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.

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 .

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 r_inner and r_outer 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
r_inner = inner radius or inradius
r_outer = outer radius or circumradius
r_avg = (r_inner + r_outer) / 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, r_inner and r_outer) by other sources, so don't rely on it yet.

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
r_inner: s / 12 * √6
r_outer: s / 4 * √6
h: s / 3 * √6

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
r_inner: s * 9 / 44 * √22
r_outer: s / 4 * √22

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
r_inner: s / 6 * √6
r_outer: s / 2 * √2

Half of an octahedron is the classic pyramid.

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
r_inner: s / 2
r_outer: 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.

Zonohedron , Wikipedia

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)
r_inner: s * 3/4
r_outer: s

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)
r_inner: s * 9/20 * √10
r_outer: s / 2 * √10

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)
r_inner: s / 17 * (5 + 2 * √2 * √(7 + 4 * √2))
r_outer: s / 2 * √(7 + 4 * √2)

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)
r_inner: s / 17 * (6 + √2) * √(5 + 2 * √2)
r_outer: s / 2 * √(5 + 2 * √2)

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)
r_inner: s * 3/97 * (14 + √2) * √(13 + 6 * √2)
r_outer: s / 2 * √(13 + 6 * √2)

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)
r_inner: s * √(abs(1-t)/(4*(2-t)))
r_outer: s * √((3-t)/(4*(2-t)))

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
r_inner: s / 12 * (3 * √3 + √15)
r_outer: s / 4 * √(10 + 2 * √5)

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

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)
r_inner: s / 20 * √(250 + 110 * √5)
r_outer: s / 4 * (√15 + √3)

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))
r_inner: s / 8 * (5 + 3 * √5)
r_outer: s * (1 + √5) / 2

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))
r_inner: s * 9/872 * (21 + √5) * √(58 + 18 * √5)
r_outer: s / 4 * √(58 + 18 * √5)

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))
r_inner: s * 5/488 * (17 * √2 + 3 * √10) * √(37 + 15 * √5)
r_outer: s / 4 * √(74 + 30 * √5)

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)))))
r_inner: s / 41 * (15 + 2 *√5) * √(11 + 4 * √5)
r_outer: s / 2 * √(11 + 4 * √5)

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))))
r_inner: s * 1/241 * (105 + 6 * √5 * √(31 + 12 * √5))
r_outer: s / 2 * √(31 + 12 * √5)

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))))
r_inner: s * 2.03987315
r_outer: s * 2.15583737

(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).

Platonic Solid , Wikipedia
Platonic Solid , Mathworld
Archimedean Solid , Wikipedia
Archimedean Solid , Mathworld
Platonic & Archimedean Solid Paper Models

Zonohedron , convex polyhedron tiling space without "holes", Wikipedia
Deltahedron , Mathworld

Zvi Har'El's Homepage , author of Uniform Solution for Uniform Polyhedra and kaleido
Uniform Polyhedra by Paul Bourke
Uniform Polyhedra povray files prepared by Russel Towle, which I used to render the models
Woven Paper Polyhedra , using woven paper to construct polyhedra

In 1966 Norman Johnson

published a paper "Convex Solids with Regular Faces" (Canadian Journal of Mathematics, Vol. 18, 1966, pp. 169-200) in which he classified and named 92 types of convex polyhedra which are composed by same side length additionally to the platonic and archimedean solids:

