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Concept
16-cell
Summary
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. It is also called C16, hexadecachoron, or hexdecahedroid .
It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes, and is analogous to the octahedron in three dimensions. It is Coxeter's polytope. Conway's name for a cross-polytope is orthoplex, for orthant complex. The dual polytope is the tesseract (4-cube), which it can be combined with to form a compound figure. The 16-cell has 16 cells as the tesseract has 16 vertices.
The 16-cell is the second in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).
Each of its 4 successor convex regular 4-polytopes can be constructed as the convex hull of a polytope compound of multiple 16-cells: the 16-vertex tesseract as a compound of two 16-cells, the 24-vertex 24-cell as a compound of three 16-cells, the 120-vertex 600-cell as a compound of fifteen 16-cells, and the 600-vertex 120-cell as a compound of seventy-five 16-cells.
The 16-cell is the 4-dimensional cross polytope, which means its vertices lie in opposite pairs on the 4 axes of a (w, x, y, z) Cartesian coordinate system.
The eight vertices 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 edge length is .
The vertex coordinates form 6 orthogonal central squares lying in the 6 coordinate planes. Squares in opposite planes that do not share an axis (e.g. in the xy and wz planes) are completely disjoint (they do not intersect at any vertices).
The 16-cell constitutes an orthonormal basis for the choice of a 4-dimensional reference frame, because its vertices exactly define the four orthogonal axes.
The Schläfli symbol of the 16-cell is {3,3,4}, indicating that its cells are regular tetrahedra {3,3} and its vertex figure is a regular octahedron {3,4}.
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