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In Book V of his COLLECTION Pappus claims that these13 semiregular solids were first described by Archimedes and so are named in his honor. The Archimedean, or semi-regular polyhedra, are 'facially' regular. Every face is a regular polygon, though the faces are not all of the same kind. Every vertex, however, is to be congruent to every other vertex, i.e. the faces must be arranged in the same order around each vertex.
The number of faces, vertices, and edges.
Most of the following material is taken from the booklet ARCHIMEDEAN AND PLATONIC SOLIDS by Mark S. Adams, Geodesic Publications, Baton Rouge, Louisiana. This booklet accompanied a paper he presented at the International Congress of Mathematicians on 5 August, 1986.
The symbol used to describe the regular polyhedra is a combination of integers and exponents. The base indecates the number of sides of the regular polygonal face: 3.4.5 would mean the a triangle, a square and a pentagon were in volved. The exponent indicates the number of these ploygonal faces meeting at a vertex.
Since it has squares meeting at each vertex the cube is written 43.
A snub cube (which is # 7 in the list) is written 34.4
because 4 triangles (3) meet one suare (4) at each vertex.
13 Archimedean solids
It can be proven that there are only 13 Archimedean solids, two of which occur in two forms. These two are the two 'snubs', and the two forms of each are related to one another like a left-hand and a righthand glove: they are enanttomorphic. The set of thirteen is illustrated below.
One of these solids, the truncated tetrahedron, can be inscribed in a regular tetrahedron. The next six can be inscribed in either a cube or an octahedron, and the last six in either a dodecahedron or an icosahedron. The 'truncated' solids are so called because each can be constructed by cutting off the corners of some other solid, but the truncated cuboctahedron and icosidodecahedron require a distortion in addition to convert rectangles into squares. So the better names for these two solids are 'Great Rhombicuboctahedron' and 'Great Rhombicosidodecahedron'. The solids 34.4 and 3.4.5.4 can then bear the prefix 'small'. The syllable 'rhomb-' shows that one set of faces lies in the planes of the rhombic dodecahedron and rhombic triacontahedron respectively. All Archimedean solids are inscribable in a sphere.
| Faces, vertices and edges | |
|---|---|
|
The type of face is indicated by a subscript to F, e.g. F3 represents a triangle; F4a square. The number of these faces in a given polyhedron is the coeficient: e.g. 6 F4 for a cube. The number of Vertices and Edges are also given as coeficients e.g. a cube would have 8V and 12E |
 |
References
Adams, Mark S. Archimedean and Platonic Solids. self published, 1985
Cundy Martyn Mathematical Models. Oxford: Clarendon Press, 1961
Eves, Howard A Survey of Geometry. Boston: Allyn and Bacon, 1972
Hartley, Miles C. Patterns of Polyhedra. Ann Arbor MI: Edward Bros., 1957
Hilbert, D. and Cohn-vossen, S. Geometry and the Imagination. New York: Chelsea, 1952
The five Platonic regular solids
| Name | faces sides | #/kind of faces | Vertices | Edges |
|---|---|---|---|---|
| Tetrahedron | 33 | 4 F3 | 4V | 6E |
| Hexahedron (cube) | 43 | 6 F4 | 8V | 12E |
| Octahedron | 34 | 8 F3 | 6V | 12E |
| Dodecahedron | 53 | 12 F5 | 20V | 30E |
| Icosahedron | 35 | 20 F3 | 12V | 30E |
The 13 Archimedean semiregular solids
| Name | faces sides | #/kind of faces | Vertices | Edges |
|---|---|---|---|---|
| 1 Truncated tetrahedron | 3.62 | 4 F3, 4 F6 | 12V | 18E |
| 2 Cuboctahedron | (3.4)2 | 8 F3, 6 F4 | 12V | 24E |
| 3 Truncated cube | 3.82 | 8 F3, 6 F8 | 24V | 36E |
| 4 Truncated octahedron | 4.62 | 6 F4, 8 F6 | 24V | 36E |
| 5 Small rhombicu-boctahedron | 3.43 | 8 F3, 18 F4 | 24V | 48E |
| 6 Great rhombicuboctahedron or Truncated cuboctahedron | 4.6.8 | 12 F4, 8 F6, 6 F8 | 48V | 72E |
| 7 Snub cube | 34.4 | 32 F3, 6 F4 | 24V | 60E |
| 8 Icosidodecahedron | (3.5)2 | 20 F3, 12 F5 | 30V | 60E |
| 9 Truncated dodecahedron | 3.102 | 20 F3, 12 F10 | 60V | 90E |
| 10 Truncated Icosahedron | 5.62 | 12 F5, 20 F6 | 60V | 90E |
| 11 Small rhombicosidodecahedron | 3.4.5.4 | 20 F3, 30 F4, 12 F5 | 60V | 120E |
| 12 Truncated Icosidodecahedron | 4.6.10 | 30 F4, 20 F6, 12 F10 | 120V | 180E |
| 13 Snub dodecahedron | 34.5 | 80 F3, 12 F5 | 60V | 150E |
The 13 semiregular polyhedra
| #1) 4F3, 4 F6, 12V, 18E | #2) 8F3, 6 F4, 12V, 24E | #3) 8F3, 6 F8, 24V, 36E | #4) 6F4, 8 F6, 24V, 36E |
|---|---|---|---|
 |  |  |  |
| #5) 8F3, 18 F4, 24V, 48E | #6) 12F4, 8 F6, 6F8, 48V, 72E | #7) 32F3, 6 F4, 24V, 60E | #8) 20F3, 12 F5, 30V, 60E |
 |  |  |  |
| #9) 20F3, 12 F10, 60V, 90E | #10) 12F5, 20 F6, 60V, 90E | #11) 20F3, 30 F4, 12 F5, 60V, 120E | #12) 30F4, 20 F6, 12F10, 120V, 180E |
 |  |  |  |
| #13) 80F3, 12 F5, 60V, 150E | |||
 | |||
Some geometry, theorems and course syllabi
Geometry
Six types of Ruled Surfaces
Half Twist Ruled Surfaces
p/q Twist Ruled Surfaces
Saddle (hypar) Surfaces
Geometry of Bridge construction
The Seven Wonders of the Ancient World
The 13 Archimedian semiregular polyhedra
Theorems
Certain Periodic Polar Curves
Monge's Twist-surface Theorems
Hyperpower Function xxx . . .
Theorems of Girolamo Sacceri, S.J. and his hyperbolic geometry
Saccheri's Solution to Euclid's BLEMISH
Course syllabi
Analysis III
Ordinary differential Equations
