The following tables list various properties of the isotoxal polyhedra and compounds. Each polyhedron has a label derived from its edge type. Columns "F", "E", and "V" contain counts of all the faces, edges, and vertices respectively. Columns "Fn" contain counts of faces of type n.
For the first two tables, links go to a separate page showing details of that polyhedron. Others proceed directly to an interactive application that lets you explore virtual models of all of these in detail. You might need to enable WebGL in your browser in order to use it.
Isotoxal polyhedra that are neither vertex-transitive nor face-transitive.
| Polyhedron | Face Type 1 | Face Type 2 | F1 | F2 | F | E | V | Sides | Genus |
| O32b_1 | aligned [8/3] | [4] | 6 | 12 | 18 | 48 | 20 | 1 | 12 |
| O32b_2 | [6/2] | [4] | 8 | 12 | 20 | 48 | 20 | 2 | 5 |
| I52c_1 | [6/2] | [4] | 20 | 30 | 50 | 120 | 42 | 1 | 30 |
| I52c_2 | aligned [10/3] | [4] | 12 | 30 | 42 | 120 | 42 | 1 | 38 |
| I52c_3 | aligned [10/4] | [4] | 12 | 30 | 42 | 120 | 42 | 1 | 38 |
| I52c_4 | [6/2] | [4] | 20 | 30 | 50 | 120 | 42 | 1 | 30 |
| I32d_1 | outer [10/4] | [4] | 12 | 30 | 42 | 120 | 50 | 1 | 30 |
| I32d_2 | inner [10/3] | [4] | 12 | 30 | 42 | 120 | 50 | 1 | 30 |
| I22a_1 | aligned [10] | medial [10] | 12 | 12 | 24 | 120 | 60 | 1 | 38 |
| I22b_1 | inner [6] | inner [6] | 20 | 20 | 40 | 120 | 60 | 1 | 22 |
| I22c_1 | overlapped [10/3] | outer [10/2] | 12 | 12 | 24 | 120 | 60 | 1 | 38 |
Vertex-intransitive isotoxal polyhedra that are face-transitive.
| Polyhedron | Face Type | F | E | V | Sides | Genus | Density |
| O43a_1 | [4] | 12 | 24 | 14 | 2 | 0 | 1 |
| I55a_1 | [4] | 30 | 60 | 24 | 2 | 4 | 3 |
| I55a_2 | inner [6] | 20 | 60 | 24 | 2 | 9 | 4 |
| I53a_1 | aligned [10] | 12 | 60 | 32 | 2 | 9 | 4 |
| I53a_2 | outer [6] | 20 | 60 | 32 | 2 | 5 | 2 |
| I53a_3 | [4] | 30 | 60 | 32 | 2 | 0 | 1 |
| I53a_4 | aligned [10] | 12 | 60 | 32 | 2 | 9 | 2 |
| I53b_1 | [4] | 30 | 60 | 32 | 2 | 0 | 7 |
| I53b_2 | overlapped [10/3] | 12 | 60 | 32 | 2 | 9 | 4 |
| I53c_1 | overlapped [10/3] | 12 | 60 | 32 | 2 | 9 | 10 |
| I53c_2 | [6/2] | 20 | 60 | 32 | 2 | 5 | 6 |
Vertex-transitive isotoxal polyhedra. These are very well known but their isotoxal nature is rarely mentioned.