Square Pyramid	Pentagonal Pyramid	Triangular Cupola	Square Cupola
Pentagonal Cupola	Pentagonal Rotunda	Elongated Triangular Pyramid	Elongated Square Pyramid
Elongated Pentagonal Pyramid	Gyroelongated Square Pyramid	Gyroelongated Pentagonal Pyramid	Triangular Dipyramid
Pentagonal Dipyramid	Elongated Triangular Dipyramid	Elongated Square Dipyramid	Elongated Pentagonal Dipyramid
Gyroelongated Square Dipyramid	Elongated Triangular Cupola	Elongated Square Cupola	Elongated Pentagonal Cupola
Elongated Pentagonal Rotunda	Gyroelongated Triangular Cupola	Gyroelongated Square Cupola	Gyroelongated Pentagonal Cupola
Gyroelongated Pentagonal Rotunda	Gyrobifastigium	Triangular Orthobicupola	Square Orthobicupola
Square Gyrobicupola	Pentagonal Orthobicupola	Pentagonal Gyrobicupola	Pentagonal Orthocupolarontunda
Pentagonal Gyrocupolarotunda	Pentagonal Orthobirotunda	Elongated Triangular Orthobicupola	Elongated Triangular Gyrobicupola
Elongated Square Gyrobicupola	Elongated Pentagonal Orthobicupola	Elongated Pentagonal Gyrobicupola	Elongated Pentagonal Orthocupolarotunda
Elongated Pentagonal Gyrocupolarotunda	Elongated Pentagonal Orthobirotunda	Elongated Pentagonal Gyrobirotunda	Gyroelongated Triangular Bicupola
Gyroelongated Square Bicupola	Gyroelongated Pentagonal Bicupola	Gyroelongated Pentagonal Cupolarotunda	Gyroelongated Pentagonal Birotunda
Augmented Triangular Prism	Biaugmented Triangular Prism	Triaugmented Triangular Prism	Augmented Pentagonal Prism
Biaugmented Pentagonal Prism	Augmented Hexagonal Prism	Parabiaugmented Hexagonal Prism	Metabiaugmented Hexagonal Prism
Triaugmented Hexagonal Prism	Augmented Dodecahedron	Parabiaugmented Dodecahedron	Metabiaugmented Dodecahedron
Triaugmented Dodecahedron	Metabidiminished Icosahedron	Tridiminished Icosahedron	Augmented Tridiminished Icosahedron
Augmented Truncated Tetrahedron	Augmented Truncated Cube	Biaugmented Truncated Cube	Augmented Truncated Dodecahedron
Parabiaugmented Truncated Dodecahedron	Metabiaugmented Truncated Dodecahedron	Triaugmented Truncated Dodecahedron	Gyrate Rhombicosidodecahedron
Parabigyrate Rhombicosidodecahedron	Metabigyrate Rhombicosidodecahedron	Trigyrate Rhombicosidodecahedron	Diminished Rhombicosidodecahedron
Paragyrate Diminished Rhombicosidodecahedron	Metagyrate Diminished Rhombicosidodecahedron	Bigyrate Diminished Rhombicosidodecahedron	Parabidiminished Rhombicosidodecahedron
Metabidiminished Rhombicosidodecahedron	Gyrate Bidiminished Rhombicosidodecahedron	Tridiminished Rhombicosidodecahedron	Snub Disphenoid
Snub Square Antiprism	Sphenocorona	Augmented Sphenocorona	Sphenomegacorona
Hebesphenomegacorona	Disphenocingulum	Bilunabirotunda	Triangular Hebesphenorotunda

As the time goes by I will comment those forms which look promising for a temporary building.

Square Pyramid

Johnson Solid: J1
Vertices: 5
Faces: 5
Edges: 8
V: s³ / 6 * √2
A: s² * (1 + √3)

General Pyramid
The general pyramid:

q = √((s/2)² + h²)
l = √((s/2)² + q²)
A = 2 s q + s²
A_wall/roof = 2 s q
V = s² h / 3

The Pyramids of Giza (courtesy Bruno Girin/DHD Multimedia Gallery

)

Most obvious association with pyramids comes from the egyptian pyramids in Giza and those of the mayans in south america. It is also apparent those pyramids have been built with enormous effort and skills which is now honored by the durability of thousands of years.

The largest pyramid in Giza is the Khufu Pyramid, with following values:

s = 231m (original) (now: 230.4m)
h = 146.6m (original) (now: 138.8m)

thereby

q = 186.632m
l = 219.478m

Interesting ratio:

s/h = 1.57571.. ~ π/2 ~ 11/7

Read more at Great Pyramid of Giza .

