Impossible Triangle

The Penrose family and impossible objects

In 1958, the mathematician Roger Penrose and his father Lionel Penrose published a short paper in the British Journal of Psychology describing two impossible figures: the “impossible tribar” (now called the Penrose triangle) and an impossible staircase that appears to ascend or descend endlessly while forming a closed loop. They were not the first to discover such figures — the Swedish artist Oscar Reutersvärd had drawn an impossible triangle in 1934, composed of a cluster of cubes arranged in a paradoxical configuration — but the Penroses formalized the concept and brought it to the attention of psychologists and mathematicians. Roger Penrose later sent his paper to M.C. Escher, who was inspired to create two of his most famous lithographs: Waterfall (1961), based on the impossible triangle, and Ascending and Descending (1960), based on the impossible staircase. The cross-pollination between mathematics and art that flowed through Penrose and Escher remains one of the most celebrated intellectual collaborations of the twentieth century.

Why impossible figures work

The human visual system interprets two-dimensional images by applying a set of heuristics for recovering three-dimensional structure — parallel lines that converge suggest depth, T-junctions suggest occlusion, shading gradients suggest curvature. These heuristics are locally reliable: each corner, each junction, each shaded face of an impossible figure is individually valid. The eye parses each local region and assigns it a coherent three-dimensional interpretation. The paradox arises because the global combination of these local interpretations is inconsistent. As your eye traces around the Penrose triangle, each bar appears to recede in depth, but after three turns you arrive back where you started having “descended” three times — which is geometrically impossible in Euclidean 3-space. The brain cannot reconcile the conflict, and the result is a compelling sense of visual paradox. The figure feels three-dimensional at every point, but the three-dimensionality does not close into a consistent whole.

Impossible objects made real

Intriguingly, it is possible to construct physical three-dimensional sculptures that look exactly like a Penrose triangle — but only from one specific viewing angle. The trick is to introduce a hidden gap or twist in the structure that is invisible from the correct vantage point. A famous example is the “Impossible Triangle” sculpture in East Perth, Western Australia, designed by Brian McKay and Ahmad Abas. From a marked viewing spot, the three separate beams appear to connect seamlessly into a Penrose triangle. Step a few meters to either side, and the illusion collapses: you see that the beams are at different depths and do not actually touch. A similar sculpture exists in Gotschuchen, Austria. These physical realizations underscore the key insight: impossible figures exploit the projection from three dimensions to two. The impossibility is not in any single piece of the object — it is in the assumption that the flat image faithfully represents a single coherent 3D scene.