Penrose Stairs

What is happening

A staircase seems to run around the top of a square block. Pick any step and follow it clockwise. You go up one step at a time, yet after four corners you are back at the step you started from. If the picture were true, you could climb forever without getting any higher.

Why it looks this way

This picture also shows a solid that can actually be built. Over one lap the steps rise little by little, and at the same time they drift sideways and backwards. When the rise and the drift are exactly the same size, the step after one lap lands in exactly the same spot as the first step — but only when seen from one direction, diagonally from above.

A flat picture does not record how far back something is. If a higher, nearer step and a lower, farther step land in the same place, we take them to be the same step. Every pair of neighbouring steps looks right, which makes the contradiction in the loop as a whole hard to notice.

How to check

The answer view turns the same staircase sideways. The two columns painted orange are the first and last steps that seemed to join in the normal view. Turned, the last step stops far above the first one. Compare it with the triangle drawn by the same trick.

Try the Penrose triangle too

For reference: Penrose stairs (Wikipedia). The figure here is an original implementation.

First described

After Lionel & Roger Penrose, 1958

A geneticist father and mathematician son introduced it with the triangle in their 1958 paper. It inspired Escher’s print Ascending and Descending (1960).

See it in the timeline