5-cell


In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is the 4-simplex, the simplest possible convex regular 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The pentachoron is a four dimensional pyramid with a tetrahedral base.
The regular 5-cell is bounded by 5 regular tetrahedra, and is one of the six regular convex 4-polytopes, represented by Schläfli symbol.
The 5-cell is a solution to the problem: Make 10 equilateral triangles, all of the same size, using 10 matchsticks, where each side of every triangle is exactly one matchstick. No solution exists in three dimensions.
The convex hull of the 5-cell and its dual is the disphenoidal 30-cell, dual of the bitruncated 5-cell.

Alternative names

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, or approximately 75.52°.

As a configuration

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 simplex's matrix is identical to its 180 degree rotation.

Construction

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 simplest set of coordinates is:,,,,, with edge length 2, where φ is the golden ratio.
The Cartesian coordinates of the vertices of an origin-centered regular 5-cell having edge length 2 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:
The vertices of a 4-simplex can be more simply constructed on a hyperplane in 5-space, as permutations of or ; in these positions it is a facet of, respectively, the 5-orthoplex or the rectified penteract.

Boerdijk–Coxeter helix

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.

Projections

The A4 Coxeter plane projects the 5-cell into a regular pentagon and pentagram.

Irregular 5-cell

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.
Many uniform 5-polytopes have tetrahedral pyramid vertex figures:
Schlegel
diagram
Name
Coxeter		×
×
×
t
t
t

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.

Compound

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. = ∩.
This compound can be seen as the 4D analogue of the 2D hexagram {} and the 3D compound of two tetrahedra.

Related polytopes and honeycomb

The pentachoron is the simplest of 9 uniform polychora constructed from the Coxeter group.
It is in the sequence of regular polychora: the tesseract, 120-cell, of Euclidean 4-space, and hexagonal tiling honeycomb of hyperbolic space. All of these have a tetrahedral vertex figure.
It is one of three regular 4-polytopes with tetrahedral cells, along with the 16-cell, 600-cell. The order-6 tetrahedral honeycomb of hyperbolic space also has tetrahedral cells.

Citations