Pentagonal Pyramid

Johnson Solid: J2
Vertices: 6
Faces: 6
Edges: 10
V: s³ / 24 * (5 + √5)
A: s² / 2 * √(5/2 * (10 + √10 + √(75+30*√5))

The roof isn't very steep, unless those roof struts would be made longer - it would be a very simplistic habitat.

Triangular Cupola

Johnson Solid: J3
Vertices: 9
Faces: 8
Edges: 15
V: s³ * (5/(5*√2))
A: s² * (3+5/2*√3)

Square Cupola

Johnson Solid: J4
Vertices: 12
Faces: 10
Edges: 20

Also cap of a Rhombicuboctahedron.

Pentagonal Cupola

Johnson Solid: J5
Vertices: 15
Faces: 12
Edges: 25
V: s³ / 6 * (5+4*√5)
A: s² / 4 * (20+√(10*(80+31*√5+√(2175+950*√5))))

Also cap of a Rhombicosidodecahedron.

Pentagonal Rotunda

Johnson Solid: J6
Vertices: 20
Faces: 17
Edges: 35
V: s³ / 12 * (45+17*√5)
A: s² * 5/2 * (√3+√(26+(58/√5)))
h: s * √(1/5*(5+2*√5))

Also 1/2 of a Icosidodecahedron.

Elongated Triangular Pyramid

Johnson Solid: J7
Vertices: 7
Faces: 7
Edges: 12

Elongated Square Pyramid

Johnson Solid: J8
Vertices: 9
Faces: 9
Edges: 16

Elongated Pentagonal Pyramid

Johnson Solid: J9
Vertices: 11
Faces: 11
Edges: 20

Gyroelongated Square Pyramid

Johnson Solid: J10
Vertices: 9
Faces: 13
Edges: 20

Gyroelongated Pentagonal Pyramid

Johnson Solid: J11
Vertices: 11
Faces: 16
Edges: 25

Triangular Dipyramid

Johnson Solid: J12
Vertices: 5
Faces: 6
Edges: 9

Pentagonal Dipyramid

Johnson Solid: J13
Vertices: 7
Faces: 10
Edges: 15
V: s³ / 12 * (5 + √5)
A: s² * 5/12 * √3
r_outer: s / 10 * √(50+10*√5)
h: s / 10 * √(50-10*√5)

Elongated Triangular Dipyramid

Johnson Solid: J14
Vertices: 8
Faces: 9
Edges: 15

Elongated Square Dipyramid

Johnson Solid: J15
Vertices: 10
Faces: 12
Edges: 20

Elongated Pentagonal Dipyramid

Johnson Solid: J16
Vertices: 12
Faces: 15
Edges: 25

Gyroelongated Square Dipyramid

Johnson Solid: J17
Vertices: 10
Faces: 16
Edges: 24
V: s³ * 2^(1/4) / 4 * (1 + √2 + 2^(1/4) )
A: s² * 4 * √3

Elongated Triangular Cupola

Johnson Solid: J18
Vertices: 15
Faces: 14
Edges: 27

Elongated Square Cupola

Johnson Solid: J19
Vertices: 20
Faces: 18
Edges: 36

Elongated Pentagonal Cupola

Johnson Solid: J20
Vertices: 25
Faces: 22
Edges: 45

Elongated Pentagonal Rotunda

Johnson Solid: J21
Vertices: 30
Faces: 27
Edges: 55

Very suitable for a habitat, the top smaller pentagon as skylight, and eventually the side pentagons further triangulated each.

