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| 16-cell (4-orthoplex) | |
|---|---|
 Schlegel diagram (vertices and edges) | |
| Type | Convex regular 4-polytope 4-orthoplex 4-demicube |
| Schläfli symbol | {3,3,4} |
| Coxeter diagram |     |
| Cells | 16 {3,3}  |
| Faces | 32 {3}  |
| Edges | 24 |
| Vertices | 8 |
| Vertex figure |  Octahedron |
| Petrie polygon | octagon |
| Coxeter group | B4, [3,3,4], order 384 D4, order 192 |
| Dual | Tesseract |
| Properties | convex, isogonal, isotoxal, isohedral, regular, Hanner polytope |
| Uniform index | 12 |
In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century.[1] It is also called C16, hexadecachoron,[2] or hexdecahedroid [sic?] .[3]
It is the 4-dimesional member of an infinite family of polytopes called cross-polytopes, orthoplexes, or hyperoctahedrons which are analogous to the octahedron in three dimensions. It is Coxeter's polytope.[4] The dual polytope is the tesseract (4-cube), which it can be combined with to form a compound figure. The cells of the 16-cell are dual to the 16 vertices of the tesseract.
- ^ Coxeter 1973, p. 141, § 7-x. Historical remarks.
- ^ N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite Symmetry Groups, 11.5 Spherical Coxeter groups, p.249
- ^ Matila Ghyka, The Geometry of Art and Life (1977), p.68
- ^ Coxeter 1973, pp. 120=121, § 7.2. See illustration Fig 7.2B.