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Archimedean solids

Seven of the archimedean solids are derived from the platonic solids by theprocess of truncation, literally cutting off the corners of each of theplatonic solids. The first five are the truncated tetrahedron, truncated

octahedron, truncated hexahedron, truncated icosahedron and thetruncated dodecahedron and these are arrived at by dividing the edges intothirds and cutting off the vertices of these points. The other two aresomewhat special and are so presented as the Two Special Archimedeans.There are three other couples of archimedeans solids and all are outlinedbelow.

Archimedean Classification:

the truncation of the five platonic solidsthe two quasi regular polyhedrathe rhombicuboctahedron and the rhombicosidodecahedronthe truncation of the two quasi regular polyhedrathe snub versions

Did you know?

All these solids were described by Archimedes, although, his originalwritings on the topic were lost and only known of second-hand. Variousartists gradually rediscovered all but one of these polyhedra during theRenaissance, and Kepler finally reconstructed the entire set. Reference tohis work can be found in the writings of Pappus, a Mathematician of theThird Century A.D.. (Magnus J. Wenninger, Polyhedron Models)

A key characteristic of the archimedean solids is that each face is a regularpolygon, and around every vertex, the same polygons appear in the same

sequence, for example, hexagon - hexagon triangle in the truncatedtetrahedron as can be seen from the picture at the start of the section.Two or more different polygons appear in each of the archimedean solids,unlike the platonic solids which each contain only one single type ofpolygon.

Truncated Tetrahedron

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The notation used to describe these semi-regular polyhedra is such that itdescribes the faces that meet at any one vertex. The notation {3, 8, 8}(Truncated Hexahedron) means each vertex contains a triangle (3), anoctagon (8) and another octagon (8) in cyclic order. In many of the solids itis important to note the order of the notation describing the polyhedrons.When the number of faces meeting at a vertex exceeds three the order ofthe description becomes important. For example the description {3,4,3,4} isvery different from {3,3,4,4}. In the latter the two triangles share acommon edge and the two squares share a common edge, where as in{3,4,3,4} the triangle is bounded by two squares and likewise the square isbounded by two triangles.

Truncated Hexahedron

{3,8,8}

The notation associated with this solid shows that each vertex contains atriangle (3), an octagon (8), and another octagon (8).

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Although, the accepted polyhedron names are less than ideal, there iscertain logic to the names of these semi-regular polyhedra. They areadopted from Keplers Latin Terminology. The term snub refers to a processof surrounding each polygon with a border of triangles as a way of deriving,for example, the snub cube from the cube. The term truncated refers to

cutting off of corners and results in the addition of a new face for eachpreviously existing vertex for e.g. it replaces the square face (4 edges) withan octagonal face (8 edges). You get octagons instead of squares.Truncation of any polyhedron then results in replacing faces of an n-sidedpolygon with 2n-sided ones.

Preview of the thirteen semi-regular polyhedra:

The Five Truncations

Quasi-regular Polyhedra Small-Rhombi Polyhedra

Truncated-Quasiregular Snub Polyhedra

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