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f5 = 7, f4 = 21, C = 35, F = 35, E = 21, V = 7 (χ=0)

Coxeter group A6, [35], order 5040

Bowers name and (acronym) Heptapeton (hop)

Vertex figure 5-simplex

Circumradius 0.645497

Properties convex, isogonal self-dual

In geometry, a 6-simplex
6-simplex
is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex
5-simplex
5-faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°.

Contents

1 Alternate names 2 As a configuration 3 Coordinates 4 Images 5 Related uniform 6-polytopes 6 Notes 7 References 8 External links

Alternate names[edit] It can also be called a heptapeton, or hepta-6-tope, as a 7-facetted polytope in 6-dimensions. The name heptapeton is derived from hepta for seven facets in Greek and -peta for having five-dimensional facets, and -on. Jonathan Bowers gives a heptapeton the acronym hop.[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]

[

7

6

15

20

15

6

2

21

5

10

10

5

3

3

35

4

6

4

4

6

4

35

3

3

5

10

10

5

21

2

6

15

20

15

6

7

]

displaystyle begin bmatrix begin matrix 7&6&15&20&15&6\2&21&5&10&10&5\3&3&35&4&6&4\4&6&4&35&3&3\5&10&10&5&21&2\6&15&20&15&6&7end matrix end bmatrix

Coordinates[edit] The Cartesian coordinates for an origin-centered regular heptapeton having edge length 2 are:

(

1

/

21

,  

1

/

15

,  

1

/

10

,  

1

/

6

,  

1

/

3

,   ± 1

)

displaystyle left( sqrt 1/21 , sqrt 1/15 , sqrt 1/10 , sqrt 1/6 , sqrt 1/3 , pm 1right)

(

1

/

21

,  

1

/

15

,  

1

/

10

,  

1

/

6

,   − 2

1

/

3

,   0

)

displaystyle left( sqrt 1/21 , sqrt 1/15 , sqrt 1/10 , sqrt 1/6 , -2 sqrt 1/3 , 0right)

(

1

/

21

,  

1

/

15

,  

1

/

10

,   −

3

/

2

,   0 ,   0

)

displaystyle left( sqrt 1/21 , sqrt 1/15 , sqrt 1/10 , - sqrt 3/2 , 0, 0right)

(

1

/

21

,  

1

/

15

,   − 2

2

/

5

,   0 ,   0 ,   0

)

displaystyle left( sqrt 1/21 , sqrt 1/15 , -2 sqrt 2/5 , 0, 0, 0right)

(

1

/

21

,   −

5

/

3

,   0 ,   0 ,   0 ,   0

)

displaystyle left( sqrt 1/21 , - sqrt 5/3 , 0, 0, 0, 0right)

(

12

/

7

,   0 ,   0 ,   0 ,   0 ,   0

)

displaystyle left(- sqrt 12/7 , 0, 0, 0, 0, 0right)

The vertices of the 6-simplex
6-simplex
can be more simply positioned in 7-space as permutations of:

(0,0,0,0,0,0,1)

This construction is based on facets of the 7-orthoplex. Images[edit]

orthographic projections

Ak Coxeter plane A6 A5 A4

Graph

Dihedral symmetry [7] [6] [5]

Ak Coxeter plane A3 A2

Graph

Dihedral symmetry [4] [3]

Related uniform 6-polytopes[edit] The regular 6-simplex
6-simplex
is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.

A6 polytopes

t0

t1

t2

t0,1

t0,2

t1,2

t0,3

t1,3

t2,3

t0,4

t1,4

t0,5

t0,1,2

t0,1,3

t0,2,3

t1,2,3

t0,1,4

t0,2,4

t1,2,4

t0,3,4

t0,1,5

t0,2,5

t0,1,2,3

t0,1,2,4

t0,1,3,4

t0,2,3,4

t1,2,3,4

t0,1,2,5

t0,1,3,5

t0,2,3,5

t0,1,4,5

t0,1,2,3,4

t0,1,2,3,5

t0,1,2,4,5

t0,1,2,3,4,5

Notes[edit]

^ Klitzing, (x3o3o3o3o3o - hop) ^ Coxeter, Regular Polytopes, sec 1.8 Configurations ^ Coxeter, Complex Regular Polytopes, p.117

References[edit]

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. "6D uniform polytopes (polypeta) x3o3o3o3o - hix". 

External links[edit]

Olshevsky, George. "Simplex". Glossary for Hyperspace. Archived from the original on 4 February 2007.  Polytopes of Various Dimensions 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 polyt

.