Penrose Stairs (The Impossible Staircase)

Climbing forever

The Penrose stairs are a square staircase of four flights. Walk round it clockwise and every step goes up; after four flights you're back where you started, having climbed the whole way. Walk the other way and every step goes down, round and round.

Each flight is drawn as an ordinary staircase would be, with no shading tricks or strange perspective. It's only the loop as a whole, four honestly drawn upward flights that somehow return to their own starting height, that can't exist.

Each flight checks out

Depth in a flat drawing comes from local cues: how each tread and riser is drawn, which edges converge, which surfaces cover others. The visual system reads them one flight at a time, and at every stage "upwards" is the only sensible answer. Nothing in any single flight warns of trouble two corners later. When the loop closes, the contradiction gets absorbed rather than flagged, because the visual system is much better at extending structure that makes sense locally than at checking that the whole thing adds up.

That's how the brain assembles a scene in general. Depth and orientation are worked out region by region and stitched together, on the safe assumption that real objects won't contradict themselves between one corner and the next. The stairs are a deliberate exception. The Shepard tone does the same thing with sound: each step sounds higher than the last, yet the scale keeps returning to where it began.

From the Penroses to Escher

The psychiatrist and geneticist Lionel Penrose and his son Roger, then a young mathematician, published the endless staircase alongside the impossible triangle in their 1958 paper on "impossible objects" in the British Journal of Psychology (Penrose & Penrose, 1958). The Swedish artist Oscar Reutersvärd had been drawing impossible figures since the 1930s, but it was the Penroses' version that entered the psychology literature and popular culture. Their paper reached M. C. Escher, who built his 1960 lithograph Ascending and Descending around it: a monastery roof where two lines of monks trudge endlessly up and down the same staircase (Ernst, 1976).

Building one for real

You can't build the stairs as a closed loop, but you can build an object that looks like them from one particular viewpoint, with a gap or a hidden drop that lines up with your line of sight. Models and video-game levels do exactly that, which is why they only work from a fixed camera angle. The Penrose triangle page has a model you can turn to see the same trick: from one exact direction, three separate beams look like a closed, impossible triangle. Escher's Waterfall applies the same loop to flowing water.