The Archimedean polyhedra are polyhedra with regular polygon faces.
Faces
may be of different types but all the vertices are identical. There are
thirteen of then, shown below.
Except for the truncated tetrahedron, lower right, all the Archimedean polyhedra are modifications of the cube-octahedron pair or the dodecahedron-icosahedron pair. The above figure groups the polyhedra according to their geometric similarities.
The top two rows in the figure above shows how the Archimedean polyhedra can be obtained by modifying the Platonic solids. Progressive truncation of a cube can lead ultimately to an octahedron and vice-versa. Likewise.progressive truncation of a dodecahedron can lead ultimately to an icosahedron and vice-versa.
In the bottom two rows, additional faces are added between the faces of a cuboctahedron or icosidodecahedron, with appropriate modifications of polygon shape. The tetrahedron and truncated tetrahedron are shown in the last pair. Further truncation of the tetrahedron would eventually lead to eight triangular faces, an octahedron.
Snub polyhedra have triangular faces
not created by a threefold symmetry axis. In each case in the last row
the threefold axis face is colored differently from the extra or snub faces.
The vertex description denotes snub faces with s. If we attempt to construct
a snub tetrahedron, we have four triangular tetrahedron faces, four triangular
faces at the threefold symmetry axes, and pairs of triangles at each of
the six tetrahedron edges, a total of 4 + 4 + 2*6 = 20. A solid with 20
equilateral triangle faces is an icosahedron. Thus, a snub tetrahedron
is an icosahedron. An icosahedron colored to show this relationship is
shown.
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Created 2 Oct. 1997, Last Update 2 Oct. 1997
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