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By Schlepkin A.K.

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The tetrahedron consists of four equilateral triangular faces, four vertices and six edges. 1]. The tetrahedron has seven axes of rotational symmetry: four axes of threefold rotation connect each vertex with the midpoints of the opposite faces, and three axes of two-fold rotation pass through midpoints of the opposite edges. In addition to rotational symmetry, the tetrahedron possesses six planes of reflection passing through axes of two-fold rotation and the edges of the tetrahedron. As noted previously, the tetrahedron is its own dual polyhedron and therefore connecting the centres of the faces of a tetrahedron forms another tetrahedron.

1 Lattice structures It has been established in earlier sections that all-over patterns exhibit symmetries, which combine to produce seventeen possibilities across the plane. Proof of the existence of only seventeen all-over pattern classes is provided by Weyl [1952], Coxeter [1969] and Martin [1982]. Designs possessing the same symmetry combinations are said to belong to the same symmetry class, and may be classified accordingly. Further accounts of the classification and construction of all-over patterns were given by Woods [1935], Schattschneider [1978], Stevens [1984], Washburn and Crowe [1988] and Hann and Thomson [1992].

51]. 158]. 294-5]. 51]. It is from Plato’s name that the term Platonic solid is derived. 1]. Propositions thirteen to seventeen of Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron and dodecahedron respectively. 419]. 195], attributed the discovery of the thirteen semi-regular polyhedra to Archimedes. Although Archimedes’ original manuscript on the semi-regular solids was lost, these thirteen solids were named the Archimedean polyhedra in tribute. 81]. Seven of the Archimedean solids can be obtained from the Platonic solids by slicing off either vertices or edges with a cutting plane.

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