5simplex
In fivedimensional geometry, a 5simplex is a selfdual regular 5polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5cell facets. It has a dihedral angle of cos^{−1}(1/5), or approximately 78.46°.
The 5simplex 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 hexa5tope, as a 6facetted polytope in 5dimensions. The name hexateron is derived from hexa for having six facets and teron (with ter being a corruption of tetra) for having fourdimensional facets.
By Jonathan Bowers, a hexateron is given the acronym hix.^{[1]}
As a configuration
This configuration matrix represents the 5simplex. The rows and columns correspond to vertices, edges, faces, cells and 4faces. The diagonal numbers say how many of each element occur in the whole 5simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This selfdual simplex's matrix is identical to its 180 degree rotation.^{[2]}^{[3]}
[math]\displaystyle{ \begin{bmatrix}\begin{matrix}6 & 5 & 10 & 10 & 5 \\ 2 & 15 & 4 & 6 & 4 \\ 3 & 3 & 20 & 3 & 3 \\ 4 & 6 & 4 & 15 & 2 \\ 5 & 10 & 10 & 5 & 6 \end{matrix}\end{bmatrix} }[/math]
Regular hexateron cartesian coordinates
The hexateron can be constructed from a 5cell by adding a 6th vertex such that it is equidistant from all the other vertices of the 5cell.
The Cartesian coordinates for the vertices of an origincentered regular hexateron having edge length 2 are:
 [math]\displaystyle{ \begin{align} &\left(\tfrac{1}\sqrt{15},\ \tfrac{1}\sqrt{10},\ \tfrac{1}\sqrt{6},\ \tfrac{1}\sqrt{3},\ \pm1\right)\\[5pt] &\left(\tfrac{1}\sqrt{15},\ \tfrac{1}\sqrt{10},\ \tfrac{1}\sqrt{6},\ \tfrac{2}\sqrt{3},\ 0\right)\\[5pt] &\left(\tfrac{1}\sqrt{15},\ \tfrac{1}\sqrt{10},\ \tfrac\sqrt{3}\sqrt{2},\ 0,\ 0\right)\\[5pt] &\left(\tfrac{1}\sqrt{15},\ \tfrac{2\sqrt 2}\sqrt{5},\ 0,\ 0,\ 0\right)\\[5pt] &\left(\tfrac\sqrt{5}\sqrt{3},\ 0,\ 0,\ 0,\ 0\right) \end{align} }[/math]
The vertices of the 5simplex can be more simply positioned on a hyperplane in 6space as permutations of (0,0,0,0,0,1) or (0,1,1,1,1,1). These construction can be seen as facets of the 6orthoplex or rectified 6cube respectively.
Projected images
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. 
Lower symmetry forms
A lower symmetry form is a 5cell pyramid ( )v{3,3,3}, with [3,3,3] symmetry order 120, constructed as a 5cell base in a 4space hyperplane, and an apex point above the hyperplane. The five sides of the pyramid are made of 5cell cells. These are seen as vertex figures of truncated regular 6polytopes, like a truncated 6cube.
Another form is { }∨{3,3}, with [2,3,3] 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] symmetry order 36, and extended symmetry 3,2,3, order 72. It represents joining of 2 orthogonal triangles, orthogonally offset, with all pairs of vertices connected between.
The form { }∨{ }∨{ } has symmetry [3[2,2,2]], order 48.
These are seen in the vertex figures of bitruncated and tritruncated regular 6polytopes, like a bitruncated 6cube and a tritruncated 6simplex. The edge labels here represent the types of face along that direction, and thus represent different edge lengths.
The vertex figure of the omnitruncated 5simplex honeycomb, , is a 5simplx with a petrie polygon cycle of 5 long edges.
Name  ( )∨{3,3,3} 51 fusil 
{ }∨{3,3} 42 fusil 
{3}∨{3} 33 fusil 
{ }∨{ }∨{ } 222 fusil 


Symmetry  [3,3,3] Order 120 
[2,3,3] Order 48 
3,2,3 Order 72 
[3[2,2,2]] Order 48 

Diagram  
Polytope  truncated 6simplex 
bitruncated 6simplex 
tritruncated 6simplex 
333 prism 
Omnitruncated 5simplex honeycomb 
Compound
The compound of two 5simplexes in dual configurations can be seen in this A6 Coxeter plane projection, with a red and blue 5simplex vertices and edges. This compound has 3,3,3,3 symmetry, order 1440. The intersection of these two 5simplexes is a uniform birectified 5simplex. = ∩ .
Related uniform 5polytopes
It is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1_{3k} series. A degenerate 4dimensional case exists as 3sphere tiling, a tetrahedral hosohedron.
It is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3_{k1} series. A degenerate 4dimensional case exists as 3sphere tiling, a tetrahedral dihedron.
The 5simplex, as 2_{20} polytope is first in dimensional series 2_{2k}.
The regular 5simplex 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)
See also
Notes
 ↑ Klitzing, Richard. "5D uniform polytopes (polytera) x3o3o3o3o — hix". https://bendwavy.org/klitzing/dimensions/polytera.htm.
 ↑ Coxeter 1973, §1.8 Configurations
 ↑ Coxeter, H.S.M. (1991). Regular Complex Polytopes (2nd ed.). Cambridge University Press. pp. 117. ISBN 9780521394901. https://books.google.com/books?id=9BY9AAAAIAAJ&pg=PA117.
References
 Gosset, T. (1900). "On the Regular and SemiRegular Figures in Space of n Dimensions". Messenger of Mathematics. Macmillan. pp. 43–. https://books.google.com/books?id=BZo_AQAAIAAJ.
 Coxeter, H.S.M.:
 Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C. et al., eds (1995). Kaleidoscopes: Selected Writings of H.S.M. Coxeter. Wiley. ISBN 9780471010036. https://books.google.com/books?id=fUm5Mwfx8rAC&pg=PP1.
 (Paper 22)
 (Paper 23)
 (Paper 24)
 (Paper 22)
 Conway, John H.; Burgiel, Heidi; GoodmanStrass, Chaim (2008). "26. Hemicubes: 1_{n1}". The Symmetries of Things. pp. 409. ISBN 9781568812205.
 Johnson, Norman (1991). Uniform Polytopes.
 Johnson, N.W. (1966). The Theory of Uniform Polytopes and Honeycombs (PhD). University of Toronto.
External links
 Olshevsky, George. "Simplex". Glossary for Hyperspace. Archived from the original on 4 February 2007. https://web.archive.org/web/20070204075028/members.aol.com/Polycell/glossary.html#Simplex.
 Polytopes of Various Dimensions, Jonathan Bowers
 Multidimensional Glossary
Original source: https://en.wikipedia.org/wiki/5simplex.
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