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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 220.127.116.11 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
The five Platonic regular solids
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
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 18.104.22.168 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|