Book Review : Art Meets Math in ‘Kaleidocycles’

M.C. Escher Kaleidocycles by Doris Schattschneider and Wallace Walker (Pomegranate Artbooks, P.O. Box 980, Corte Madera, Calif. 94925: $13.95; 57 pages plus 17 die-cut, prescored models)

When the Dutch artist Maurits C. Escher died in 1972 at the age of 74, he left behind a body of work unique in its imaginative power and creative design.

Escher’s compelling and fascinating drawings combine mathematics and art in a way never done before or since. Years after I first saw his work, I still look at it in amazement. Escher captures and holds my attention as few other artists can.

The depth of Escher’s vision and intellectual strength is perhaps best shown by the variety of work it has spawned. Douglas R. Hofstadter won a Pulitzer Prize for his 1979 book, “Godel, Escher, Bach” (Basic Books), which demonstrated the unity of thought in logic, art and music. Now Doris Schattschneider and Wallace Walker have produced a remarkable activity book that carries Escher’s two-dimensional work into three dimensions, still full of surprises at every turn.


A large portion of Escher’s most imaginative work involved inventing two-dimensional shapes that can completely cover a surface without any gaps or overlaps. A shape that can do this--such as a square or a hexagon, but not a circle--is said to tile the plane. Escher’s tiling shapes were much more complicated, frequently involving human or animal figures that ingeniously interlocked to cover his drawings. Anyone interested in exploring this work should consult “M. C. Escher, His Life and Complete Graphic Work” by Bool, Kist, Locher and Wierda (Harry N. Abrams: 1982). Be careful: you may get hooked.

Long History

So much for two dimensions. The world of three-dimensional shapes has a long history that goes back to the ancient Greeks. A regular solid is an object that has the same regular polygon on each face (all edges and all angles are equal) and all of whose vertexes are the same. The most common example is a cube, each of whose six faces is a square.

The final proof in the final book of Euclid’s “Elements of Geometry” is the theorem that there are five and only five regular solids: the pyramid (four equilateral triangles), the cube (six squares), the octahedron (eight equilateral triangles), the dodecahedron (12 pentagons) and the icosahedron (20 equilateral triangles).


But that does not begin to exhaust everything to be known about solid shapes. New facts are still being discovered. In the 1950s, Walker, one of the authors of this book, invented “kaleidocycles,” which are three-dimensional rings put together in such a way that they can be infinitely rotated in on themselves through a central hole, which can be as small as a point. The word kaleidocycle was coined from the Greek kalos (beautiful) plus eidos (form) plus kyklos (ring). These are extremely interesting forms in their own right.

The upshot of all this is that Schattschneider, a mathematician at Moravian College in Bethlehem, Pa., put together the two-dimensional drawings of Escher with the three-dimensional shapes of Walker to create objects of sublime and uncommon beauty.

Included in this set is a slim book that explains the theory of Escher and Walker along with 17 models that are easily assembled and glued together to bring the theory to life.

“The kaleidoscopically designed geometric forms in this collection are a continuation and extension of Escher’s own work,” Schattschneider writes. “Your involvement is required also! A casual glance cannot reveal the surprises to be discovered in Escher’s prints. So, too, the secrets to be discovered in our models are only revealed by your creating the forms, examining them, and yes, playing with them!”

Putting the models together was not hard. (If you do it, use Yes glue, which is available at an artists supply store. It is easy to work with, and it will hold, while rubber cement does not hold together.) I put the models together in an evening, and I have been fiddling with them since--to my considerable delight.

Repeating Patterns

As Schattschneider explains in the book, getting the Escher drawings on these shapes was not just a matter of slapping them on. They had to be put on in such a way that no matter how the shapes were rotated, the repeating patterns remained continuous. This involved studies of the various axes of symmetry of the drawings, which the book explains.

In addition to the kaleidocycles, there are models of the five regular solids, which have also been covered with Escher prints that form continuous designs even though they do not move. There is also one semi-regular solid, the cuboctahedron, whose faces are triangles and squares.


The resulting objects are beautifully colored and intriguing, and they provide a new way of looking at and appreciating Escher’s fantastic work. They allow you to take his designs and make new designs out of them. Like a wonderful poem or great work of literature, Escher’s drawings have unbounded depth. The work of Schattschneider and Walker points the way to new discoveries.