All these solids were described by Archimedes, although, his
original writings on the topic were lost and only known of second-hand. Various artists
gradually rediscovered all but one of these polyhedra during the Renaissance, and Kepler
finally reconstructed the entire set. Reference to his work can be found in the writings
of Pappus, a Mathematician of the Third Century A.D.. (Magnus J. Wenninger, Polyhedron
Models)
A key characteristic of the archimedean solids is that each face
is a regular polygon, and around every vertex, the same polygons appear in the same
sequence, for example, hexagon - hexagon – triangle in the truncated tetrahedron 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 of polygon.
Truncated Tetrahedron
The notation used to describe these semi-regular polyhedra is
such that it describes the faces that meet at any one vertex. The notation {3, 8, 8} (Truncated Hexahedron) means each vertex contains a triangle (3), an
octagon (8) and another octagon (8) in cyclic order. In many of the solids it is important
to note the order of the notation describing the polyhedrons. When the number of faces
meeting at a vertex exceeds three the order of the description becomes important. For
example the description {3,4,3,4} is very different from {3,3,4,4}. In the latter the two
triangles share a common 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 is bounded by two
triangles.
Truncated Hexahedron
{3,8,8}
The notation associated with this solid
shows that each vertex contains a triangle (3), an octagon (8), and another octagon (8).
Although, the accepted polyhedron names are less than ideal,
there is certain logic to the names of these semi-regular polyhedra. They are adopted from
Kepler’s Latin Terminology. The term snub refers to a process of 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 each previously existing vertex for e.g. it replaces the square face (4
edges) with an octagonal face (8 edges). You get octagons instead of squares. Truncation
of any polyhedron then results in replacing faces of an n-sided polygon with 2n-sided
ones.
Preview of the thirteen semi-regular
polyhedra:
