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In five-dimensional geometry , a 5-simplex
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°.

CONTENTS

* 1 Alternate names * 2 Regular hexateron cartesian coordinates * 3 Projected images * 4 Related uniform 5-polytopes * 5 Other forms * 6 Notes * 7 References * 8 External links

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.

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
Cartesian coordinates
for the vertices of an origin-centered regular hexateron having edge length 2 are: ( 1 15 , 1 10 , 1 6 , 1 3 , 1 ) ( 1 15 , 1 10 , 1 6 , 2 3 , 0 ) ( 1 15 , 1 10 , 3 2 , 0 , 0 ) ( 1 15 , 2 2 5 , 0 , 0 , 0 ) ( 5 3 , 0 , 0 , 0 , 0 ) {displaystyle {begin{aligned}&left({tfrac {1}{sqrt {15}}}, {tfrac {1}{sqrt {10}}}, {tfrac {1}{sqrt {6}}}, {tfrac {1}{sqrt {3}}}, pm 1right)\&left({tfrac {1}{sqrt {15}}}, {tfrac {1}{sqrt {10}}}, {tfrac {1}{sqrt {6}}}, -{tfrac {2}{sqrt {3}}}, 0right)\&left({tfrac {1}{sqrt {15}}}, {tfrac {1}{sqrt {10}}}, -{tfrac {sqrt {3}}{sqrt {2}}}, 0, 0right)\&left({tfrac {1}{sqrt {15}}}, -{tfrac {2{sqrt {2}}}{sqrt {5}}}, 0, 0, 0right)\ width:27.898ex; height:29.843ex;" alt="{displaystyle {begin{aligned}&left({tfrac {1}{sqrt {15}}}, {tfrac {1}{sqrt {10}}}, {tfrac {1}{sqrt {6}}}, {tfrac {1}{sqrt {3}}}, pm 1right)\&left({tfrac {1}{sqrt {15}}}, {tfrac {1}{sqrt {10}}}, {tfrac {1}{sqrt {6}}}, -{tfrac {2}{sqrt {3}}}, 0right)\&left({tfrac {1}{sqrt {15}}}, {tfrac {1}{sqrt {10}}}, -{tfrac {sqrt {3}}{sqrt {2}}}, 0, 0right)\&left({tfrac {1}{sqrt {15}}}, -{tfrac {2{sqrt {2}}}{sqrt {5}}}, 0, 0, 0right)\"> E 7 {displaystyle {tilde {E}}_{7}} =E7+ T 8 {displaystyle {bar {T}}_{8}} =E7++

Coxeter diagram

SYMMETRY

]

ORDER 48 720 23,040 2,903,040 ∞

GRAPH

- -

NAME 13,-1 130 131 132 133 134

It is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter
Coxeter
as 3k1 series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron .

3k1 dimensional figures SPACE FINITE EUCLIDEAN HYPERBOLIC

N 4 5 6 7 8 9

Coxeter group A3A1 A5 D6 E7 E 7 {displaystyle {tilde {E}}_{7}} =E7+ T 8 {displaystyle {bar {T}}_{8}} =E7++

Coxeter diagram

SYMMETRY

]

ORDER 48 720 46,080 2,903,040 ∞

GRAPH

- -

NAME 31,-1 310 311 321 331 341

The 5-simplex, as 220 polytope is first in dimensional series 22k.

22k figures of n dimensions SPACE FINITE EUCLIDEAN HYPERBOLIC

N 5 6 7 8

Coxeter group A5 E6 E 6 {displaystyle {tilde {E}}_{6}} =E6+ E6++

Coxeter diagram

GRAPH

∞ ∞

NAME 220 221 222 223

The regular 5-simplex
5-simplex
is one of 19 uniform polytera based on the Coxeter group
Coxeter group
, all shown here in A5 Coxeter plane
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 t1 t2 t0,1 t0,2 t1,2 t0,3

t1,3 t0,4 t0,1,2 t0,1,3 t0,2,3 t1,2,3 t0,1,4

t0,2,4 t0,1,2,3 t0,1,2,4 t0,1,3,4 t0,1,2,3,4

OTHER FORMS

The 5-simplex
5-simplex
can also be considered a 5-cell pyramid , 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.

NOTES

* ^ Klitzing, (x3o3o3o3o - hix)

REFERENCES

* T. Gosset : On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900

* H.S.M. Coxeter</