In this Build Week activity, we dug into the math behind the art of M. C. Escher.

M. C. Escher’s images of impossible staircases, distorted reflections, and parallelograms metamorphosing into birds and fishes are well-known. Though Escher claimed not to be a mathematician, he kept his eyes open to sources of mathematical inspiration, from the exquisite symmetries of Islamic tilework and carvings in the Alhambra to a diagram of the Poincaré hyperbolic disc model in a contemporary geometry textbook. For one afternoon this past Build Week, we dug into the math behind Escher’s works and created some “Escherian” drawings of our own.

We began by trying to identify all the polyhedra in Escher’s “Stars”. The Platonic solids, familiar to middle schoolers from Maker Studio, are all there, which led to an impromptu lesson on how to draw a good-looking icosahedron step by step. Harder to spot—and to draw—are various truncated, stellated, and intersecting pairs of polyhedra; we had a debate about whether the cuboctahedron is a cube with the corners shaved off, or an octahedron subjected to the same treatment. (It’s both.)

Next, we examined “Sphere Spirals,” which gives a surprising answer to this question: If you set a course at a constant compass bearing, say 30° north of east, what path would you trace out around the globe? A due north course will take you straight to the North Pole along a meridian; a due east course will follow a parallel of latitude; but try any noncardinal direction, and you’ll spiral in ever-tightening coils as you approach the Pole. We considered questions such as: Will you circle the Pole finitely or infinitely many times? How far will you have traveled when you get there? To show this spiral path as a straight line, what kind of distortion must a world map engage in? What’s the shortest path between two points on a spherical Earth?

Many of Escher’s drawings are based on what he called “regular divisions of the plane”—a.k.a. tessellations. These begin with simple tilings, usually triangle- or parallelogram-based, which are then modified to become sheets of interlocking crabs or seahorses or lizards. We looked at “Regular Division of the Plane I,” which demystifies the process by showing it in 12 steps: beginning with a featureless fog in the first panel, Escher shows how to carve the grid, detach small indents and reattach them elsewhere, and finally transform the tiles into creatures. By viewing many of Escher’s tessellations, we observed that the underlying grid doesn’t fully determine the pattern; there are many possible systems of symmetries, known as the 17 wallpaper groups. Using a schematic chart, students identified which group was represented by each of Escher’s symmetry drawings. Along the way, many of us recognized a type of symmetry we’d never had a name for—glide reflection.

To create the illusion of infinity in a bounded space, Escher based some of his tessellations on hyperbolic geometry. Members of our group who took Sachi’s Aspects of Geometry course in Block 3 were able to explain to us how this works!

Finally, we explored the distortions Escher used in works such as “Balcony,” “Hand with Reflecting Sphere,” and “Print Gallery.” The last of these, depicting a museum with a framed hanging of a city that bursts through its frame to encompass the museum itself, is so deviously complex that Escher was forced to leave a blank spot at its center; it would be almost 50 years before a group of mathematicians would analyze Escher’s technique and figure out what belongs in that space.

A common technique for copying a drawing is to superimpose a grid on the original, then copy it out square by square onto a target grid. In a twist, I gave students grids with various distortions applied to them: one- and two-point perspective, a “peephole” bulge, spherical reflection, and so on. The students then chose their own drawings to copy to Escherian effect. Although we ran short of time in the end, the few finished sketches (including two Sigma Cats) were successful beyond my expectations, and I regret that I didn’t get to see the final result of one student’s attempt to scrutinize a certain presidential candidate in a fish-eye lens.

--Austin Shapiro