Tetrahedron	Cube	Octahedron	Truncated tetrahedron	Dodecahedron	Truncated octahedron	Icosahedron	Rhombicuboctahedron. Cantellated cube

Truncated icosahedron	Snub cube	Truncated rhombic triacontahedron	Pentakis icosidodecahedron

Face  Vert  Edge  Gold-  Geo-   Plat-  Archi-  Spher-  Face
                  berg   desic  onic   median  icity   type

   4    4     6            *      *            .671     3    Tetrahedron
   6    8    12                   *            .806     4    Cube
   8    6    12            *      *            .846     3    Octahedron
   8   12    18                          *     .775     6 3  Truncated tetrahedron
  10   16    24     
  12    8    18            *                            3    Triakistetrahedron
  12   20    30    *              *            .910     5    Dodecahedron
  14   12    24                          *     .905     3 4  Cuboctahedron +
  14   24    36                          *     .910     6 4  Truncated octahedron
  14   24    36                          *     .849          Truncated cube +
  16   10    24            *                            3
  18   32    48     
  20   12    30            *      *            .939     3    Icosahedron
  20   36    54     
  24   14    36            *                            3
  26   24    48                          *     .954     4 3  Rhombicuboctahedron. Cantellated cube
  26   48    72                          *     .943          Truncated cuboctahedron
  30   32    60                                              Rhombic triacontahedron. Catalan solid
  32   18    48            *                            3
  32   30    60                          *     .951     5 3  Icosidodecahedron
  32   60    90                          *     .926     3 10 Truncated dodecahedron
  32   60    90
  32   90    60    *                     *     .967     6 5  Truncated icosahedron
  34   64    96
  36   20    54            *                            3
  38   24    60                          *     .965     3 4  Snub cube
  38   72   108
  42   80   120    *                                    6 5  Truncated rhombic triacontahedron
  50   96   144
  52  100   150
  60   32    90            *                            3
  62   60   120                          *     .979          Rhombicosidodecahedron
  62  120   180                          *     .970          Truncated icosidodecahedron
  64   34    96            *                            3
  92   60   150                          *     .982          Snub dodecahedron
  66  128   192
  72   38   108            *                            3
  72  140   210    *
  74  144   216
  80   42   120            *                            3    Pentakis icosidodecahedron
  92  180   270    *                                    6 5
  96   50   144            *                            3
 100   52   150            *                            3
 102  200   300    *
 122               *
 128   66   192            *                            3
 140   72   210            *                            3
 144   74   216            *                            3
 162               *

Euler formula:  Faces + Vertices - Edges = 2

12	32	42	72	92	122	132	162

A Goldberg polyhedron has faces that are either pentagons or hexagons. There are always 12 pentagons. The Goldberg polyhedra with ≤162 faces are shown above.

Face  Vert  Edge  Class  Spher-
                         icity

  12   20    30     1    .910
  32   60    90     2    .967
  42   80   120     1
  72  140   210     3
  92  180   270     1
 122  240   320     2
 132  260   390     3
 162  320   480     1

Vertex truncation

Truncation       Shave vertices
Cantellation     Shave vertices and edges
Chamfer          Shave edges
Runcination      Shave faces
Rectification    Truncation down to the to the midpoint of the edges

Catalan solids are the duals of Archimedian solids.

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© Jason Maron, all rights reserved.

Data from Wikipedia unless otherwise specified.