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User:Tomruen/List of rotational symmetry polyhedral sets

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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. Cnv - n-fold pyramidal symmetry
  2. Dnh - n-fold prismatic symmetry
  3. Dnv - 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