Gyroelongated Triangular Cupola

Johnson Solid: J22
Vertices: 15
Faces: 20
Edges: 33

Gyroelongated Square Cupola

Johnson Solid: J23
Vertices: 20
Faces: 26
Edges: 44

Gyroelongated Pentagonal Cupola

Johnson Solid: J24
Vertices: 25
Faces: 32
Edges: 55

Gyroelongated Pentagonal Rotunda

Johnson Solid: J25
Vertices: 30
Faces: 37
Edges: 65

Alike the "Elongated Pentagonal Rotunda", suitable and increased stability for the vertical walls being triangles.

Gyrobifastigium

Johnson Solid: J26
Vertices: 8
Faces: 8
Edges: 14

Triangular Orthobicupola

Johnson Solid: J27
Vertices: 12
Faces: 14
Edges: 24

Square Orthobicupola

Johnson Solid: J28
Vertices: 16
Faces: 18
Edges: 32

Square Gyrobicupola

Johnson Solid: J29
Vertices: 16
Faces: 18
Edges: 32

Pentagonal Orthobicupola

Johnson Solid: J30
Vertices: 20
Faces: 22
Edges: 40

Pentagonal Gyrobicupola

Johnson Solid: J31
Vertices: 20
Faces: 22
Edges: 40

Pentagonal Orthocupolarontunda

Johnson Solid: J32
Vertices: 25
Faces: 27
Edges: 50

Pentagonal Gyrocupolarotunda

Johnson Solid: J33
Vertices: 25
Faces: 27
Edges: 50

Pentagonal Orthobirotunda

Johnson Solid: J34
Vertices: 30
Faces: 32
Edges: 60

Elongated Triangular Orthobicupola

Johnson Solid: J35
Vertices: 18
Faces: 20
Edges: 36

Elongated Triangular Gyrobicupola

Johnson Solid: J36
Vertices: 18
Faces: 20
Edges: 36

Elongated Square Gyrobicupola

Johnson Solid: J37
Vertices: 24
Faces: 26
Edges: 48

Elongated Pentagonal Orthobicupola

Johnson Solid: J38
Vertices: 30
Faces: 32
Edges: 60

Elongated Pentagonal Gyrobicupola

Johnson Solid: J39
Vertices: 30
Faces: 32
Edges: 60

Elongated Pentagonal Orthocupolarotunda

Johnson Solid: J40
Vertices: 35
Faces: 37
Edges: 70

Elongated Pentagonal Gyrocupolarotunda

Johnson Solid: J41
Vertices: 35
Faces: 37
Edges: 70

Elongated Pentagonal Orthobirotunda

Johnson Solid: J42
Vertices: 40
Faces: 42
Edges: 80

Elongated Pentagonal Gyrobirotunda

Johnson Solid: J43
Vertices: 40
Faces: 42
Edges: 80

Gyroelongated Triangular Bicupola

Johnson Solid: J44
Vertices: 18
Faces: 26
Edges: 42

Gyroelongated Square Bicupola

Johnson Solid: J45
Vertices: 24
Faces: 34
Edges: 56

Gyroelongated Pentagonal Bicupola

Johnson Solid: J46
Vertices: 30
Faces: 42
Edges: 70

Gyroelongated Pentagonal Cupolarotunda

Johnson Solid: J47
Vertices: 35
Faces: 47
Edges: 80

Gyroelongated Pentagonal Birotunda

Johnson Solid: J48
Vertices: 40
Faces: 52
Edges: 90

Augmented Triangular Prism

Johnson Solid: J49
Vertices: 7
Faces: 8
Edges: 13

Biaugmented Triangular Prism

Johnson Solid: J50
Vertices: 8
Faces: 11
Edges: 17

Triaugmented Triangular Prism

Johnson Solid: J51
Vertices: 9
Faces: 14
Edges: 21

Augmented Pentagonal Prism

Johnson Solid: J52
Vertices: 11
Faces: 10
Edges: 19

Biaugmented Pentagonal Prism

Johnson Solid: J53
Vertices: 12
Faces: 13
Edges: 23

Augmented Hexagonal Prism

Johnson Solid: J54
Vertices: 13
Faces: 11
Edges: 22

Parabiaugmented Hexagonal Prism

Johnson Solid: J55
Vertices: 14
Faces: 14
Edges: 26