| Vertex Configuration | Face Type1 | Face Type2 | F1 | F2 | F | E | V | Sides | Genus | Density |
| (3,3,3) | {3} | 4 | 4 | 6 | 4 | 2 | 0 | 1 | ||
| (4,4,4) | {4} | 6 | 6 | 12 | 8 | 2 | 0 | 1 | ||
| (3,3,3,3) | {3} | 8 | 8 | 12 | 6 | 2 | 0 | 1 | ||
| (5,5,5) | {5} | 12 | 12 | 30 | 20 | 2 | 0 | 1 | ||
| (3,3,3,3,3) | {3} | 20 | 20 | 30 | 12 | 2 | 0 | 1 | ||
| (5/2,5/2,5/2,5/2,5/2) | {5/2} | 12 | 12 | 30 | 12 | 2 | 4 | 3 | ||
| (5,5,5,5,5)/2 | {5} | 12 | 12 | 30 | 12 | 2 | 4 | 3 | ||
| (5/2,5/2,5/2) | {5/2} | 12 | 12 | 30 | 20 | 2 | 0 | 7 | ||
| (3,3,3,3,3)/2 | {3} | 20 | 20 | 30 | 12 | 2 | 0 | 7 | ||
| (3,±4,-3,±4) | {3} | {4} | 4 | 3 | 7 | 12 | 6 | 1 | 1 | -- |
| (3,4,3,4) | {3} | {4} | 8 | 6 | 14 | 24 | 12 | 2 | 0 | 1 |
| (3,±6,-3,±6) | {3} | {6} | 8 | 4 | 12 | 24 | 12 | 2 | 1 | -- |
| (4,±6,-4,±6) | {4} | {6} | 6 | 4 | 10 | 24 | 12 | 1 | 4 | -- |
| (3,5,3,5) | {3} | {5} | 20 | 12 | 32 | 60 | 30 | 2 | 0 | 1 |
| (3,±10,-3,±10) | {3} | {10} | 20 | 6 | 26 | 60 | 30 | 1 | 6 | -- |
| (5,±10,-5,±10) | {5} | {10} | 12 | 6 | 18 | 60 | 30 | 1 | 14 | -- |
| (5/2,5,5/2,5) | {5/2} | {5} | 12 | 12 | 24 | 60 | 30 | 2 | 4 | 3 |
| (5/2,±6,-5/2,±6) | {5/2} | {6} | 12 | 10 | 22 | 60 | 30 | 1 | 10 | -- |
| (5,±6,-5,±6) | {5} | {6} | 12 | 10 | 22 | 60 | 30 | 1 | 10 | -- |
| (5/2,3,5/2,3) | {5/2} | {3} | 12 | 20 | 32 | 60 | 30 | 2 | 0 | 7 |
| (5/2,±10/3,-5/2,±10/3) | {5/2} | {10/3} | 12 | 6 | 18 | 60 | 30 | 1 | 14 | -- |
| (3,±10/3,-3,±10/3) | {3} | {10/3} | 20 | 6 | 26 | 60 | 30 | 1 | 6 | -- |
| (5/2,3,5/2,3,5/2,3) | {5/2} | {3} | 12 | 20 | 32 | 60 | 20 | 2 | 5 | 2 |
| (-5/2,5,-5/2,5,-5/2,5) | {5/2} | {5} | 12 | 12 | 24 | 60 | 20 | 2 | 9 | 4 |
| (3,5,3,5,3,5)/2 | {3} | {5} | 20 | 12 | 32 | 60 | 20 | 2 | 5 | 6 |
Isotoxal compounds.
| Compound | Face Type 1 | Face Type 2 | F1 | F2 | F | E | V | Binding |
| 5 Rhombic Dodecahedra | [4] | 60 | 60 | 120 | 50 | full | ||
| 2 Tetrahedra | {3} | 8 | 8 | 12 | 8 | free | ||
| 5 Tetrahedra | {3} | 20 | 20 | 30 | 20 | free | ||
| 10 Tetrahedra | {3} | 40 | 40 | 60 | 20 | full | ||
| 5 Cubes | {4} | 30 | 30 | 60 | 20 | full | ||
| 5 Octahedra | {3} | 40 | 40 | 60 | 30 | free | ||
| 5 Tetrahemihexahedra | {3} | {4} | 20 | 15 | 35 | 60 | 30 | free |
| 5 Cuboctahedra | {3} | {4} | 40 | 30 | 70 | 120 | 60 | free |
| 5 Cubohemioctahedra | {4} | {6} | 30 | 20 | 50 | 120 | 60 | free |
| 5 Octahemioctahedra | {3} | {6} | 40 | 20 | 60 | 120 | 60 | free |
| Home | Enumeration (PDF) | Nomenclature |