Dome Geodesic Dome Geodesic Dome Notes & Calculator Geodesic Dome Diary Geodesic Polyhedra Polyhedra Notes Bow Dome Triangulated Bow Dome Star Dome Star Dome Diary Wigwam Zome Helix Zome Low Cost Dome (PVC) Miscellaneous Domes

Site Search

Enter term & press ENTER

If you found the information useful, consider to make a donation:

USD, EUR, BTC

Page << Prev | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Next >>

# Polyhedra Notes

written by Rene K. Mueller, Copyright (c) 2007, last updated Wed, January 9, 2008

 UpdatesTue, 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
• 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

Page << Prev | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Next >>

Content:

 Home · About · Tipi · Yurt · Dome · Features · Gallery · Links

 Print · Contact · Bookmark

Creative Commons (CC) BY, SA, NC 2005-2015, developed, designed and written by René K. Müller,
Graphics & illustrations made with Inkscape, Tgif, Gimp, PovRay, GD.pm