written by Rene K. Mueller, Copyright (c) 2007, last updated Wed, January 9, 2008
Updates
Tue, April 17, 2007: 7 origins of Waterman polyhedra (CCP) included (W1-100, plus a few more for origin 1, 2, 3, 3*, 4, 5, 6) and an interactive viewer for those.
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Fri, April 6, 2007: Added Waterman Polyhedra, a parametrical created polyhedra of a defined complexity, which is a nice feature.
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Sat, February 10, 2007: Added Johnson Solid, names & models rendered, listing vertices, edges & faces for now only (no calculators).
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Sun, February 4, 2007: Compiled information for Platonic and Archimedean Solids (subset of Uniform Polyhedra) from various sources and additionally listed V, A, and rinner and router with a calculator for each platonic & archimedean solid.
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Introduction
After a some research I composed following comprehensive overview:
5 Platonic Solids, regular faces: triangle, square or pentagon only
13 Archimedean Solids, semi-regular faces: triangle, square and pentagon
92 Johnson Solids, semi-regular faces: triangle, square, penta-, hexa-, octa- and decagons
80 Uniform Polyhedra, incl. platonic & archimedean solid and many concave forms not suitable for habitats
Waterman Polyhedra, parametrically created, which include also some platonic and archimedean solids
So a total of 110 convex and 62 concave polyhedra plus apprx. 500 convex parametrical created Waterman polyhedra are listed on the next pages of this document.
A hint on name convention:
1: mono
2: di
3: tri
4: tetra
5: penta
6: hexa
7: hepta
8: octa
9: ennéa
10: deca
11: hendeca
12: dodeca
13: triskaideca
14: tetrakaideca
20: ico
24: icotetra
30: triaconta
60: hexaconta
thereby polygons ('gon' from greek 'gonu' (knee or angle)) of n-sides:
1: monogon
2: digon
3: trigon, triangle
4: tetragon, quadrilateral
5: pentagon
6: hexagon
7: heptahon
8: octagon
9: enneagon
10: decagon
11: hendecagon
12: dodecagon
13: triskaidecagon
14: tetrakaidecagon, tetradecagon
15: pentakaidecagon, pentadecagon
16: hexakaidecagon, hexadecagon
17: heptakaidecagon
18: octakaidecagon
19: enneakaidecagon
20: icosagon
21: icosikaihenagon, icosihenagon
22: icosikaidigon
23: icosikaitrigon
24: icosikaitetragon
25: icosikaipentagon
26: icosikaihexagon
27: icosikaiheptagon
28: icosikaioctagon
29: icosikaienneagon
30: triacontagon
31: triacontakaihenagon
32: triacontakaidigon
33: triacontakaitrigon
34: triacontakaitetragon
35: triacontakaipentagon
36: triacontakaihexagon
37: triacontakaiheptagon
38: triacontakaioctagon
39: triacontakaienneagon
40: tetracontagon
41: tetracontakaihenagon
42: tetracontakaidigon
43: tetracontakaitrigon
44: tetracontakaitetragon
45: tetracontakaipentagon
46: tetracontakaihexagon
47: tetracontakaiheptagon
48: tetracontakaioctagon
49: tetracontakaienneagon
50: pentacontagon ...
60: hexacontagon ...
70: heptacontagon ...
80: octacontagon ...
90: enneacontagon ...
100: hectogon, hecatontagon
1000: chiliagon
10000: myriagon
Based on the study here of suitable solids or polyhedra I extract geodesic variants and from there I sort out those finally which are suitable for dome construction.
Note: The page structure might change depending how much info I will include in the future, e.g. multiple pages or separate pages for each form. Let's see.
They are called "platonic" as Plato (400 BC) described them in Timaeus, but those forms have been discovered in Scotland and are dated 2000-3200 BC and relate to the "neolithic" or "new stone age" people of that time (see also George Hart: Neolithic Carved Stone Polyhedra).
For more infos see Google: Carved Stone Balls.
The base information is compiled from Wikipedia and Mathworld and "Uniform solution for uniform polyhedra" by Zvi Har'El, merged all that information and additionally listed V, A, and rinner and router with a calculator.
I also plan to comment on each form, and suggest usage for a temporary building, especially if further triangulation like with the icosahedron to geodesic domes.
Symbols:
s = strut length
V = volume
A = surface area
rinner = inner radius or inradius
router = outer radius or circumradius
ravg = (rinner + router) / 2
Duals of a solid is when the solids' vertices become faces and vice-versa.
Edit the fields with yellow background and hit ENTER or TAB to (re)calculate the other values.
Note: I still need to cross-check all expressions (V, A, rinner and router) by other sources, so don't rely on it yet.
Tetrahedron
Tetrahedron
Uniform Polyhedron: U1
Platonic Solid
Platonic Element: Fire
Vertices: 4
Edges: 6
Faces: 4
Wythoff symbol: 3|2 3
Symmetry Group: {3, 3, 3}
Vertex Configuration: tetrahedral
Dual: tetrahedron
V: s3 / 12 * √2
A: s2 * √3
rinner: s / 12 * √6
router: s / 4 * √6
h: s / 3 * √6
Truncated Tetrahedron
Truncated Tetrahedron
Uniform Polyhedron: U2
Archimedean Solid: A13
Vertices: 12
Edges: 18
Faces: 8
Wythoff symbol: 2 3|3
Symmetry Group: tetrahedral
Vertex Configuration: {6, 6, 3}
Dual: triakis tetrahedron
V: s3 * 23/12 * √2
A: s2 * 7 * √3
rinner: s * 9 / 44 * √22
router: s / 4 * √22
Octahedron
Octahedron
Uniform Polyhedron: U5
Platonic Solid
Platonic Element: Air
Vertices: 6
Edges: 12
Faces: 8
Wythoff symbol: 4|2 3
Symmetry Group: octahedral
Vertex Configuration: {3, 3, 3, 3}
Dual: cube
V: s3 / 3 * √2
A: s2 * 8 / 4 * √3
rinner: s / 6 * √6
router: s / 2 * √2
Half of an octahedron is the classic pyramid.
Cube
Cube
Uniform Polyhedron: U6
aka Hexahedron
Platonic Solid
Platonic Element: Earth
Vertices: 8
Edges: 12
Faces: 6
Wythoff symbol: 3|2 4
Symmetry Group: octahedral
Vertex Configuration: {4, 4, 4}
Dual: octahedron
V: s3
A: s2 * 6
rinner: s / 2
router: s / 2 * √3
It's one of the main forms of western architecture, and one of the main zonohedra (aka parallelohedron), the ability to tile space without holes. There are many more possible, more complex with more faces.
, but those forms have been discovered in Scotland and are dated 2000-3200 BC and relate to the "neolithic" or "new stone age" people of that time (see also George Hart: Neolithic Carved Stone Polyhedra 
