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Polyhedra Notes

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.

Fri, April 6, 2007: Added Waterman Polyhedra, a parametrical created polyhedra of a defined complexity, which is a nice feature.

Sat, February 10, 2007: Added Johnson Solid, names & models rendered, listing vertices, edges & faces for now only (no calculators).

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.

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.

Platonic Solids


Tetrahedron

Octahedron

Cube

Icosahedron

Dodecahedron


Neolithic Carved Stones
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 .

Archimedean Solids


Truncated Tetrahedron

Cuboctahedron

Truncated Octahedron

Truncated Cube

Rhombicuboctahedron

Truncated Cuboctahedron

Snub Cube

Icosidodecahedron

Truncated Icosahedron

Truncated Dodecahedron

Rhombicosidodecahedron

Truncated Icosidodecahedron

Snub Dodecahedron

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


s = , V = , A = , rinner = , router = , ravg =

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


s = , V = , A = , rinner = , router = , ravg =

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


s = , V = , A = , rinner = , router = , ravg =

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


s = , V = , A = , rinner = , router = , ravg =

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.

Cuboctahedron


Cuboctahedron
  • Uniform Polyhedron: U7
  • Archimedean Solid: A1
  • Vertices: 12
  • Edges: 24
  • Faces: 14
  • Wythoff symbol: 2|3 4
  • Symmetry Group: octahedral
  • Vertex Configuration: {3, 4, 3, 4}
  • Dual: rhombic dodecahedron
  • V: s3 * 5/3 * √2
  • A: s2 * (6 + 2 * √3)
  • rinner: s * 3/4
  • router: s


s = , V = , A = , rinner = , router = , ravg =

Truncated Octahedron


Truncated Octahedron
  • Uniform Polyhedron: U8
  • Archimedean Solid: A12
  • Vertices: 24
  • Edges: 36
  • Faces: 14
  • Wythoff symbol: 2 4|3
  • Symmetry Group: octahedral
  • Vertex Configuration: {6, 6, 4}
  • Dual: tetrakis hexahedron
  • V: s3 * 8 * √2
  • A: s2 * (6 + 12 * √3)
  • rinner: s * 9/20 * √10
  • router: s / 2 * √10


s = , V = , A = , rinner = , router = , ravg =

Truncated Cube


Truncated Cube
  • Uniform Polyhedron: U9
  • Archimedean Solid: A9
  • Vertices: 24
  • Edges: 36
  • Faces: 14
  • Wythoff symbol: 2 3|4
  • Symmetry Group: octahedral
  • Vertex Configuration: {8, 8, 3}
  • Dual: triakis octahedron
  • V: s3 / 3 * (21 + 14 * √2)
  • A: s2 * 2 * (6 + 6 * √2 + √3)
  • rinner: s / 17 * (5 + 2 * √2 * √(7 + 4 * √2))
  • router: s / 2 * √(7 + 4 * √2)


s = , V = , A = , rinner = , router = , ravg =

Rhombicuboctahedron


Rhombicuboctahedron
  • Uniform Polyhedron: U10
  • aka Small Rhombicuboctahedron
  • Archimedean Solid: A6
  • Vertices: 24
  • Edges: 48
  • Faces: 26
  • Wythoff symbol: 3 4|2
  • Symmetry Group: octahedral
  • Vertex Configuration: {4, 3, 4, 4}
  • Dual: deltoidal icositetrahedron
  • V: s3 / 3 * (12 + 10 * √2)
  • A: s2 * (18 + 2 * √3)
  • rinner: s / 17 * (6 + √2) * √(5 + 2 * √2)
  • router: s / 2 * √(5 + 2 * √2)


