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In 7-dimensional geometry, a 7-simplex
7-simplex
is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell
5-cell
5-faces, 28 5-simplex
5-simplex
6-faces, and 8 6-simplex
6-simplex
7-faces. Its dihedral angle is cos−1(1/7), or approximately 81.79°.

Contents

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

Alternate names[edit] It can also be called an octaexon, or octa-7-tope, as an 8-facetted polytope in 7-dimensions. The name octaexon is derived from octa for eight facets in Greek and -ex for having six-dimensional facets, and -on. Jonathan Bowers gives an octaexon the acronym oca.[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]

[

8

7

21

35

35

21

7

2

28

6

15

20

15

6

3

3

56

5

10

10

5

4

6

4

70

4

6

4

5

10

10

5

56

3

3

6

15

20

15

6

28

2

7

21

35

35

21

7

8

]

displaystyle begin bmatrix begin matrix 8&7&21&35&35&21&7\2&28&6&15&20&15&6\3&3&56&5&10&10&5\4&6&4&70&4&6&4\5&10&10&5&56&3&3\6&15&20&15&6&28&2\7&21&35&35&21&7&8end matrix end bmatrix

Coordinates[edit] The Cartesian coordinates of the vertices of an origin-centered regular octaexon having edge length 2 are:

(

1

/

28

,  

1

/

21

,  

1

/

15

,  

1

/

10

,  

1

/

6

,  

1

/

3

,   ± 1

)

displaystyle left( sqrt 1/28 , sqrt 1/21 , sqrt 1/15 , sqrt 1/10 , sqrt 1/6 , sqrt 1/3 , pm 1right)

(

1

/

28

,  

1

/

21

,  

1

/

15

,  

1

/

10

,  

1

/

6

,   − 2

1

/

3

,   0

)

displaystyle left( sqrt 1/28 , sqrt 1/21 , sqrt 1/15 , sqrt 1/10 , sqrt 1/6 , -2 sqrt 1/3 , 0right)

(

1

/

28

,  

1

/

21

,  

1

/

15

,  

1

/

10

,   −

3

/

2

,   0 ,   0

)

displaystyle left( sqrt 1/28 , sqrt 1/21 , sqrt 1/15 , sqrt 1/10 , - sqrt 3/2 , 0, 0right)

(

1

/

28

,  

1

/

21

,  

1

/

15

,   − 2

2

/

5

,   0 ,   0 ,   0

)

displaystyle left( sqrt 1/28 , sqrt 1/21 , sqrt 1/15 , -2 sqrt 2/5 , 0, 0, 0right)

(

1

/

28

,  

1

/

21

,   −

5

/

3

,   0 ,   0 ,   0 ,   0

)

displaystyle left( sqrt 1/28 , sqrt 1/21 , - sqrt 5/3 , 0, 0, 0, 0right)

(

1

/

28

,   −

12

/

7

,   0 ,   0 ,   0 ,   0 ,   0

)

displaystyle left( sqrt 1/28 , - sqrt 12/7 , 0, 0, 0, 0, 0right)

(

7

/

4

,   0 ,   0 ,   0 ,   0 ,   0 ,   0

)

displaystyle left(- sqrt 7/4 , 0, 0, 0, 0, 0, 0right)

More simply, the vertices of the 7-simplex
7-simplex
can be positioned in 8-space as permutations of (0,0,0,0,0,0,0,1). This construction is based on facets of the 8-orthoplex. Images[edit]

7- Simplex
Simplex
in 3D

Model created using straws (edges) and plasticine balls (vertices) in triakis tetrahedral envelope

7- Simplex
Simplex
as an Amplituhedron
Amplituhedron
Surface

7-simplex
7-simplex
to 3D with camera perspective showing hints of its 2D Petrie projection

orthographic projections

Ak Coxeter plane A7 A6 A5

Graph

Dihedral symmetry [8] [7] [6]

Ak Coxeter plane A4 A3 A2

Graph

Dihedral symmetry [5] [4] [3]

Related polytopes[edit] This polytope is a facet in the uniform tessellation 331 with Coxeter-Dynkin diagram:

This polytope is one of 71 uniform 7-polytopes with A7 symmetry.

A7 polytopes

t0

t1

t2

t3

t0,1

t0,2

t1,2

t0,3

t1,3

t2,3

t0,4

t1,4

t2,4

t0,5

t1,5

t0,6

t0,1,2

t0,1,3

t0,2,3

t1,2,3

t0,1,4

t0,2,4

t1,2,4

t0,3,4

t1,3,4

t2,3,4

t0,1,5

t0,2,5

t1,2,5

t0,3,5

t1,3,5

t0,4,5

t0,1,6

t0,2,6

t0,3,6

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

t1,2,3,5

t0,1,4,5

t0,2,4,5

t1,2,4,5

t0,3,4,5

t0,1,2,6

t0,1,3,6

t0,2,3,6

t0,1,4,6

t0,2,4,6

t0,1,5,6

t0,1,2,3,4

t0,1,2,3,5

t0,1,2,4,5

t0,1,3,4,5

t0,2,3,4,5

t1,2,3,4,5

t0,1,2,3,6

t0,1,2,4,6

t0,1,3,4,6

t0,2,3,4,6

t0,1,2,5,6

t0,1,3,5,6

t0,1,2,3,4,5

t0,1,2,3,4,6

t0,1,2,3,5,6

t0,1,2,4,5,6

t0,1,2,3,4,5,6

Notes[edit]

^ Klitzing, Richard. "7D uniform polytopes (polyexa) x3o3o3o3o3o - oca".  ^ Coxeter, Regular Polytopes, sec 1.8 Configurations ^ Coxeter, Complex Regular Polytopes, p.117

External links[edit]

Glossary for hyperspace, George Olshevsky. 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

.