User:Tomruen/List of rotational symmetry polyhedral sets

There are an unlimited number of polyhedral sets which can be defined with rotational symmetry.

This list shows the some of the simplest sets which appear within the convex uniform polyhedra, Johnson solids and related duals.

Each set has one polyhedron for each polygon, and each is prefixed by its generating polygon: triangular, square, pentagonal, hexagonal, etc. Some have degenerate (linear) forms as well where the symmetry has collapsed to an edge.

There are three symmetry classes:

1. C_nv - n-fold pyramidal symmetry
2. D_nh - n-fold prismatic symmetry
3. D_nv - n-fold antiprismatic symmetry

Set	Symmetry	V	E	F	Example(s)	Dual set
Pyramids	Cnv	n+1	2n	n+1	Pentagonal pyramid	Self-dual
Frustums (Truncated pyramids)	Cnv	2n	3n	n+2	(Pentagonal frustum)	Bipyramids
Cupolae	Cnv	3n	5n	2n+2	Pentagonal cupola	--
Prisms	Dnh	2n	3n	n+2	Pentagonal prism	Bipyramids
Bipyramids	Dnh	n+2	3n	2n	Octahedron	Prisms
--	Dnh	3n+2	6n	3n	Rhombic dodecahedron --	(below)
--	Dnh	3n	6n	3n+2	Cuboctahedron	--
Bifrustums (Truncated bipyramids)	Dnh	3n	5n	2n+2	Pentagonal bifrustum	Elongated bipyramids
Elongated dipyramids	Dnh	2n+2	5n	3n	Elongated hexagonal dipyramid J14, J15, J16	Bifrustums
Orthobicupolae	Dnh	4n	8n	4n+2	J27, J28, J30	(below)
Dual orthobicupolae	Dnh	4n+2	8n	4n	File:Trapezo-rhombic dodecahedron	Orthobicupolae
Antiprisms	Dnd	2n	4n	2n+2	Pentagonal antiprism	Trapezohedra
Trapezohedra	Dnd	2n+2	4n	2n	Pentagonal trapezohedron	Antiprisms
Truncated trapezohedra	Dnd	4n	6n	2n+2	Dodecahedron (Truncated pentagonal trapezohedron)	Gyroelongated dipyramids
Gyroelongated dipyramids	Dnd	2n+2	6n	4n	Icosahedron (Gyroelongated pentagonal dipyramid) (J17)	Truncated trapezohedra
Gyrobicupolae	Dnd	4n	8n	4n+2	Cuboctahedron (Triangular gyrobicupola) J29, J31	(below)
Dual gyrobicupolae	Dnd	4n+2	8n	4n	Rhombic dodecahedron	Gyrobicupolae