The Penrose triangle

The impossible triangle

Three straight bars, three right angles, and a loop that can’t exist.

How this site works. Every fact is in our own words and linked to its source, beside it. Where good sources disagree, we show each version in a “Sources differ” box. Every drawing is our own.

The Penrose triangle, or tribar (our drawing). Follow any face round and it changes from top to side.Drawing: our own

What it is

The Penrose triangle is also called the tribar, the impossible tribar or the impossible triangle. MathWorld

The impossible triangle could only exist if the rules of Euclidean geometry were broken: its bottom bar is drawn both in front of and behind the top corner. Illusions Index

Trace it from the top corner: one side seems to run away from you, the other towards you, yet they meet on the same level at the bottom. Illusions Index

In the Escher museum’s words: the three right angles are normal, but they are joined up in an impossible way. Escher in The Palace

The Penroses

Lionel and Roger Penrose published it in the British Journal of Psychology, volume 49, in February 1958. Escher in The Palace The article was called “Impossible objects: a special type of visual illusion”. Escher in The Palace

The Penroses came up with the triple bar after Roger Penrose visited Escher’s one-man show at the Stedelijk Museum in 1954. Escher in The Palace MathWorld notes that Escher both inspired the triangle and later made it famous. MathWorld

Lionel Penrose (1898–1972) was a British psychiatrist, geneticist and mathematician; his son Roger was born in 1931. Illusions Index Roger Penrose was born on 8 August 1931 in Colchester, Essex. Britannica

He later shared the 2020 Nobel Prize for Physics for his work on black holes. Britannica He also discovered Penrose tiling, a set of shapes that covers a flat surface without a repeating pattern. Britannica He was knighted in 1994. Britannica

Sources differ: who drew it first?

Oscar Reutersvärd had drawn the shape earlier, though the Penroses didn’t know. Escher in The Palace
The Illusions Index says Reutersvärd created it first and the Penroses discovered it independently and made it popular. Illusions Index
In 1934, aged only 18, he drew an impossible triangle while sketching cubes in perspective. Escher in The Palace
Michael Bach notes the first impossible object is often credited to Reutersvärd, but Josef Albers and Hogarth were clearly earlier. Michael Bach

So: the Penroses published it and made it famous; Reutersvärd drew it first, and Michael Bach finds earlier impossible pictures still. See the timeline.

Bigger loops

There are impossible figures with four sides and more, too. Illusions Index The idea stretches to many-sided barred shapes, including the “tribox”. MathWorld

A chain of cubes that closes up impossibly, in the style of Reutersvärd’s first figure (our drawing, not a copy of his).Drawing: our own

Can you build one?

In 1968 Richard Gregory made a wooden object that looks like a real impossible triangle from one position only; from anywhere else it is clearly three joined struts. Illusions Index

The real object behind the illusion: three straight bars and a gap. From one exact spot, the gap disappears behind the bar.Drawing: our own

Try it yourself in the find-the-angle toy, then visit the real ones.

Monument Valley, a 2014 game from Ustwo Games, is built around impossible figures. Illusions Index

Next: the Penrose stairs →