| Symmetry | Triangular dihedral symmetry
 | Tetrahedral
 | Octahedral
 | Icosahedral
 | p6m symmetry
 | [3,7] symmetry
 | [3,8] symmetry
 |
|---|
Starting solid
Operation | Symbol
{p,q}
    | Triangular hosohedron
{2,3}
 | Triangular dihedron
{3,2}
 | Tetrahedron
{3,3}
 | Cube
{4,3}
 | Octahedron
{3,4}
 | Dodecahedron
{5,3}
 | Icosahedron
{3,5}
 | Hexagonal tiling
{6,3}
 | Triangular tiling
{3,6}
 | Heptagonal tiling
{7,3}
 | Order-7 triangular tiling
{3,7}
 | Octagonal tiling
{8,3}
 | Order-8 triangular tiling
{3,8}
 |
|---|
| Truncation (t) | t{p,q}
| triangular prism
 | truncated triangular dihedron
 (Half of the "edges" count as degenerate digon faces. The other half are normal edges.) | truncated tetrahedron
 | truncated cube
 | truncated octahedron
 | truncated dodecahedron
 | truncated icosahedron
 | Truncated hexagonal tiling
 | Truncated triangular tiling
 | Truncated heptagonal tiling
 | Truncated order-7 triangular tiling
 | Truncated octagonal tiling
 | Truncated order-8 triangular tiling
 |
|---|
Rectification (r)
Ambo (a) | r{p,q}
| tridihedron
(All of the "edges" count as degenerate digon faces.) | tetratetrahedron
 | cuboctahedron
 | icosidodecahedron
 | Trihexagonal tiling
 | Triheptagonal tiling
 | Trioctagonal tiling
 |
|---|
Bitruncation (2t)
Dual kis (dk) | 2t{p,q}
| truncated triangular dihedron
(Half of the "edges" count as degenerate digon faces. The other half are normal edges.) | triangular prism
| truncated tetrahedron
 | truncated octahedron
| truncated cube
| truncated icosahedron
| truncated dodecahedron
| truncated triangular tiling
 | truncated hexagonal tiling
 | Truncated order-7 triangular tiling
| Truncated heptagonal tiling
| Truncated order-8 triangular tiling
| Truncated octagonal tiling
|
|---|
Birectification (2r)
Dual (d) | 2r{p,q}
| triangular dihedron
{3,2}
| triangular hosohedron
{2,3}
| tetrahedron
 | octahedron
| cube
| icosahedron
| dodecahedron
| triangular tiling
| hexagonal tiling
| Order-7 triangular tiling
| Heptagonal tiling
| Order-8 triangular tiling
| Octagonal tiling
|
|---|
Cantellation (rr)
Expansion (e) | rr{p,q}
| triangular prism
(The "edge" between each pair of tetragons counts as a degenerate digon face. The other edges (the ones between a trigon and a tetragon) are normal edges.) | rhombitetratetrahedron
 | rhombicuboctahedron
 | rhombicosidodecahedron
 | rhombitrihexagonal tiling
 | Rhombitriheptagonal tiling
 | Rhombitrioctagonal tiling
 |
|---|
Snub rectified (sr)
Snub (s) | sr{p,q}
 | triangular antiprism
 (Three yellow-yellow "edges", no two of which share any vertices, count as degenerate digon faces. The other edges are normal edges.) | snub tetratetrahedron
 | snub cuboctahedron
 | snub icosidodecahedron
 | snub trihexagonal tiling
 | Snub triheptagonal tiling
 | Snub trioctagonal tiling
 |
|---|
Cantitruncation (tr)
Bevel (b) | tr{p,q}
| hexagonal prism
 | truncated tetratetrahedron
 | truncated cuboctahedron
 | truncated icosidodecahedron
 | truncated trihexagonal tiling
 | Truncated triheptagonal tiling
 | Truncated trioctagonal tiling
 |
|---|
