M.C. Escher

Maurits Cornelis Escher (1898–1972) never considered himself a mathematician, and for much of his career the art world wasn't sure what to make of him either. He trained as a graphic artist in Haarlem in the Netherlands, and his early work, made while living in Italy in the 1920s and early 1930s, was largely conventional: landscapes and studies of Mediterranean hill towns.

The turn came from the Alhambra in Granada, which he visited in 1922 and again, more decisively, in 1936. He was fascinated by the Moorish tilework, interlocking patterns that cover a surface with no gaps. The tiles are abstract; Escher wondered what would happen if the same principles were applied to birds, fish, lizards and people. His notebooks of "regular divisions of the plane" show him working this out with the rigour of a mathematician, though he had no mathematical training (Schattschneider, 1990).

Pictures that can't be built

Over the 1940s and 1950s that interest in pattern turned into something stranger: testing how much a flat picture can get away with. Escher began making prints that look like ordinary architecture but contain a contradiction somewhere, a structure that can be drawn consistently line by line yet could never be built.

Belvedere (1958) shows a tower whose upper storey is turned at right angles to the lower one, held up by pillars that join front to back. On a bench sits a young man holding an impossible cube, with a drawing of a Necker cube at his feet, the ambiguous figure from which the impossible one is made. Ascending and Descending (1960) puts monks on a staircase that climbs forever, the Penrose stairs. Waterfall (1961) builds an aqueduct on the Penrose triangle, so that water flows downhill all the way round a loop and back to the top of its own waterfall (Ernst, 1976).

The Penrose connection

The connection with the Penroses ran both ways. Roger Penrose saw Escher's work at an exhibition held alongside the International Congress of Mathematicians in Amsterdam in 1954, and the experience led him and his father Lionel to their impossible figures. Their 1958 paper was sent to Escher, who corresponded with them and built his next two impossible prints on its figures (Penrose & Penrose, 1958). Few artists have taken ideas so directly from a psychology journal, or given so much back to it.

Why it matters to perception

Escher's impossible prints are precise. Each joint, corner and shadow follows the ordinary rules of drawing when you look at it alone; the impossibility only appears when the visual system tries to join the pieces into one solid object and can't. That's exactly why impossible figures interest vision scientists: they show that the brain builds three-dimensional shape piece by piece and can be led to stitch together pieces that don't add up. The Waterfall demo on this site lets you trace the water round the loop and find the joint that can't exist.

Escher also explored figure and ground (in Sky and Water I, 1938, a row of birds turns into a row of fish through shapes that serve as both), as well as recursive images such as Drawing Hands (1948). He approached these as design problems rather than experiments, yet they repeatedly paralleled what perception researchers were studying at the same time. He remains both a fixture of psychology textbooks and one of the twentieth century's most distinctive printmakers.