ Johnson 2018, p. 249.
- ^ Ghyka 1977, p. 68.
- ^ Coxeter 1973, p. 120, §7.2. see illustration Fig 7.2A.