16-cell

16-cell
(4-orthoplex)
Schlegel diagram
(vertices and edges)
TypeConvex regular 4-polytope
4-orthoplex
4-demicube
Schläfli symbol{3,3,4}
Coxeter diagram
Cells16 {3,3}
Faces32 {3}
Edges24
Vertices8
Vertex figure
Octahedron
Petrie polygonoctagon
Coxeter groupB4, [3,3,4], order 384
D4, order 192
DualTesseract
Propertiesconvex, isogonal, isotoxal, isohedral, regular, Hanner polytope
Uniform index12

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.

  1. ^ Coxeter 1973, p. 141, § 7-x. Historical remarks.
  2. ^ 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
  3. ^ Matila Ghyka, The Geometry of Art and Life (1977), p.68
  4. ^ Coxeter 1973, pp. 120=121, § 7.2. See illustration Fig 7.2B.