June 21, 2024

The Penrose Triangle and Other Impossible Objects

They cannot be constructed, they serve no practical purpose, they are mysterious and baffling; welcome to the strange world of Impossible Objects.

One of the impossible objects: the impossible cube
One of the impossible objects: the impossible cube

The term ‘Impossible Object’ is applied to objects that can be drawn in two dimensions, but never physically rendered. An example is the ‘Impossible Cube’, shown above.

Our subconscious is full of preconceived ideas about the way the world around us should look. That a line at a particular angle will always indicate ‘x’; height or depth or scale. Impossible Objects play with perspective and so confound these ideas, pitting what we can see against our assumptions.

They are optical illusions, and also a type of art. They have fascinated psychologists, physicists, mathematicians, graphic designers, and regular folk throughout the past century.

They are cool to look at.

Roger Penrose teaching physics
Physicist Roger Penrose

Roger Penrose is one of the most acclaimed names in 20th century physics.

The British scientist was born in 1931 to an academic family, already prominent in psychiatry and genetics. Penrose would find his calling in  cosmology; in the 1960s he produced a series of papers that explained the curvature of space-time, and in the 1970s he collaborated with Stephen Hawking, analysing Black Holes and the Big Bang Theory.

But in the 1950s, when still an impressionable youth, he encountered the artwork of M.C. Escher.

M.C. Escher, self portrait
M.C. Escher, self portrait

Maurits Cornelis Escher was a Dutch born artist and graphic designer, who liked to incorporate mathematical principles into his work.

Born in 1898 in Leeuwarden, Escher was a sickly child, and something of a loner. He showed an aptitude for drawing from a young age, but was a lacklustre student.

Placed in a technical college when he was 13, Escher was to be trained as a draftsman. But he failed most of his subjects, and soon switched to the Haarlem School of Architecture and Decorative Arts, where he studied graphic design.

An early landscape drawing by M.C.Escher

An early landscape drawing by M.C.Escher
Two of Escher's landscape drawings from the early 1920s, both from Italy.

Escher travelled extensively through Europe in the 1920s, and his early works capture what he saw.

Using woodblock printing techniques, he created a series of landscape prints. These are striking pictures with a high contrast; simple, yet still conveying a lot of visual information.

As Escher developed as an artist, he also began to add distinctive, stylistic flourishes; surreal elements that moved his art away from the everyday.

'Möbius Strip II', by M.C. Escher
'Möbius Strip II', by M.C. Escher

While he never studied the subject formally, Escher had always been interested in mathematics.

Mathematical shapes, like the ‘Mobius Strip’, began to appear in his drawings. He also began to experiment with perspective, based on books he read about how the eye and brain interpret depth, and three-dimensional shapes.

In the 1930s and 40s, Escher’s work evolved again. They now seemed to be set in some kind of hyper reality; an imaginary dimension with its own laws and logic.

'Relativity' by M.C. Escher, a drawing employing impossible objects
'Relativity' by M.C. Escher

The lithograph ‘Relativity’ gives a good example of Escher’s developing technique.

This picture presents a scene (or, possibly, scenes) from an everyday household. Only, rather than show the image from one perspective, Escher has rotated different sections of the picture on different axis’, so that none of them are aligned on one plane.

‘It is impossible for the inhabitants of different worlds to walk or sit or stand on the same floor, because they have differing conceptions of what is horizontal and what is vertical.

Contact between them is out of the question because they live in different worlds and therefore can have no knowledge of each other's existence.’

- M.C. Escher

Visually, the image is striking. Conceptually, it conveys the artists sense of loneliness, and the difficulty he had communicating with other people, outside of his work.

A 'Penrose Triangle', another of the impossible objects
A 'Penrose Triangle'

Roger Penrose first encountered Escher’s art at the at the International Congress of Mathematicians in Amsterdam, in 1954.

As a fan of puzzles and optical illusions, he was excited by what he saw, writing to his father that he was ‘spellbound’ by Escher’s drawings. Returning to England, Penrose was inspired to try to create his own visual puzzle. Doodling absently, he created the ‘Impossible Triangle’.

