Objects in space are here considered from the point of view of geometrical symmetry. Accordingly, only their shape or form is important, and their size or position in space may be ignored. By comparing the symmetry of a given object with that of nearby objects of the same kind, we can identify those objects that are ‘perfect’ of their kind in a precise, technical sense. Among convex polyhedra with given incidence structure, there are exactly nine instances of such perfection of form.
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References
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RobertsonS. A., Classifying Triangles and Quadrilaterals, Math. Gaz. 61, 38–41 (1977).
2.
RobertsonS. A. and CarterS., On the Platonic and Archimedean Solids, J. London Math. Soc. 22, 125–132 (1970).
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ReevesJ., The Blackbird in the Lilac, Oxford University Press, London (1952).
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AldersonB., Cakes and Custard(illustrated by Helen Oxenbury), Heinemann, London (1974).
5.
There is a choice of highly readable books on polyhedra. My favourite is H. S. M. Coxeter's Regular Polytopes which first appeared in 1948 (A third edition is available, very cheaply, under the Dover imprint). Another deservedly well-known text is that of L. Fejes Tóth on Regular Figures (Pergamon, 1964), Coxeter has recently completed a companion volume on Regular Complex Polytopes (Cambridge, 1974). For an extensive account of the combinatorial theory, see B. Grünbaum's Convex Polytopes (Interscience, New York, 1967).
6.
An English translation of Pascal's Pensées is available in the Penguin Classics and the same series includes Plato's Timaeus, where the regular solids are discussed at some length. Kepler's Mysterium Cosmographicum, which was published in 1596, describes his attempt to relate the regular solids to the distances of the planets from the Sun. (See Opera Omnia, Heyder and Zimmer, Frankfurt, 1864.) T. L. Heath's edition of Euclid's Elements is available in a Dover reprint. Two books of great interest in this field, both designed for the general reader, are Hermann Weyl's Symmetry (Princeton 1952), and D. Pedoe's Geometry and the Liberal Arts (Penguin Books, 1976).
