5-simplex Hexateron (hix) | ||
---|---|---|
Type | uniform 5-polytope | |
Schläfli symbol | {3^{4}} | |
Coxeter diagram | ||
4-faces | 6 | 6 {3,3,3} |
Cells | 15 | 15 {3,3} |
Faces | 20 | 20 {3} |
Edges | 15 | |
Vertices | 6 | |
Vertex figure | 5-cell | |
Coxeter group | A_{5}, [3^{4}], order 720 | |
Dual | self-dual | |
Base point | (0,0,0,0,0,1) | |
Circumradius | 0.645497 | |
Properties | convex, isogonal regular, self-dual |
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}(1/5), 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.
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 (with ter- being a corruption of tetra-) for having four-dimensional facets.
By Jonathan Bowers, a hexateron is given the acronym hix.^{[1]}
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.^{[2]}^{[3]}
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 (0,0,0,0,0,1) or (0,1,1,1,1,1). These constructions can be seen as facets of the 6-orthoplex or rectified 6-cube respectively.
A_{k} Coxeter plane |
A_{5} | A_{4} |
---|---|---|
Graph | ||
Dihedral symmetry | [6] | [5] |
A_{k} Coxeter plane |
A_{3} | A_{2} |
Graph | ||
Dihedral symmetry | [4] | [3] |
Stereographic projection 4D to 3D of Schlegel diagram 5D to 4D of hexateron. |
A lower symmetry form is a 5-cell pyramid {3,3,3}∨( ), with [3,3,3] 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 {3,3}∨{ }, with [3,3,2,1] symmetry order 48, the joining of an orthogonal digon and a tetrahedron, orthogonally offset, with all pairs of vertices connected between. Another form is {3}∨{3}, with [3,2,3,1] symmetry order 36, and extended symmetry [[3,2,3],1], order 72. It represents joining of 2 orthogonal triangles, orthogonally offset, with all pairs of vertices connected between.
The form { }∨{ }∨{ } has symmetry [2,2,1,1], order 8, extended by permuting 3 segments as [3[2,2],1] or [4,3,1,1], 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 isomophic to dihedral group Dih_{6} or simple rotation group [6,2]^{+}, order 12.
Join | {3,3,3}∨( ) | {3,3}∨{ } | {3}∨{3} | { }∨{ }∨{ } | |
---|---|---|---|---|---|
Symmetry | [3,3,3,1] Order 120 |
[3,3,2,1] Order 48 |
[[3,2,3],1] Order 72 |
[3[2,2],1,1]=[4,3,1,1] Order 48 |
~[6] or ~[6,2]^{+} Order 12 |
Diagram | |||||
Polytope | truncated 6-simplex |
bitruncated 6-simplex |
tritruncated 6-simplex |
3-3-3 prism |
Omnitruncated 5-simplex honeycomb |
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 has [[3,3,3,3]] symmetry, order 1440. The intersection of these two 5-simplexes is a uniform birectified 5-simplex. = ∩ .
It is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1_{3k} series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.
Space | Finite | Euclidean | Hyperbolic | |||
---|---|---|---|---|---|---|
n | 4 | 5 | 6 | 7 | 8 | 9 |
Coxeter group |
A_{3}A_{1} | A_{5} | D_{6} | E_{7} | =E_{7}^{+} | =E_{7}^{++} |
Coxeter diagram |
||||||
Symmetry | [3^{−1,3,1}] | [3^{0,3,1}] | [3^{1,3,1}] | [3^{2,3,1}] | [[3^{3,3,1}]] | [3^{4,3,1}] |
Order | 48 | 720 | 23,040 | 2,903,040 | ∞ | |
Graph | - | - | ||||
Name | 1_{3,-1} | 1_{30} | 1_{31} | 1_{32} | 1_{33} | 1_{34} |
It is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3_{k1} series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral dihedron.
Space | Finite | Euclidean | Hyperbolic | |||
---|---|---|---|---|---|---|
n | 4 | 5 | 6 | 7 | 8 | 9 |
Coxeter group |
A_{3}A_{1} | A_{5} | D_{6} | E_{7} | =E_{7}^{+} | =E_{7}^{++} |
Coxeter diagram |
||||||
Symmetry | [3^{−1,3,1}] | [3^{0,3,1}] | [[3^{1,3,1}]] = [4,3,3,3,3] |
[3^{2,3,1}] | [3^{3,3,1}] | [3^{4,3,1}] |
Order | 48 | 720 | 46,080 | 2,903,040 | ∞ | |
Graph | - | - | ||||
Name | 3_{1,-1} | 3_{10} | 3_{11} | 3_{21} | 3_{31} | 3_{41} |
The 5-simplex, as 2_{20} polytope is first in dimensional series 2_{2k}.
Space | Finite | Euclidean | Hyperbolic | ||
---|---|---|---|---|---|
n | 4 | 5 | 6 | 7 | 8 |
Coxeter group |
A_{2}A_{2} | A_{5} | E_{6} | =E_{6}^{+} | E_{6}^{++} |
Coxeter diagram |
|||||
Graph | ∞ | ∞ | |||
Name | 2_{2,-1} | 2_{20} | 2_{21} | 2_{22} | 2_{23} |
The regular 5-simplex is one of 19 uniform polytera based on the [3,3,3,3] Coxeter group, all shown here in A_{5} Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)
A5 polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
t0 t_{1} |
t_{2} |
t_{0,1} |
t_{0,2} |
t_{1,2} |
t_{0,3} | ||||||
t_{1,3} |
t_{0,4} |
t_{0,1,2} |
t_{0,1,3} |
t_{0,2,3} |
t_{1,2,3} |
t_{0,1,4} | |||||
t_{0,2,4} |
t_{0,1,2,3} |
t_{0,1,2,4} |
t_{0,1,3,4} |
t_{0,1,2,3,4} |