The next article for **Plane Geometry** is on **Penrose Tiling**.

Unlike the earlier discussion of Rep-Tiles, which are periodic, Penrose Tilings are nonperiodic, and were first discovered by Roger Penrose. Periodic tilings can be made by shifting the pattern in specific directions without rotation or reflection. These are used extensively in the pictures by M. C. Escher. Nonperiodic tilings, at a minimum then, involve rotating or mirror reflecting the tiles. A very simple nonperiodic tile is the triangle. put two triangles back to back to form either a larger triangle, or a 4-sided rhomboid; the tiles themselves get rotated or flipped, but the patterns they make may themselves be periodic. Periodic tiles can be made nonperiodic by giving the edges different colors and requiring that only tiles with matching colors can be adjacent to each other.

Gardner laments that Escher died before learning about Penrose tiles, but I think M.C.’s butterflies print qualifies. Which is interesting to me, because Roger’s father, geneticist L.S. Penrose, invented the unending **Penrose Staircase**, which Escher depicted in his “Ascending and Descending” woodblock print. (Roger himself worked in general relativity and quantum mechanics.) In 1973, Roger found 6 tiles that are nonperiodic, with notches and tabs that forced the tiles to only fit one way. That was lowered to two triangular tiles, called “kites” and “darts”. Before making them public, he filed patents in the U.K., U.S. and Japan. You can now buy them as a game from Kadon Enterprises.

Penrose tiling, from the wiki entry. (All rights belong to their owners. Images used here for review purposes only.)

In 1993, John Conway came up with a 3D object that could be used to tile an enclosed volume. Called a “biprism”, these objects stack on each other, but in an irregular pattern. Currently referred to as a Schmitt-Conway-Danzer biprism, the cut-out pattern shown below was created for Gardner by Doris Schattschneider, co-creator of the M. C. Escher Kaleidocycles, and first female editor of *Mathematics Magazine*.

An example of tiling Penrose chickens (from the Martin Gardner book)

The Penrose Tiles are tied to the discovery that quasicrystals, crystals with five-fold symmetry, are possible.

I tried to make 4 biprisms, and all four failed in exactly the same way. Granted, I was using very flimsy copy paper, but the final fold when I glued the arms together, always pointed at least 5 degrees off-center, making me think that maybe there’s something wrong with the original pattern. It might be better to use thicker paper stock, and then cut the sides into separate pieces and hold then together with cloth tape. Anyway, the instructions to make these are – use a dried-out pallpoint pen to score along all the lines. Then fold the arms marked “u” upward (valley folds) so that those arms pair up to make a prism on the top of the central rhomboid. And, fold the arms marked “d” downward (mountain folds) to have a matching biprism on the bottom side of the rhomboid.

You may have more fun playing with the Kadon Enterprises game sets, or with your own set of Penrose chickens.