Metabiaugmented Hexagonal Prism

Johnson Solid: J56
Vertices: 14
Faces: 14
Edges: 26

Triaugmented Hexagonal Prism

Johnson Solid: J57
Vertices: 15
Faces: 17
Edges: 30

Augmented Dodecahedron

Johnson Solid: J58
Vertices: 21
Faces: 16
Edges: 35

Parabiaugmented Dodecahedron

Johnson Solid: J59
Vertices: 22
Faces: 20
Edges: 40

Metabiaugmented Dodecahedron

Johnson Solid: J60
Vertices: 22
Faces: 20
Edges: 40

Triaugmented Dodecahedron

Johnson Solid: J61
Vertices: 23
Faces: 24
Edges: 45

Metabidiminished Icosahedron

Johnson Solid: J62
Vertices: 10
Faces: 12
Edges: 20

Tridiminished Icosahedron

Johnson Solid: J63
Vertices: 9
Faces: 8
Edges: 15

Augmented Tridiminished Icosahedron

Johnson Solid: J64
Vertices: 10
Faces: 10
Edges: 18

Augmented Truncated Tetrahedron

Johnson Solid: J65
Vertices: 15
Faces: 14
Edges: 27

Augmented Truncated Cube

Johnson Solid: J66
Vertices: 28
Faces: 22
Edges: 48

Biaugmented Truncated Cube

Johnson Solid: J67
Vertices: 32
Faces: 30
Edges: 60

Augmented Truncated Dodecahedron

Johnson Solid: J68
Vertices: 65
Faces: 42
Edges: 105

Parabiaugmented Truncated Dodecahedron

Johnson Solid: J69
Vertices: 70
Faces: 52
Edges: 120

Metabiaugmented Truncated Dodecahedron

Johnson Solid: J70
Vertices: 70
Faces: 52
Edges: 120

Triaugmented Truncated Dodecahedron

Johnson Solid: J71
Vertices: 75
Faces: 62
Edges: 135

Gyrate Rhombicosidodecahedron

Johnson Solid: J72
Vertices: 60
Faces: 62
Edges: 120

Parabigyrate Rhombicosidodecahedron

Johnson Solid: J73
Vertices: 60
Faces: 62
Edges: 120

Metabigyrate Rhombicosidodecahedron

Johnson Solid: J74
Vertices: 60
Faces: 62
Edges: 120

Trigyrate Rhombicosidodecahedron

Johnson Solid: J75
Vertices: 60
Faces: 62
Edges: 120

Diminished Rhombicosidodecahedron

Johnson Solid: J76
Vertices: 55
Faces: 52
Edges: 105

Paragyrate Diminished Rhombicosidodecahedron

Johnson Solid: J77
Vertices: 55
Faces: 52
Edges: 105

Metagyrate Diminished Rhombicosidodecahedron

Johnson Solid: J78
Vertices: 55
Faces: 52
Edges: 105

Bigyrate Diminished Rhombicosidodecahedron

Johnson Solid: J79
Vertices: 55
Faces: 52
Edges: 105

Parabidiminished Rhombicosidodecahedron

Johnson Solid: J80
Vertices: 50
Faces: 42
Edges: 90

Metabidiminished Rhombicosidodecahedron

Johnson Solid: J81
Vertices: 50
Faces: 42
Edges: 90

Gyrate Bidiminished Rhombicosidodecahedron

Johnson Solid: J82
Vertices: 50
Faces: 42
Edges: 90

Tridiminished Rhombicosidodecahedron

Johnson Solid: J83
Vertices: 45
Faces: 32
Edges: 75

Snub Disphenoid

Johnson Solid: J84
Vertices: 8
Faces: 12
Edges: 18

Snub Square Antiprism

Johnson Solid: J85
Vertices: 16
Faces: 26
Edges: 40

Sphenocorona

Johnson Solid: J86
Vertices: 10
Faces: 14
Edges: 22

Augmented Sphenocorona

Johnson Solid: J87
Vertices: 11
Faces: 17
Edges: 26

Sphenomegacorona

Johnson Solid: J88
Vertices: 12
Faces: 18
Edges: 28

Hebesphenomegacorona

Johnson Solid: J89
Vertices: 14
Faces: 21
Edges: 33

Disphenocingulum

Johnson Solid: J90
Vertices: 16
Faces: 24
Edges: 38

Bilunabirotunda

