Escher’s Art and Penrose Tiles

I nearly fell down the rabbit hole last week writing about the Golden Ratio. I tried to cram in something about the art of M.C. Escher. But then I realized that Escher’s art was so inspired by mathematics, it needed a blog of its own this week.

Maurits Cornelis (M.C.) Escher (1898-1972) was a Dutch graphic artist who created lithographs, wood cuts, and wood engraving. His art played with ideas like symmetry, transformation, and impossible constructions.

Escher investigated symmetry with patterns (https://mcescher.com/gallery/symmetry/). He might repeat the pattern by merely translating it around the page without rotation or reflection. In other cases, he might repeat patterns ad infinitum at different scales. These are fractals, which in more mathematical language we describe as “infinitely complex patterns that are self-similar across different scales”. Many of his works were boldly colored to highlight their symmetry, whether it be rectilinear, circular, or something else.

Then there are the Escher transformations (https://mcescher.com/gallery/transformation-prints/). The “transformation” pieces morph one reality to the next. Some use just black and white, while others include shades of grey, or even colors. For example, Metamorphosis II changes words and lines into checkboard squares, then lizards, hexagons, bees, other insects, fish, birds, cubes, a village, a game of chess, checkerboard squares, and back to the original words and lines.

Still other Escher graphic art involves staircases that climb inexorably to a pinnacle, where you realize you are back at the starting point at the bottom of the stairs. They present impossible constructions of reality which make no sense in the context of the real, Euclidean world (https://mcescher.com/gallery/impossible-constructions/). In fact, Escher got the idea for the staircase graphic after reading a paper about impossible objects like this staircase in a 1958 journal by the mathematician Roger Penrose and his father Lionel.

A great deal of Escher’s art can be described as tiling. A tiling, or tessellation, is a filling of a plane with a set of plane figures (tiles), with no overlaps or gaps between tiles. Escher was inspired by the tiles he had seen at the Alhambra in Granada, Spain, and adopted the idea of patterned tiling in his graphic art. He studied the geometric patterns of these Moorish tiles, and replaced them with his own motifs, most often animals. This fundamental work taught him about transformations that preserved congruences, like translations, rotations, and reflections. Many of Escher’s designs display periodic properties, where shapes or animals repeat in recognizable patterns. A tiling is said to be periodic when you can identify a region which when repeated will fill the plane only by translation, not rotation nor reflection.

In general, some shapes tile only periodically. Other shapes can tile periodically and non-periodically. And a third set of shapes tile only non-periodically. In the 1970s, Roger Penrose discovered several sets of shapes which form only non-periodic tilings. One set of Penrose tiles includes a kite and dart shape, derived from a rhombus with angles of 72 and 108 degrees. By the way, in all three shapes, the Golden Ratio pops out everywhere! To learn more about this special set of tiles, see Martin Gardiner’s 1988 book “Penrose Tiles to Trapdoor Ciphers” at https://www.maa.org/sites/default/files/pdf/pubs/focus/Gardner_PenroseTilings1-1977.pdf.

Escher never thought of himself as a mathematician, but mathematics inspired his art. His fascination with beautiful patterns lead him to do a lot of his own mathematical research, to study technical papers, and to communicate directly with mathematicians and scientists. His breathtaking images still inspire us today with artistic visions of the possible. And impossible.