Shapes, Space, and Symmetry

by Alan Holden

"Altogether a most instructive, entertaining, and esthetically pleasing book." — Science.
Since the ancient Greeks, the visualization of space has been a challenge that has intrigued men of learning. Through centuries of thoughtful looking, a number of three-dimensional figures or polyhedral, as the Greeks called them, have been discovered, admired, and wondered at for their mathematical elegance and beauty. And they have been put to use in remarkably diverse ways by engineers and builders, chemists and crystallographers, architects and sculptors.
This book describes very clearly and simply, and illustrates with beautiful photographs of models, a great number of three-dimensional figures, all but a few consisting of plane faces bounded by straight lines. It examines the nine regular solids — the five commonly called Platonic, described by Theaetetus in the fourth century B.C., and the four called Kepler-Poinsot, two each of which were discovered by Kepler and Poinsot many centuries later. And it examines many variations obtained by truncation, stellation, dualization, and compounding.
Writing for the layman as well as the student or professional in mathematics, Alan Holden explains the structure of the figures and demonstrates how they can be used to explain mathematics visually rather than by symbol systems, an effort hailed by Scientific American magazine as "a victory of clear, connected thinking over the theorematic method." At the end of the book the author includes a section containing instructions for constructing cardboard models.

Genres: Mathematics, Nonfiction

208 pages, Paperback

First published April 1, 1973

Reviews

[Andrew · Rating 4 out of 5]

I wish that I had glanced through this book while studying Crystallography way back when. Holden starts with familiar solids such as the cube and octahedron, and proceeds by steps to solids that can fit inside others, solids that result from cutting corners or edges off of more familiar solids, solids that are made of stars, solids that interpenetrate, collections of polygons that have symmetry in three dimensions but couldn't hold water (the "nolids") -- whew. And more. Holden even gives advice on how to build models of these polyhedra, which are extensively photographed.

The gray-scale illustrations, vintage 1971, were described by one reviewer as beautiful, but today seem drab, grainy, and out of focus. I remember being excited by books like this, however; few books in the seventies had photos on every page. My impression is that one could learn a good deal more about three-dimensional visualization by handling the models than by reading the book, and I had the opportunity to do that with wooden models in Crystallography class. Ultimately the book is not so much about solid polyhedra as about patterns and symmetry relationships, and these are powerful tools in crystallography. The cube and the octahedron look different, but they are really two aspects of one pattern of symmetries.

Toward the end of the book, Holden explores the polyhedra that can pack together to fill three-dimensional space. We are all familiar with the idea that sugar cubes can be packed into a box with no space left over. Rectangular boxes can fill a closet or a room, and they can even be squashed into parallelepipeds and still fill the room. Holden reminded me of a beautiful fact, that boxes can be the shape of garnet crystals -- rhombic dodecahedra -- and still fill three dimensions with no space left over. And he informed me that the boxes don't even have to be convex! You can place a pyramid on each of the faces of a rhombic dodecahedron, making "the first stellation of the rhombic dodecahedron," and, remarkably, these solids will fill space neatly and completely. I'm not planning on storing old photos or clothing in such strangely shaped boxes, but they
would make a fine novelty item.

[Harry Van Der Velde · Rating 5 out of 5]

When I found this book in an Amsterdam Architecture bookshop around 1985, I was thrilled. Finally a visual feast on those fascinating forms and spaces that I could not really wrap my mind around in math class. Today it looks dated, but it is still unsurpassed in it´s hands on accessibility.
I keep going back to grok the intricacies of space and form.

[Peter Sandwall · Rating 4 out of 5]

A unique deep dive into the stellation of objects and solid geometry.

[Røbert · Rating 5 out of 5]

This is just about the perfect book for anyone interested in polyhedra. Starting with the basic Platonics, the book works its way through many other exotic polyhedra, in particular how they can be derived from one another. Richly illustrated throughout and with an informal text that takes you through this journey, this is an deep but enjoyable romp through the subject. Much less mathematical than Cromwell, and less formal than Wenninger, this is the book to go for for both fun, instruction and inspiration.