Johnson Solid: J91
Vertices: 14
Faces: 14
Edges: 26

Triangular Hebesphenorotunda

Johnson Solid: J92
Vertices: 18
Faces: 20
Edges: 36

Johnson Solid , Mathworld
Johnson Solid , Wikipedia
George Hart , very interesting web-site, also from where I've got the wrl-files and converted into .pov to render the models

For sake of completeness I list all "uniform polyhedra", which include the platonic and archimedean solids but additionally cover als the concave (non-convex) polyhedra which aren't suitable for habitat development.

Tetrahedron

Truncated Tetrahedron

Octahemioctahedron

Tetrahemihexahedron

Octahedron

Cube

Cuboctahedron

Truncated Octahedron

Truncated Cube

Rhombicuboctahedron

Truncated Cuboctahedron

Snub Cube

Small Cubicuboctahedron

Great Cubicuboctahedron

Cubohemioctahedron

Cubitruncated Cuboctahedron

Great Rhombicuboctahedron

Small Rhombihexahedron

Stellated Truncated Hexahedron

Great Truncated Cuboctahedron

Icosahedron

Dodecahedron

Icosidodecahedron

Truncated Icosahedron

Truncated Dodecahedron

Rhombicosidodecahedron

Truncated Icosidodecahedron

Snub Dodecahedron

Small Ditrigonal Icosidodecahedron

Small Icosicosidodecahedron

Small Snub Icosicosidodecahedron

Small dodecicosidodecahedron

Small Stellated Dodecahedron

Great Dodecahedron

Dodecadodecahedron

Truncated Great Dodecahedron

Rhombidodecadodecahedron

Small Rhombidodecahedron

Snub Dodecadodecahedron

Ditrigonal Dodecadodecahedron

Great Ditrigonal Dodecicosidodecahedron

Small Ditrigonal Dodecicosidodecahedron

Icosidodecadodecahedron

Icositruncated Dodecadodecahedron

Snub Icosidodecadodecahedron

Great Ditrigonal Icosidodecahedron

Great Icosicosidodecahedron

Small Icosihemidodecahedron

Small Dodecicosahedron

Small Dodecahemidodecahedron

Great Stellated Dodecahedron

Great Icosahedron

Great Icosidodecahedron

Great Truncated Icosahedron

Rhombicosahedron

Great Snub Icosidodecahedron

Small Stellated Truncated Dodecahedron

Truncated Dodecadodecahedron

Inverted Snub Dodecadodecahedron

Great Dodecicosidodecahedron

Small Dodecahemicosahedron

Great Dodecicosahedron

Great Snub Dodecicosidodecahedron

Great Dodecahemicosahedron

Great Stellated Truncated Dodecahedron

Great Rhombicosidodecahedron

Great Truncated Icosidodecahedron

Great Inverted Snub Icosidodecahedron

Great Dodecahemidodecahedron

Great Icosihemidodecahedron

Small Retrosnub Icosicosidodecahedron

Great Rhombidodecahedron

Great Retrosnub Icosidodecahedron

Great Dirhombicosidodecahedron

Pentagonal Prism

Pentagonal Antiprism

Pentagrammic Prism

Pentagrammic Antiprism

Pentagrammic Crossed Antiprism

Uniform Polyhedron , Mathworld
Uniform Polyhedron , Wikipedia
List of Uniform Polyhedra , Wikipedia

Waterman polyhedra exist in mathematics since about 1990, a rather recent discovery. The concept of Waterman polyhedra was developed by Steve Waterman

and relate to "Cubic Close Packing" (CCP) of spheres, also known as IVM (Isotropic Vector Matrix) by R. Buckminster Fuller.

