# 5-cell

The 5-cell is a solution to the problem: No solution exists in three dimensions.

*Make 10 equilateral triangles, all of the same size, using 10 matchsticks, where each side of every triangle is exactly one matchstick.*

The convex hull of the 5-cell and its dual (assuming that they are congruent) is the disphenoidal 30-cell, dual of the bitruncated 5-cell.

The 5-cell is self-dual, and its vertex figure is a tetrahedron. Its maximal intersection with 3-dimensional space is the triangular prism. Its dihedral angle is cos^{−1}(1/4), or approximately 75.52°.

This configuration matrix represents the 5-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 5-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual polytope's matrix is identical to its 180 degree rotation.^{[5]}

The 5-cell can be constructed from a tetrahedron by adding a 5th vertex such that it is equidistant from all the other vertices of the tetrahedron. (The 5-cell is a 4-dimensional pyramid with a tetrahedral base and four tetrahedral sides.)

The simplest set of coordinates is: (2,0,0,0), (0,2,0,0), (0,0,2,0), (0,0,0,2), (φ,φ,φ,φ), with edge length 2√2, where φ is the golden ratio.^{[6]}

The Cartesian coordinates of the vertices of an origin-centered regular 5-cell having edge length 2 and radius √1.6 are:

Another set of origin-centered coordinates in 4-space can be seen as a hyperpyramid with a regular tetrahedral base in 3-space, with edge length 2√2 and radius √3.2:

The vertices of a 4-simplex (with edge √2 and radius 1) can be more simply constructed on a hyperplane in 5-space, as (distinct) permutations of (0,0,0,0,1) *or* (0,1,1,1,1); in these positions it is a facet of, respectively, the 5-orthoplex or the rectified penteract.

A 5-cell can be constructed as a Boerdijk–Coxeter helix of five chained tetrahedra, folded into a 4-dimensional ring. The 10 triangle faces can be seen in a 2D net within a triangular tiling, with 6 triangles around every vertex, although folding into 4-dimensions causes edges to coincide. The purple edges represent the Petrie polygon of the 5-cell.

The A_{4} Coxeter plane projects the 5-cell into a regular pentagon and pentagram.

There are many lower symmetry forms, including these found in uniform polytope vertex figures:

The **tetrahedral pyramid** is a special case of a **5-cell**, a polyhedral pyramid, constructed as a regular tetrahedron base in a 3-space hyperplane, and an apex point *above* the hyperplane. The four *sides* of the pyramid are made of tetrahedron cells.

Other uniform 5-polytopes have irregular 5-cell vertex figures. The symmetry of a vertex figure of a uniform polytope is represented by removing the ringed nodes of the Coxeter diagram.

The compound of two 5-cells in dual configurations can be seen in this A5 Coxeter plane projection, with a red and blue 5-cell vertices and edges. This compound has [[3,3,3]] symmetry, order 240. The intersection of these two 5-cells is a uniform bitruncated 5-cell. = ∩ .

The pentachoron (5-cell) is the simplest of 9 uniform polychora constructed from the [3,3,3] Coxeter group.