Shapes, Space, and Symmetry by Alan Holden, and: The Divine Proportion: A Study in Mathematical Beauty by H. E. Huntley (review)

Books 79 alternating between the realm of mathematical fact and the realm that speaks of'The interplay ofcosmic polarities with their breathing reciprocity ....' All this we find in the final chapter, entitled 'Geometry [sic] of the Twentieth Century'. However, in the closing pages of the penultimate chapter, something from modern geometries is touched upon, namely the good old 'Moebius Leaf', and the Real Projective Plane ('Boys' Surface'). Concerning the latter, there is a most suggestive illustration, recalling a sculpture by Hans Arp (Fig. 3)-which in no way conveys to the reader what is really at stake; for this the reader will have to consult Hilbert, Cohn Vossen, or Leitzman's 'Visual Topology' or, for that matter, Lewis Caroll in 'Sylvie and Bruno'. (For the last reference I am indebted to H.S.M. Coxeter, author, amongst other mathematical classics, of one entitled 'The Real Projective Plane'.) Shapes, Space, and Symmetry. Alan Holden. Columbia University Press, New York, 1971. 200 pp., illus. £5.25. The Divine Proportion: A Study in Mathematical Beauty. H. E. Huntley. Dover, New York, 1970. 186 pp., illus. $2.50. Reviewed by: William Moser* There is no contents list, no subdivision into chapters and sections, no numbering of photos to clutter Holden's book. Just a fine collection of photographs of polyhedra and a simple text. With one exception (p. 20) every page contains a photograph -202 in all. The accompanying text has the 'feel' ofa narrator's script, as if the photographs are 'frames' from a film. The photographs of Holden's beautiful models are by Doug Kendall. It is a labour of love, for one can sense, indeed see, the huge amount of thought, patient work and skill that must have gone into the book's construction and the photographing. Platonic solids, their dual relationships, symmetries , sections of them by planes, how they 'fit' into each other, compound solids, semi-regular (Archimedean) solids, star-polyhedra (but not zonohedra), truncation, stellation, faceting, spaccfilling , relation to growth forms and more are described but the photographs of the models are the jewels. The last few pages describe briefly the tools, materials and techniques used to construct the models. Not unexpectedly, there are photographs ofthe author in his workshop surrounded by models. There is a short bibliography, which could have been longer, for example, the following important ones are missing: W. W. R. Ball and H. S. M. Coxeter, 'Mathematical Recreations and Essays' and H.S.M. Coxeter, P. Duval, H. T. Flather and J. F. Petrie, 'The Fifty-nine Icosahedra'. The sub-title of Huntley's book deserves equal billing with the title. The author wanders back and *University College, University of London, W.C.t, England. forth, in and out and around, literally, dozens of topics are covered, for example, emotional activity, aesthetics, creativity, pleasure, patterns, unity in variety, teleology and mathematics, particularly the golden ratio and its many fascinating properties and occurrences in mathematics. Reading this book helped me to pass a most enjoyable few hours of a transatlantic flight. The prose is simple (the author is unashamedly 'open' and 'warm') and there is much to enjoy, much to agree with and some to argue about, for example, the author states that (in his own opinion) the act of creation and the act of appreciation of beauty are not in essence distinguishable. The Geometry of Environment. L. March and P. Steadman. R.I.B.A. Publications, London, 1971. 360 pp., illus. £6.00. Reviewed by: Michael Challinor* The authors of this excellent book are architects. Their stated two-fold purpose is to introduce 'new' mathematics to an 'older generation' and to show younger readers that architecture can be an exciting subject, not wholly concerned with ancient styles or stresses in beams. This seems to me to underestimate the usefulness of the book, which can be of value also to civil engineers, historians of architecture and, more especially, to artists whose works have a basis in mathematics. New mathematics is not so much a departure from the old but a re-ordering of the manner of studying mathematical subjects. The authors introduce symbolic logic, group theory, matrices, vectors, modular arithmetic and graph theory in a readable...