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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-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces. Its dihedral angle is cos−1(1/7), or approximately 81.79°.

CONTENTS

* 1 Alternate names * 2 Coordinates * 3 Images * 4 Related polytopes * 5 Notes * 6 External links

ALTERNATE NAMES

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.

COORDINATES

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
8-orthoplex
.

IMAGES

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

AK COXETER PLANE A4 A3 A2

GRAPH

DIHEDRAL SYMMETRY

RELATED POLYTOPES

This polytope is a facet in the uniform tessellation 331 with Coxeter-Dynkin diagram
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

* ^ Klitzing, Richard. "7D uniform polytopes (polyexa) x3o3o3o3o3o - oca".