Escher's Waterfall (The Impossible Waterfall)

Water that falls forever

M. C. Escher's 1961 lithograph Waterfall shows a stone aqueduct zigzagging between two towers. The water runs along the channel, falls over a waterfall, turns a mill wheel at the bottom, then flows along the channel again and arrives back at the top of the same waterfall. It's a closed loop that keeps the same water falling forever, with no pump and no hidden pipe.

Trace the channel one stretch at a time and each part is reasonable: a trough running level or gently downhill, as an aqueduct should. Only when you follow the whole loop do you realise the water has somehow climbed back up by the height of the waterfall. Tick "Trace the water" in the demo to follow the route and find the one joint that can't exist.

Built on the Penrose triangle

Escher based Waterfall on the impossible triangle published by Lionel and Roger Penrose in 1958, which the Penroses sent him (Penrose & Penrose, 1958). The zigzagging aqueduct is built from the same trick as the Penrose triangle: a series of joints that each make sense in three dimensions but can't all be true at once when you follow them round a loop.

What makes the print so convincing is everything around the geometry. Escher put the impossible structure in an ordinary-looking scene, with terraced gardens, stonework and a woman hanging out washing. The detail tells the visual system that this is a real, solid place, and the contradiction hides in plain sight inside it.

A loop with no uphill stretch

The print is especially effective because it shows movement, and we have strong expectations about flowing water: it runs downhill or stays level. Escher never shows an uphill section. Every visible stretch slopes the right way, just as the Penrose stairs never show a downward step in their endless climb. The impossibility lies in how the pieces connect, never in any single piece, which is why it survives a second and a third look.

Escher and the Penroses

Escher was already known for prints about tessellation, reflection and paradoxical space when the Penroses' paper reached him, and he corresponded with them about it. He used the Penrose stairs in Ascending and Descending (1960), then the triangle in Waterfall the following year (Ernst, 1976). His earlier Relativity (1953) plays with space in a related way, with three different directions of gravity in one building, but doesn't rely on the Penrose figures. Waterfall is now one of his most reproduced prints. The Kanizsa triangle reaches a shape that isn't there by a completely different route: from missing information rather than contradictory information.