Shapes, Space, and Symmetry

“No ideas but in things.” (William Carlos Williams)

One of the reasons I love this small book is that it is redolent of a different time. It smells like a basement workshop: like X-Acto knives, the Edmund Scientific catalog and Estes rockets.

If someone were to assemble a basic 3D geometry text like Shapes, Space, and Symmetry in 2015, the illustrations (in the hardcopy, paper version at least) would be created using 3D modeling software. The PDF or e-book version might go even further, with interactive graphics that would truly allow one to see how the forms interlace, metamorphose and reflect.

Instead, Alan Holden’s 1971 book (of which this Dover version is a reprint) has black and white photographs of models, constructed of cutouts made of black and white cardboard and wire, painstakingly put together with Elmer’s glue. There’s even a section at the back of the book about how the models were created. There is a sense of craftsmanship and a whiff of dusty things.

Shapes, Space and Symmetry is an informal, heavily-illustrated sonata on the regular polyhedra — and some not-so-regular. In one 200-page slab of words and pictures (there are no chapter divisions), Holden takes the reader from the Platonic solids — deftly explaining how they pair off — and on to ever-more-complicated forms, all without a single equation. Along the way he covers concepts like symmetry, space-filling, convexity, reflection and more.

Holden’s method is simplicity itself, but the physical setup for it must have cost countless hours of prep time (model building and photography). Here is a typical sequence in the text:

“When you chop off the tips of the big black cube at the left, you replace its eight corners with little white equilateral triangles. The chopping is called truncation. Chopping further, you reach a stage at which the triangles meet at their vertices; still further, the triangles become hexagons ...

This description of how to cut a cube down to an octahedron is accompanied by a photograph of a marching row of solids — black cube at one end, white octahedron at the other, with the three intermediate forms clarified by the use of contrasting black and white cardboard.

This book could engender 3D visualization skills and fire a passion for geometry in young or old. Holden’s text is not always easy reading, because one has to constantly check to make sure one is following the instructions correctly, but along the way one learns how to see. This is a book that will change one’s mind — and by that I do not mean change a given opinion or modify a view — rather, it will change the way one sees, perceives and understands.

Would 3D graphics created with software be superior pedagogically? Well — maybe. They’d be more accurate and uniform, probably full color, and the interactivity (in an e-book) would make visualization a snap. The flip side of that coin, however, is that with the present old-fashioned text, the reader has important work to do using the software installed in her own head. That sort of mental exercise has very real benefits.

Probably the only thing that could have made this reprint “more perfect” would have been a return to the old Dover production standards, à la sewn bindings and “[t]his is a permanent book” — but those days are long gone, and they will not return. One must be grateful to Dover for reprinting volumes like this at all.

That said, the reproduction of the black and white photos in my copy run a bit muddy and dim, which renders some of the shapes difficult to see. That is a shame in a book where the images are as important as they are here.

Glenn Becker is a staff member at the Harvard-Smithsonian Center for Astrophysics in Cambridge, MA, where he toils in the data archive of the Chandra X-Ray telescope. He is a “reborn astronomy and mathematics fellow traveler” who spent far too many years getting advanced degrees in theater, only to ultimately abandon the entire discipline.