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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]. A further geometrical element of importance to pattern structure is the underlying framework or lattice, as noted previously. Each lattice (of which there are five distinct types) is comprised of unit 51 cells of identical size, shape and content, which contain the essential repeating unit or element of the pattern or tiling, as well as the symmetry instructions for the pattern’s construction.

4] The polygons that join to form polyhedra are called faces, these faces meet at edges, and edges come together at vertices. 5]. 16]. 53]. The focus of attention in this monograph is with the former group, the Platonic solids. These are as follows: the (regular) tetrahedron, the (regular) octahedron, the cube (or regular hexahedron), the (regular) icosahedron and the (regular) dodecahedron, as illustrated in Figure 24. 41]. There are a total of thirteen semi-regular polyhedra, more commonly known as the Archimedean solids, as shown in Figure 25.

Gerretsen and Verdenduin [1974] determined the symmetry groups of all polyhedra with respect to their vertices; those of the Platonic solids are listed in Table 2 in association with their Schläfli notation. 8]. 43 Descriptions of the symmetry properties of each individual solid are provided in the sections below, together with relevant illustrative material. A key for the diagrams illustrating the symmetry characteristics of the Platonic solids is given below. 4]. The tetrahedron consists of four equilateral triangular faces, four vertices and six edges.

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