# Penrose Tiles

### Mathematics, Penrose Tiles, Roger Penrose

———————

In 1974, Roger Penrose, a British mathematician, created a revolutionary set of tiles that could be used to cover an infinite plane in a pattern that never repeats. In 1982, Daniel Shechtman, an Israeli crystallographer, discovered a metallic alloy whose atoms were organized unlike anything ever observed in materials science. Penrose garnered public renown on a scale rarely seen in mathematics. Shechtman won the Nobel Prize. Both scientists defied human intuition and changed our basic understanding of nature’s design, revealing how infinite variation could emerge within a highly ordered environment.

At the heart of their breakthroughs is “forbidden symmetry,” so-called because it flies in the face of a deeply ingrained association between symmetry and repetition. Symmetry is based on axes of reflection—whatever appears on one side of a line is duplicated on the other. In math, that relationship is reflected in tiling patterns. Symmetrical shapes such as rectangles and triangles can cover a plane with neither gap nor overlap, and in an ever-repeating pattern. Repeated patterns are called “periodic” and are said to have “translational symmetry.” If you move a pattern from place to place, it looks the same.

Penrose, a bold, ambitious scientist, was interested less in identical patterns and repetition, and more in infinite variation. To be precise, he was interested in “aperiodic” tiling, or sets of tiles that can cover an infinite plane with neither gap nor overlap, without the tiling pattern ever repeating itself. That was a challenge because he couldn’t use tiles with two, three, four, or six axes of symmetry—rectangles, triangles, squares, and hexagons—because on an infinite plane they would result in periodic or repeated patterns. That meant he had to rely on shapes believed to leave gaps in the tiling of a plane—those with forbidden symmetries.

Penrose turned to five-axis symmetry, the pentagon, to create his plane of non-repeating patterns, in part, he has said, because pentagons “are just nice to look at.” What was remarkable about Penrose tiles was that even though he derived his tiles from the lines and angles of pentagons, his shapes left no awkward gaps. They snugged together perfectly, twisting and turning across the plane, always coming close to repetition, but never quite getting there.

Penrose tiling captured public attention for two major reasons. First, he found a way to generate infinitely changing patterns using just two types of tiles. Second, and even more spectacular, his tiles were simple, symmetrical shapes that on their own betrayed no sign of their unusual properties.

Penrose made several versions of his aperiodic tile sets. One of his most famous is known as the “kite” and the “dart.” The kite looks like the kids’ toy of the same name, and the dart looks like a simplified outline of a stealth bomber. Both divide cleanly along axes of symmetry and each has two simple, symmetrical arcs on their surface. Penrose established one placing rule: for a “legal” tile placement these arcs must match up, creating contiguous curves. Without this rule, kites and darts can be placed together in repeating patterns. With this rule, repetition never comes. The kite and the dart tile forever, dancing around their five axes, creating starbursts and decagons, winding curves, butterflies and flowers. Shapes recur but new variations keep creeping in.

Edmund Harriss, an assistant clinical professor in mathematical studies at the University of Arkansas, who wrote his Ph.D. thesis on Penrose tiles, offers a comparison. “Imagine you’re on a world that is just made up of squares,” Harriss says. “You start walking, and when you get to the edge of the square, and the next square is exactly the same, you know what you’re going to see if you walk forever.” Penrose tiling has the exact opposite nature. “No matter how much information you have, how much you’ve seen of the tiling, you’ll never be able to predict what happens next. It will be something that you’ve never seen before.”