s = , V = , A = , rinner = , router = , ravg =

Truncated Cuboctahedron


Truncated Cuboctahedron
  • Uniform Polyhedron: U11
  • aka Great Rhombicuboctahedron
  • Archimedean Solid: A3
  • Vertices: 48
  • Edges: 72
  • Faces: 26
  • Wythoff symbol: 2 3 4|
  • Symmetry Group: octahedral
  • Vertex Configuration: {4, 6, 8}
  • Dual: disdyakis dodecahedron
  • V: s3 * (22 + 14 * √2)
  • A: s2 * 12 * (2 + √2 + √3)
  • rinner: s * 3/97 * (14 + √2) * √(13 + 6 * √2)
  • router: s / 2 * √(13 + 6 * √2)


s = , V = , A = , rinner = , router = , ravg =

Snub Cube


Snub Cube
  • Uniform Polyhedron: U12
  • aka Cubus Simus
  • aka Snub Cuboctahedron
  • Archimedean Solid: A7
  • Vertices: 24
  • Edges: 60
  • Faces: 38
  • Wythoff symbol: |2 3 4
  • Symmetry Group: octahedral
  • Vertex Configuration: {3, 3, 3, 3, 4}
  • Dual: pentagonal icositetrahedron
  • t: 1/3 * (1 + (10-3*√33)(1/3) + (19+3*√33)(1/3) )
  • V: s3 * ( 8/3 * √(3 * (3-t)/(4*(2-t)) - 1) + √(4 * (3-t)/(4*(2-t)) -2) )
  • V: s3 * √((613 * t + 203)/(9*(35*t-62)))
  • A: s2 * (6 + 8 * √3)
  • rinner: s * √(abs(1-t)/(4*(2-t)))
  • router: s * √((3-t)/(4*(2-t)))


s = , V = , A = , rinner = , router = , ravg =

Icosahedron


Icosahedron
  • Uniform Polyhedron: U22
  • Platonic Solid
  • Platonic Element: Water
  • Vertices: 12
  • Edges: 30
  • Faces: 20
  • Wythoff symbol: 5|2 3
  • Symmetry Group: icosahedral
  • Vertex Configuration: {3, 3, 3, 3, 3}
  • Dual: dodecahedron
  • V: s3 * 5/12 * (3 + √5)
  • A: s2 * 20 / 4 * √3
  • rinner: s / 12 * (3 * √3 + √15)
  • router: s / 4 * √(10 + 2 * √5)


s = , V = , A = , rinner = , router = , ravg =

One variant of a geodesic dome can be derived from the Icosahedron.

Dodecahedron


Dodecahedron
  • Uniform Polyhedron: U23
  • Platonic Solid
  • Platonic Element: Ether
  • Vertices: 20
  • Edges: 30
  • Faces: 12
  • Wythoff symbol: 3|2 5
  • Symmetry Group: icosahedral
  • Vertex Configuration: {5, 5, 5}
  • Dual: icosahedron
  • V: s3 / 4 * (15 + 7 * √5)
  • A: s2 * 12 / 4 * √(25 + 10 * √5)
  • rinner: s / 20 * √(250 + 110 * √5)
  • router: s / 4 * (√15 + √3)


s = , V = , A = , rinner = , router = , ravg =

Icosidodecahedron


Icosidodecahedron
  • Uniform Polyhedron: U24
  • Archimedean Solid: A4
  • Vertices: 30
  • Edges: 60
  • Faces: 32
  • Wythoff symbol: 2|3 5
  • Symmetry Group: icosahedral
  • Vertex Configuration: {3, 5, 3, 5}
  • Dual: rhombic triacontahedron
  • V: s3 / 6 * (45 + 17 * √5)
  • A: s2 * (5 * √3 + 3 * √5 * √(5 + 2 * √5))
  • rinner: s / 8 * (5 + 3 * √5)
  • router: s * (1 + √5) / 2


