What is 16-cell?
[[File:Schlegel wireframe 16-cell.png|thumb|right|250px|A hexadecachoron]] In four dimensional [[geometry]], a '''16-cell''', is a [[regular convex polychora|regular convex polychoron]], or [[polytope]] existing in four dimensions. It is also known as the '''hexadecachoron'''. It is one of the six [[regular convex polychora]] first described by the Swiss mathematician [[Ludwig Schläfli]] in the mid-19th century. [[John Horton Conway|Conway]] calls it an '''orthoplex''' for ''orthant complex'', as well as the entire class of [[cross-polytope]]s. == Geometry == The hexadecachoron is a member of the family of polytopes called the [[cross-polytope]]s, which exist in all dimensions. As such, its dual polychoron is the [[tesseract]] (the 4-dimensional hypercube). It is bounded by 16 [[cell (mathematics)|cells]], all of which are regular [[tetrahedron|tetrahedra]]. It has 32 triangular faces, 24 edges, and 8 vertices. The 24 edges bound 6 squares lying in the 6 coordinate planes. The eight vertices of the hexadecachoron are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by edges except opposite pairs. The [[Schläfli symbol]] of the hexadecachoron is {3,3,4}. Its [[vertex figure]] is a regular octahedron. There are 8 tetrahedra, 12 triangles, and 6 edges meeting at every vertex. Its [[edge figure]] is a square. There are 4 tetrahedra and 4 triangles meeting at every edge. There is a lower symmetry form of the ''16-cell'', called a '''demitesseract''' or '''4-demicube''', a member of the [[demihypercube]] family, and represented by h{4,3,3}, and can be drawn bicolored with alternating [[tetrahedron|tetrahedral]] cells. == Images == <gallery> File:stereographic_polytope_16cell_colour.png|[[Stereographic projection]] File:Cell16-4dpolytope.svg|[[Orthographic projection (geometry)|Four orthographic projections]] File:Cross graph 4.svg|A skew orthogonal projection inside its regular [[octagon]]al [[Petrie polygon]], connecting all vertices except opposite ones. File:16-cell nets