| 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
 |