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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
5-cell
facets. It has a dihedral angle of cos−1(1/5), or approximately 78.46°.

Contents

1 Alternate names 2 As a configuration 3 Regular hexateron cartesian coordinates 4 Projected images 5 Related uniform 5-polytopes 6 Other forms 7 Notes 8 References 9 External links

Alternate names[edit] 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[edit] The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.[2][3]

[

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

]

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&6end matrix end bmatrix

Regular hexateron cartesian coordinates[edit] The hexateron can be constructed from a 5-cell
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)\[5pt]&left( tfrac 1 sqrt 15 , tfrac 1 sqrt 10 , tfrac 1 sqrt 6 , - tfrac 2 sqrt 3 , 0right)\[5pt]&left( tfrac 1 sqrt 15 , tfrac 1 sqrt 10 , - tfrac sqrt 3 sqrt 2 , 0, 0right)\[5pt]&left( tfrac 1 sqrt 15 , - tfrac 2 sqrt 2 sqrt 5 , 0, 0, 0right)\[5pt]&left(- tfrac sqrt 5 sqrt 3 , 0, 0, 0, 0right)end aligned

The vertices of the 5-simplex
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
6-orthoplex
or rectified 6-cube
6-cube
respectively. Projected images[edit]

orthographic projections

Ak Coxeter
Coxeter
plane A5 A4

Graph

Dihedral symmetry [6] [5]

Ak Coxeter
Coxeter
plane A3 A2

Graph

Dihedral symmetry [4] [3]

Stereographic projection
Stereographic projection
4D to 3D of Schlegel diagram
Schlegel diagram
5D to 4D of hexateron.

Related uniform 5-polytopes[edit] It is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter
Coxeter
as 13k series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral dihedron.

13k 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 [3−1,3,1] [30,3,1] [31,3,1] [32,3,1] [[33,3,1]] [34,3,1]

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 [3−1,3,1] [30,3,1] [[31,3,1]] = [4,3,3,3,3] [32,3,1] [33,3,1] [34,3,1]

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 [3,3,3,3] Coxeter
Coxeter
group, all shown here in A5 Coxeter
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[edit] The 5-simplex
5-simplex
can also be considered a 5-cell
5-cell
pyramid, constructed as a 5-cell
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
5-cell
cells. Notes[edit]

^ Klitzing, (x3o3o3o3o - hix) ^ Coxeter, Regular Polytopes, sec 1.8 Configurations ^ Coxeter, Complex Regular Polytopes, p.117

References[edit]

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

Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]

(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591] (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]

John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1) Norman Johnson Uniform Polytopes, Manuscript (1991)

N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)

Klitzing, Richard. "5D uniform polytopes (polytera) x3o3o3o3o - hix". 

External links[edit]

Olshevsky, George. "Simplex". Glossary for Hyperspace. Archived from the original on 4 February 2007.  Polytopes of Various Dimensions, Jonathan Bowers Multi-dimensional Glossary

v t e

Fundamental convex regular and uniform polytopes in dimensions 2–10

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn

Regular polygon Triangle Square p-gon Hexagon Pentagon

Uniform polyhedron Tetrahedron Octahedron
Octahedron
• Cube Demicube

Dodecahedron • Icosahedron

Uniform 4-polytope 5-cell 16-cell
16-cell
• Tesseract Demitesseract 24-cell 120-cell
120-cell
• 600-cell

Uniform 5-polytope 5-simplex 5-orthoplex
5-orthoplex
• 5-cube 5-demicube

Uniform 6-polytope 6-simplex 6-orthoplex
6-orthoplex
• 6-cube 6-demicube 122 • 221

Uniform 7-polytope 7-simplex 7-orthoplex
7-orthoplex
• 7-cube 7-demicube 132 • 231 • 321

Uniform 8-polytope 8-simplex 8-orthoplex
8-orthoplex
• 8-cube 8-demicube 142 • 241 • 421

Uniform 9-polytope 9-simplex 9-orthoplex
9-orthoplex
• 9-cube 9-demicube

Uniform 10-polytope 10-simplex 10-orthoplex
10-orthoplex
• 10-cube 10-demicube

Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope

Topics: Polytope
Polytope
families • Regular polytope
Regular polytope
• List of regular polytopes a

.