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}(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.
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 (with ter- being a corruption of tetra-) for having four-dimensional facets.
By Jonathan Bowers, a hexateron is given the acronym hix.^{[1]}
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.^{[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 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:
- [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 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 construction can be seen as facets of the 6-orthoplex or rectified 6-cube 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 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. It's 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 |
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 has 3,3,3,3 symmetry, order 1440. The intersection of these two 5-simplexes is a uniform birectified 5-simplex. = ∩ .
Related uniform 5-polytopes
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.
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.
The 5-simplex, as 2_{20} polytope is first in dimensional series 2_{2k}.
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)
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 Semi-Regular 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 978-0-471-01003-6. https://books.google.com/books?id=fUm5Mwfx8rAC&pg=PP1.
- (Paper 22)
- (Paper 23)
- (Paper 24)
- (Paper 22)
- Conway, John H.; Burgiel, Heidi; Goodman-Strass, Chaim (2008). "26. Hemicubes: 1_{n1}". The Symmetries of Things. pp. 409. ISBN 978-1-56881-220-5.
- 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
- Multi-dimensional Glossary
Original source: https://en.wikipedia.org/wiki/5-simplex.
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