Customer Reviews

Shapes, Space, and Symmetry

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A friend of mine lent me this book in 1975. (I still haven't given it back). Although this 200 page book is very simply and clearly written, I have never been able to sit down and read it from start to finish. Each part I read makes me stop and contemplate. After 24 years I still find new things in it! The book starts out describing the five Platonic solids. Next it explores the dualities: between the octahedron and cube, between the dodecahedron and icosohedron, and between the tetrahedron with itself. Holden talks about solids discovered by Kepler and Poinsot, space filling solids other than the cube, Nolids, lattices and a whole lot more. He also describes how to make your own models with cardboard and Elmers' glue. Doug Kendall's photographs of Holden's models are very pleasing. This is my favorite book.

Alan Holden must be a very dedicated man. This short book on polyhedra is filled with pictures of intricate paper models, all made by the author. In the final section, showing how to construct similar models, one can see a photo of his workshop with hundreds of models arrayed neatly on shelves behind him. This book was clearly a work of love.

Most of the book is occupied with a treatment of regular and semiregular polyhedra, prisms and antiprisms. These are examined in some depth--for example, all nine regular polyhedra are constructed. The last fifty pages introduce other topics, such as packing, lattices, and knots; the treatment here is very brief, somewhat disappointing and leaving a desire for more depth. The same can be said of the final section, on construction--Holden gives general guidelines but leaves the reader to compute the dimensions of all the faces of his models himself.

The prose is clear and concise, rare for a mathematics book. But the real substance lies in the photographs of polyhedra models. These are contructed in such a way that it is always easy to see the details of the solid: faces of different shapes are made of different shades of paper, complicated models are shown in intermediate stages of construction, polyhedra to be compared (such as duals) are shown as individuals and interpenetrating. The great icosidodecahedron photo on page 112 (or its companion that might go by the same name on page 98) is almost worth the price of the book by itself.

This is not a rigorous treatment of the subject, but it is a beautiful one.

This well-made and inexpensive book is brim-full of pictures of Alan Holden's models of polyhedra. It is a book for the hobbyist and the enthusiastic closet Pythagorean, more than it is for the professional mathematician. It is especially useful as an introduction to Archimedean star polyhedra, which are surely as beautiful as anything in geometry, and which were not fully catalogued until the 1950s.

If you find this material as compellingly fascinating as I do, you may want to follow up this book with these two:

"Polyhedron Models," by Magnus Wenninger, has a more thorough and systematic treatment of the Archimedean star polyhedra than Holden's book. These include some incredibly complicated models of "snub" star polyhedra -- spectacular stuff that is not included here. (On the other hand, Wenninger's book costs a good deal more.)

"Regular Polytopes," by H.S.M. Coxeter, is an elegantly written introduction to polyhedra in 3 and 4 dimensions. Coxeter himself wrote the first systematic treatment of the Archimedean star-polyhedra, and helped to discover the last few in the process. This book's illustrations are nowhere near as nice as the other ones', but this is balanced by its more rigorous mathematical treatment of the theme. Somebody needs to come up with a better way (using computer graphics?) to illustrate higher-dimensional polyhedra. In the meantime, this inexpensive book is the best I know on the subject.

This book illustrates many beautiful ideas through photographs of paper models with concise commentary. First the regular polyhedra, of course, followed by variations of the models that brings out their duality and symmetry properties. We also study how the regular polyhedra can be compounded, inscribed in each-other, and pasted together along a common cross-section ("when halves of the Platonic solids having the same cross-sections come together, monsters are born"). We also study truncating the regular polyhedra to obtain the semi-regular polyhedra, stellating to obtain non-convex regular solids, and faceting to obtain non-convex semi-regular polyhedra. There is a deplorably short but somewhat useful section with practical advice on building models. "Suitable cardboard is of the sort used for mounting photographs, and suitable glue is a 'white glue' of the sort exemplified by 'Elmer's Glue-all' ... For several hours the joints remain sufficiently flexible to permit adjustment of the dihedral angle". It's a pity that we are advised to construct the sides of the regular polyhedra by unromantic means; equilateral triangles are to be made using "a draftsman's 30-60-90 triangle" and the construction of a regular pentagon "requires a guide made of heavier cardboard, such as an illustration board, with two edges meeting at the desired angle"---obviously you will want to construct the faces using ruler-and-compass constructions.

other reviewers have used the words GEM, CHARMING, ELEGANT.
they are all RIGHT!
this is one of my favorite books on polyhedra.
beautifully illustrated.
good companion to cromwell.

This book is beautiful. It is easy to read and the ideas are beautifully illustrated through photos of models created by the author, a real feat given the 3D subject. A guide for building the models is included; as the author states, the best way to learn about these shapes is to build them. The book carefully and succinctly guides the non-mathematician through fascinating and illuminating transformations of regular polyhedra using their symmetries and duality.

For the artist or craftsman who wants to make something inspired by polyhedra, this book is perfect. This is one of my favorite books because every time I pick it up I have a new idea for something to make.