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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 **4 ^{3}**.

A snub cube (which is # 7 in the list) is written

because

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 3^{4}.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. **F _{3}** represents a triangle;

The number of these faces in a given polyhedron is the coeficient:

e.g.

The number of

e.g. a cube would have

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

** **

Name | faces sides | #/kind of faces | Vertices | Edges |
---|---|---|---|---|

Tetrahedron | 3^{3} | 4 F_{3} | 4V | 6E |

Hexahedron (cube) | 4^{3} | 6 F_{4} | 8V | 12E |

Octahedron | 3^{4} | 8 F_{3} | 6V | 12E |

Dodecahedron | 5^{3} | 12 F_{5} | 20V | 30E |

Icosahedron | 3^{5} | 20 F_{3} | 12V | 30E |

Name | faces sides | #/kind of faces | Vertices | Edges |
---|---|---|---|---|

1 Truncated tetrahedron | 3.6^{2} | 4 F_{3}, 4 F_{6} | 12V | 18E |

2 Cuboctahedron | (3.4)^{2} | 8 F_{3}, 6 F_{4} | 12V | 24E |

3 Truncated cube | 3.8^{2} | 8 F_{3}, 6 F_{8} | 24V | 36E |

4 Truncated octahedron | 4.6^{2} | 6 F_{4}, 8 F_{6} | 24V | 36E |

5 Small rhombicu-boctahedron | 3.4^{3} | 8 F_{3}, 18 F_{4} | 24V | 48E |

6 Great rhombicuboctahedron or Truncated cuboctahedron | 4.6.8 | 12 F_{4}, 8 F_{6}, 6 F_{8} | 48V | 72E |

7 Snub cube | 3^{4}.4 | 32 F_{3}, 6 F_{4} | 24V | 60E |

8 Icosidodecahedron | (3.5)^{2} | 20 F_{3}, 12 F_{5} | 30V | 60E |

9 Truncated dodecahedron | 3.10^{2} | 20 F_{3}, 12 F_{10} | 60V | 90E |

10 Truncated Icosahedron | 5.6^{2} | 12 F_{5}, 20 F_{6} | 60V | 90E |

11 Small rhombicosidodecahedron | 3.4.5.4 | 20 F_{3}, 30 F_{4}, 12 F_{5} | 60V | 120E |

12 Truncated Icosidodecahedron | 4.6.10 | 30 F_{4}, 20 F_{6}, 12 F_{10} | 120V | 180E |

13 Snub dodecahedron | 3^{4}.5 | 80 F_{3}, 12 F_{5} | 60V | 150E |

#1) 4F_{3}, 4 F_{6}, 12V, 18E |
#2) 8F_{3}, 6 F_{4}, 12V, 24E |
#3) 8F_{3}, 6 F_{8}, 24V, 36E |
#4) 6F_{4}, 8 F_{6}, 24V, 36E |
---|---|---|---|

#5) 8F_{3}, 18 F_{4}, 24V, 48E |
#6) 12F_{4}, 8 F_{6}, 6F_{8}, 48V, 72E |
#7) 32F_{3}, 6 F_{4}, 24V, 60E |
#8) 20F_{3}, 12 F_{5}, 30V, 60E |

#9) 20F_{3}, 12 F_{10}, 60V, 90E |
#10) 12F_{5}, 20 F_{6}, 60V, 90E |
#11) 20F_{3}, 30 F_{4}, 12 F_{5}, 60V, 120E |
#12) 30F_{4}, 20 F_{6}, 12F _{10}, 120V, 180E |

#13) 80F_{3}, 12 F_{5}, 60V, 150E | |||

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