Polytope families

There are several families of symmetric polytopes with irreducible symmetry which have a member in more than one dimensionality. These are tabulated here with Petrie polygon projection graphs and Coxeter–Dynkin diagrams.

Table of irreducible polytope families
Family
n
n-simplex n-hypercube n-orthoplex n-demicube 1k2 2k1 k21 pentagonal polytope
Group An Bn
I2(p) Dn
E6 E7 E8 F4 G2
Hn
2

Triangle


Square



p-gon
(example: p=7)


Hexagon


Pentagon
3

Tetrahedron


Cube


Octahedron


Tetrahedron
 

Dodecahedron


Icosahedron
4

5-cell

Tesseract



16-cell

Demitesseract



24-cell


120-cell


600-cell
5

5-simplex


5-cube


5-orthoplex


5-demicube
   
6

6-simplex


6-cube


6-orthoplex


6-demicube


122


221
 
7

7-simplex


7-cube


7-orthoplex


7-demicube


132


231


321
 
8

8-simplex


8-cube


8-orthoplex


8-demicube


142


241


421
 
9

9-simplex


9-cube


9-orthoplex


9-demicube
 
10

10-simplex


10-cube


10-orthoplex


10-demicube
 
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds