By B.G. Hann, M.A. Thomas

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**Example text**

The dual relationship between the dodecahedron and the icosahedron is shown in Figure 28. Interchanging the corners and faces of the tetrahedron produces another tetrahedron, as shown in Figure 29. The tetrahedron is therefore a self-dual as the two are geometrically similar. 14]. 30]. The Platonic solids are related to each other in many ways aside from their duality. Euclid, in Book XIII of Elements, showed how the solids were associated through the golden mean. In any convex solid, a theorem of Euclid also explains that the angles at any vertices must amount to less than 360 degrees, with the angle calculation often falling considerably short of 360 degrees.

Euclid, in Book XIII of Elements, showed how the solids were associated through the golden mean. In any convex solid, a theorem of Euclid also explains that the angles at any vertices must amount to less than 360 degrees, with the angle calculation often falling considerably short of 360 degrees. 1]. The polyhedral formula, discovered independently by Euler (in 1752) and Descartes, relating to the number of vertices, faces and edges of a polyhedron states: V + F – E = 2. 252-253]. 41 The Schläﬂi notation {p,q} can be used to describe a polyhedron, whose faces are p-gons (denoted by {p}), with q the number of faces meeting at each polyhedron vertex.

In addition to rotational symmetry, the octahedron exhibits nine planes of reﬂection. The octahedron possesses the same symmetry characteristics as the cube. 194] The dual relationship of the octahedron and the cube is shown in Figure 27. An illustration of a net for the octahedron is shown in Figure 36. 5 The dodecahedron The dodecahedron is composed of twelve pentagonal faces, thirty edges and twenty vertices. 1]. The dodecahedron exhibits thirty-one axes of rotational symmetry. Fifteen axes of two-fold rotation pass through the midpoint of opposite edges, ten axes of three-fold rotation connect opposite vertices, and six axes of ﬁve-fold rotation link the centres of opposite faces.