In fivedimensional geometry , a
5simplex
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°.
CONTENTS
* 1 Alternate names
* 2 Regular hexateron cartesian coordinates
* 3 Projected images
* 4 Related uniform 5polytopes
* 5 Other forms
* 6 Notes
* 7 References
* 8 External links
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.
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
Cartesian coordinates for the vertices of an origincentered
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
4dimensional case exists as 3sphere 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 5simplex, 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
5simplex
5simplex 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
5simplex
5simplex can also be considered a
5cell pyramid , 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.
NOTES
* ^ Klitzing, (x3o3o3o3o  hix)
REFERENCES
* T. Gosset : On the Regular and SemiRegular Figures in Space of n
Dimensions, Messenger of Mathematics, Macmillan, 1900
* H.S.M.
Coxeter</