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Platonic and Archimedian Polyhedra

The five Platonic regular polyhedra and the 13 semiregular polyhedra

Warning: Some netscape versions may not exponentiate properly: i.e. It would not be clear that x2 means x squared

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

Namefaces
sides
#/kind
of faces
VerticesEdges
Tetrahedron334 F34V6E
Hexahedron (cube) 436 F48V12E
Octahedron348 F36V12E
Dodecahedron5312 F520V30E
Icosahedron 3520 F312V30E

The 13 Archimedean semiregular solids

Namefaces
sides
#/kind
of faces
VerticesEdges
1 Truncated tetrahedron 3.624 F3, 4 F612V18E
2 Cuboctahedron(3.4)2 8 F3, 6 F412V24E
3 Truncated cube3.82 8 F3, 6 F824V36E
4 Truncated octahedron4.62 6 F4, 8 F624V36E
5 Small rhombicu-boctahedron3.43 8 F3, 18 F424V48E
6 Great rhombicuboctahedron
or Truncated cuboctahedron
4.6.812 F4, 8 F6, 6 F848V72E
7 Snub cube34.4 32 F3, 6 F424V60E
8 Icosidodecahedron(3.5)220 F3, 12 F530V60E
9 Truncated dodecahedron3.10220 F3, 12 F1060V90E
10 Truncated Icosahedron5.6212 F5, 20 F660V90E
11 Small rhombicosidodecahedron3.4.5.420 F3, 30 F4, 12 F560V120E
12 Truncated Icosidodecahedron4.6.1030 F4, 20 F6, 12 F10120V180E
13 Snub dodecahedron 34.580 F3, 12 F560V150E

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



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