Spheres are packed in a cubic manner:

x, y and z are integers,
sphere diameter is √2 thereby,
at 0,0,0 resides the first sphere,

and then those removed which have longer vector than √(2*n) from the center (0,0,0) and then a convex hull is calculated which gives a polyhedron:

Waterman W24 O1 (radius √48)

Waterman W24 O1

Waterman W1 O1 (radius √2)

Waterman W5 O1 (radius √10)

Waterman W24 O1 (radius √48)

Waterman W200 O1 (radius √400)

Waterman W1000 O1 (radius √2000)

Waterman W1 O1

Waterman W5 O1

Waterman W24 O1

Waterman W200 O1

Waterman W1000 O1

Several origins are available as summarized by Mirek Majewski :

Origin Origin Location radius Comment
1 0,0,0 √(2*n) Atom center: traditional origin of Waterman polyhedra, symmetry properties of a cube.
2 1/2,1/2,0 √(2+4*n)/2 Touch point between 2 atoms: symmetry properties of flat rectangle whose top & bottom faces are the same
3 1/3,1/3,2/3 √(6*(n+1))/3 Void center between 3 atoms: symmetry properties of flat triangle whose top & bottom faces are different
3* 1/3,1/3,1/3 √(3+6*n)/3 Void center between 3 lattice voids: same symmetry of origin 3.
4 1/2,1/2,1/2 √(3+8*(n-1))/2 Tretrahedron void center between 4 atoms: symmetry properties of a tetrahedron.
5 0,0,1/2 √(1+4*n)/2 5 atoms: symmetry properties of a flat rectangle whose top & bottom are different.
6 1,0,0 √(1+2*(n-1)) Octahedral void center between 6 atoms: symmetry properties of a cube.
7 Origins of Waterman Polyhedra by Mark Newbold

whereas n starts with 1 for all origins.

	O1	O2	O3	O3*	O4	O5	O6
W1
W2
W3
W4
W5
W6
W7
W8
W9
W10

So the nice thing is, the creation of the polyhedra is done parametrically, in other words, systematically. Needless to say, there are infinite of Waterman polyhedra, and interestingly there are doubles to be encountered, where:

the same amount of packed spheres result in the same polyhedra (14 + 16 n) m² (whereas n & m integers >= 0)
the final polyhedra from packed spheres result in a previous discovered polyhedra when normalized (e.g. W3 O1 = W24 O1)

Those are not yet removed in the overview which follows.

The Waterman Polyhedra (WP) covers also some of the platonic and archimedean solids. For sake of this comparison the WP are normalized, as W2 O1 has a different size/volume than W1 O6, but the same form of a Octahedron.

Waterman W1 O4

Waterman W2 O1

Waterman W2 O6

W1 O3* = W2 O3* = W1 O3 = W1 O4 = Tretrahedron

W2 O1 = W1 O6 = Octahedron

W2 O6 = Cube

Icosahedron and Dodecahedron have no WP representation.

Waterman W1 O1

Waterman W10 O1

Waterman W2 O4

W1 O1 = W4 O1 = Cuboctahedron

W10 O1 = Truncated Octahedron

W4 O3 = W2 O4 = Truncated Tetrahedron

The following look like archimedean solids, but they are not:

Waterman W7 O1

Waterman W3 O1

W7 O1 != Truncated Cuboctahedron W3 O1 = W12 O1 != Rhombicuboctahedron

as they are not having one length of edge, but two lengths of edges.

The others have no WP representation.

The Generalized Waterman Polyhedra (GWP) are currently still sorted by Steve Waterman, and I will include them later as they may cover more platonic and archimedean solids.