As you trace your eye around the shape, wherever you start, once you get to the third strut you are mentally forced to do something you know you could not actually do; using foreshortening and a cunning shift in perspective, the third strut twists the image, impossibly.

A 'Penrose Staircase', another of the impossible objects
A 'Penrose Staircase'

Penrose’s father, Lionel, was intrigued by his son’s creation.

Also a fan of Escher and optical illusions, he used the Impossible Triangle as the basis for his own design. His creation was the ‘Impossible Staircase’; a set of stairs that appears to be going up, but somehow loops directly from the ‘top’ back to the ‘bottom’, enclosing the steps in an endless loop.

The Penroses were pleased with their creations but were unsure what, if anything, to do with them. Did Impossible Objects have any practical value?

They eventually collaborated on a scientific paper, detailing the psychological impact of Impossible Objects, citing Escher as an inspiration for their investigation. This was published in the British Journal of Psychology, in 1958.

Lionel Penrose
Lionel Penrose

A friend of Escher’s saw the article and forwarded it to the artist.

He was flattered to have been mentioned in a reputable scientific journal, but even more taken by the Impossible Objects the Penrose’s had created. Escher was immediately determined to incorporate these designs into a new work of art:

‘A few months ago, a friend of mine sent me a photocopy of your article... Your figures 3 and 4, the 'continuous flight of steps', were entirely new to me, and I was so taken by the idea that they recently inspired me to produce a new picture, which I would like to send to you as a token of my esteem.’

- M.C. Escher, letter to Lionel Penrose

The new picture would be ‘Ascending and Descending’.

'Ascending and Descending' by M.C. Escher
'Ascending and Descending' by M.C. Escher

‘Ascending and Descending’ shows a building, a medieval looking structure, depicted from above. The top of the building is a Penrose Staircase, around which a series of men are walking endlessly.

Thematically, the drawing reflects Escher growing bleak view of humanity:

‘That staircase is a rather sad, pessimistic subject, as well as being profound and absurd.


Yes, yes, we climb up and up, we imagine we are ascending; every step is about 10 inches high, terribly tiring – and where does it all get us? Nowhere.’


- M.C. Escher

'Waterfall', by M.C. Escher, a drawing using impossible objects
'Waterfall', by M.C. Escher

Escher would use the Penrose Triangle as the basis for his picture ‘Waterfall’.

This again uses the optical illusion to create the impression of an endless loop; using the provided channel, the water somehow flows ‘up’, which seems both logical, based on the design of the image, and impossible, based on your own knowledge of gravity.

The Penroses were delighted to have contributed to Escher's art.

While Escher saw these endless loops and visual paradoxes as a pessimistic reflection of the meaninglessness of existence, they have proved enduringly popular.

Poster for the NGV's Escher show; a huge popular hit
The NGV's Escher show; a huge popular hit

The gallery opened at 8am. I got there at 8.15. There were already 100 people lined up for tickets.

It was the last weekend of an exhibition of Escher at the NGV, and demand was enormous. All of his major works were on display, including ‘Ascending and Descending’, ‘Relativity’ and ‘Waterfall’. The gallery was opening two hours earlier than usual, to accommodate the crowds.

Once we got in, we moved together through the rooms, looking at these curious puzzle pictures with rapt wonder.

A woman standing next to me, looking at ‘Relativity’, said, ‘Isn’t it wonderful? I just… love it, so much.’

She addressed these words to no one in particular. Her sentiment summarised the mood. These are artworks like no other.

Impossible objects used in the movie 'Labyrinth'
David Bowie in the Escher influenced 'Labyrinth'
Impossible objects used in the Christopher Nolan film 'Inception'
Penrose Staircase in Christopher Nolan's 'Inception'

Escher’s works have inspired many other artists. The Goblin King’s Palace, in the movie ‘Labyrinth’, and the dream staircase, in Christopher Nolan’s ‘Inception’, are two well known examples.

His works today are seen as playful, and inventive. Fun.

A far remove from the moody, pessimistic artist who created them.


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