16-cell

This article may require cleanup to meet Wikipedia's quality standards. The specific problem is: Eliminate explanatory footnotes. Screen readers can put these at the end of the article, which is confusingly out of context. Merge into main prose or drop where content is already covered by a linked article. Please help improve this article if you can. (May 2024) (Learn how and when to remove this message)

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 ${\displaystyle \beta _{4}}$ 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.