Waterman Polyhedron , very informative site by Steve Waterman, the inventor of the Waterman polyhedra
Paul Bourke has also a comprehensive website (I highly recommend to visit), his waterman.c I used to calculate those rendered Waterman polyhedra.
3D Junkyard , vast & interesting collection of links by David Eppstein

Origin	Origin Location	radius	Comment
1	0,0,0	√(2*n)	Atom center: traditional origin of Waterman polyhedra, symmetry properties of a cube.
2	1/2,1/2,0	√(2+4*n)/2	Touch point between 2 atoms: symmetry properties of flat rectangle whose top & bottom faces are the same
3	1/3,1/3,2/3	√(6*(n+1))/3	Void center between 3 atoms: symmetry properties of flat triangle whose top & bottom faces are different
3*	1/3,1/3,1/3	√(3+6*n)/3	Void center between 3 lattice voids: same symmetry of origin 3.
4	1/2,1/2,1/2	√(3+8*(n-1))/2	Tretrahedron void center between 4 atoms: symmetry properties of a tetrahedron.
5	0,0,1/2	√(1+4*n)/2	5 atoms: symmetry properties of a flat rectangle whose top & bottom are different.
6	1,0,0	√(1+2*(n-1))	Octahedral void center between 6 atoms: symmetry properties of a cube.
7 Origins of Waterman Polyhedra by Mark Newbold

Polyhedra Notes

Introduction

Platonic Solids

Archimedean Solids

Tetrahedron

Truncated Tetrahedron

Octahedron

Cube

Cuboctahedron

Truncated Octahedron

Truncated Cube

Rhombicuboctahedron

Truncated Cuboctahedron

Snub Cube

Icosahedron

Dodecahedron

Icosidodecahedron

Truncated Icosahedron

Truncated Dodecahedron

Rhombicosidodecahedron

Truncated Icosidodecahedron

Snub Dodecahedron

References

Johnson Solids

Square Pyramid

Historic Pyramids

Pentagonal Pyramid

Triangular Cupola

Square Cupola

Pentagonal Cupola

Pentagonal Rotunda

Elongated Triangular Pyramid

Elongated Square Pyramid

Elongated Pentagonal Pyramid

Gyroelongated Square Pyramid

Gyroelongated Pentagonal Pyramid

Triangular Dipyramid

Pentagonal Dipyramid

Elongated Triangular Dipyramid

Elongated Square Dipyramid

Elongated Pentagonal Dipyramid

Gyroelongated Square Dipyramid

Elongated Triangular Cupola

Elongated Square Cupola

Elongated Pentagonal Cupola

Elongated Pentagonal Rotunda

Gyroelongated Triangular Cupola

Gyroelongated Square Cupola

Gyroelongated Pentagonal Cupola

Gyroelongated Pentagonal Rotunda

Gyrobifastigium

Triangular Orthobicupola

Square Orthobicupola

Square Gyrobicupola

Pentagonal Orthobicupola

Pentagonal Gyrobicupola

Pentagonal Orthocupolarontunda

Pentagonal Gyrocupolarotunda

Pentagonal Orthobirotunda

Elongated Triangular Orthobicupola

Elongated Triangular Gyrobicupola

Elongated Square Gyrobicupola

Elongated Pentagonal Orthobicupola

Elongated Pentagonal Gyrobicupola

Elongated Pentagonal Orthocupolarotunda

Elongated Pentagonal Gyrocupolarotunda

Elongated Pentagonal Orthobirotunda

Elongated Pentagonal Gyrobirotunda

Gyroelongated Triangular Bicupola

Gyroelongated Square Bicupola

Gyroelongated Pentagonal Bicupola

Gyroelongated Pentagonal Cupolarotunda

Gyroelongated Pentagonal Birotunda

Augmented Triangular Prism

Biaugmented Triangular Prism

Triaugmented Triangular Prism

Augmented Pentagonal Prism

Biaugmented Pentagonal Prism

Augmented Hexagonal Prism

Parabiaugmented Hexagonal Prism