5-simplex
In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1, or approximately 78.46°.
The 5-simplex is a solution to the problem: ''Make 20 equilateral triangles using 15 matchsticks, where each side of every triangle is exactly one matchstick.''
Alternate names
It can also be called a hexateron, or hexa-5-tope, as a 6-facetted polytope in 5-dimensions. The name hexateron is derived from hexa- for having six facets and teron for having four-dimensional facets.By Jonathan Bowers, a hexateron is given the acronym hix.
As a configuration
This configuration matrix represents the 5-simplex. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-simplex. 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.Regular hexateron cartesian coordinates
The hexateron can be constructed from a 5-cell by adding a 6th vertex such that it is equidistant from all the other vertices of the 5-cell.The Cartesian coordinates for the vertices of an origin-centered regular hexateron having edge length 2 are:
The vertices of the 5-simplex can be more simply positioned on a hyperplane in 6-space as permutations of or. These constructions can be seen as facets of the 6-orthoplex or rectified 6-cube respectively.
Projected images
| 320px Stereographic projection 4D to 3D of Schlegel diagram 5D to 4D of hexateron. |
Lower symmetry forms
A lower symmetry form is a 5-cell pyramid ∨, with symmetry order 120, constructed as a 5-cell base in a 4-space hyperplane, and an apex point above the hyperplane. The five sides of the pyramid are made of 5-cell cells. These are seen as vertex figures of truncated regular 6-polytopes, like a truncated 6-cube.Another form is ∨, with symmetry order 48, the joining of an orthogonal digon and a tetrahedron, orthogonally offset, with all pairs of vertices connected between. Another form is ∨, with symmetry order 36, and extended symmetry
The form ∨∨ has symmetry, order 8, extended by permuting 3 segments as,1] or, order 48.
These are seen in the vertex figures of bitruncated and tritruncated regular 6-polytopes, like a bitruncated 6-cube and a tritruncated 6-simplex. The edge labels here represent the types of face along that direction, and thus represent different edge lengths.
The vertex figure of the omnitruncated 5-simplex honeycomb,, is a 5-simplex with a petrie polygon cycle of 5 long edges. Its symmetry is isomorphic to dihedral group Dih6 or simple rotation group +, order 12.
| Join | ∨ | ∨ | ∨ | ∨∨ | |
| Symmetry | Order 120 | Order 48 | Order 72 | ,1,1]= Order 48 | ~ or ~+ Order 12 |
| Diagram | |||||
| Polytope | truncated 6-simplex | bitruncated 6-simplex | tritruncated 6-simplex | 3-3-3 prism | Omnitruncated 5-simplex honeycomb |
Compound
The compound of two 5-simplexes in dual configurations can be seen in this A6 Coxeter plane projection, with a red and blue 5-simplex vertices and edges. This compound hasRelated uniform 5-polytopes
It is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.It is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3k1 series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral dihedron.
The 5-simplex, as 220 polytope is first in dimensional series 22k.
The regular 5-simplex is one of 19 uniform polytera based on the Coxeter group, all shown here in A5 Coxeter plane orthographic projections.