s = , V = , A = , rinner = , router = , ravg =

Truncated Icosahedron


Truncated Icosahedron
  • Uniform Polyhedron: U25
  • Archimedean Solid: A11
  • Vertices: 60
  • Edges: 90
  • Faces: 32
  • Wythoff symbol: 2 5|3
  • Symmetry Group: icosahedral
  • Vertex Configuration: {6, 6, 5}
  • Dual: pentakis dodecahedron
  • V: s3 / 4 * (125 + 43 * √5)
  • A: s2 * 3 * (10 * √3 + √5 * √(5 + 2 * √5))
  • rinner: s * 9/872 * (21 + √5) * √(58 + 18 * √5)
  • router: s / 4 * √(58 + 18 * √5)


s = , V = , A = , rinner = , router = , ravg =

Truncated Dodecahedron


Truncated Dodecahedron
  • Uniform Polyhedron: U26
  • Archimedean Solid: A10
  • Vertices: 60
  • Edges: 90
  • Faces: 32
  • Wythoff symbol: 2 3|5
  • Symmetry Group: icosahedral
  • Vertex Configuration: {10, 10, 3}
  • Dual: triakis icosahedron
  • V: s3 * 5/12 * (99 + 47 * √5)
  • A: s2 * 5 * (√3 + 6 * √(5 + 2 * √5))
  • rinner: s * 5/488 * (17 * √2 + 3 * √10) * √(37 + 15 * √5)
  • router: s / 4 * √(74 + 30 * √5)


s = , V = , A = , rinner = , router = , ravg =

Rhombicosidodecahedron


Rhombicosidodecahedron
  • Uniform Polyhedron: U27
  • aka Small Rhombicosidodecahedron
  • Archimedean Solid: A5
  • Vertices: 60
  • Edges: 120
  • Faces: 62
  • Wythoff symbol: 3 5|2
  • Symmetry Group: icosahedral
  • Vertex Configuration: {4, 3, 4, 5}
  • Dual: deltoidal hexecontahedron
  • V: s3 / 3 * (60 + 29 * √5)
  • A: s2 * (30 + √(30 * (10 + 3 * √5 + √(15 * (5 + 2 * √5)))))
  • rinner: s / 41 * (15 + 2 *√5) * √(11 + 4 * √5)
  • router: s / 2 * √(11 + 4 * √5)


s = , V = , A = , rinner = , router = , ravg =

Truncated Icosidodecahedron


Truncated Icosidodecahedron
  • Uniform Polyhedron: U28
  • aka Great Rhombicosidodecahedron (which actually is misleading as it also references U67 "(Uniform) Great Rhombicosidodecahedron")
  • aka Rhombitruncated Icosidodecahedron
  • aka Omnitruncated Icosidodecahedron
  • Archimedean Solid: A2
  • Vertices: 120
  • Edges: 180
  • Faces: 62
  • Wythoff symbol: 2 3 5|
  • Symmetry Group: icosahedral
  • Vertex Configuration: {4, 6, 10}
  • Dual: disdyakis triacontahedron
  • V: s3 * (95 + 50 * √5)
  • A: s2 * 30 * (1 + √(2 * (4 + √5 + √(15 + 6 * √6))))
  • rinner: s * 1/241 * (105 + 6 * √5 * √(31 + 12 * √5))
  • router: s / 2 * √(31 + 12 * √5)


s = , V = , A = , rinner = , router = , ravg =

Snub Dodecahedron


Snub Dodecahedron
  • Uniform Polyhedron: U29
  • Archimedean Solid: A8
  • Vertices: 60
  • Edges: 150
  • Faces: 92
  • Wythoff symbol: |2 3 5
  • Symmetry Group: icosahedral
  • Vertex Configuration: {3, 3, 3, 3, 5}
  • Dual: pentagonal hexecontahedron
  • V: s3 * 3.7543
  • A: s2 * √(15 * (95 + 6 * √5 + 8 * √(15 * (5 + 2 * √5))))
  • rinner: s * 2.03987315
  • router: s * 2.15583737


s = , V = , A = , rinner = , router = , ravg =

(Note: the volume for this solid is only available numerically, not symbolical - if you can provide me with the expression for V let me know).

References


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