The Penrose stairs or Penrose steps, also dubbed the impossible staircase, is an impossible object

Impossible object

An impossible object is a type of optical illusion consisting of a two-dimensional figure which is instantly and subconsciously interpreted by the visual system as representing a projection of a three-dimensional object although it is not actually possible for such an object to exist An impossible...

 created by Lionel Penrose

Lionel Penrose

Lionel Sharples Penrose, FRS was a British psychiatrist, medical geneticist, mathematician and chess theorist, who carried out pioneering work on the genetics of mental retardation. He was educated at the Quaker Leighton Park School and St...

 and his son Roger Penrose

Roger Penrose

Sir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...

. A variation on the Penrose triangle

Penrose triangle

The Penrose triangle, also known as the Penrose tribar, is an impossible object. It was first created by the Swedish artist Oscar Reutersvärd in 1934. The mathematician Roger Penrose independently devised and popularised it in the 1950s, describing it as "impossibility in its purest form". It is...

, it is a two-dimensional depiction of a staircase in which the stairs make four 90-degree turns as they ascend or descend yet form a continuous loop, so that a person could climb them forever and never get any higher. This is clearly impossible in three dimensions.

The "continuous staircase" was first presented in an article that the Penroses wrote in 1959, based on the so called "triangle of Penrose" published by Roger Penrose in the British Journal of Psychology in 1958. M. C. Escher

M. C. Escher

Maurits Cornelis Escher , usually referred to as M. C. Escher , was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints...

 then discovered the Penrose stairs in the following year and made his now famous lithography Klimmen en dalen (Ascending and Descending

Ascending and Descending

Ascending and Descending is a lithograph print by the Dutch artist M. C. Escher which was first printed in March 1960.The original print measures 14" x 11 1/4”. The lithograph depicts a large building roofed by a never-ending staircase. Two lines of identically dressed men appear on the staircase,...

) in March 1960. Penrose and Escher were informed of each other's work that same year.
Escher developed the theme further in his print Waterval (Waterfall

Waterfall (M. C. Escher)

Waterfall is a lithography print by the Dutch artist M. C. Escher which was first printed in October, 1961. It shows an apparent paradox where water from the base of a waterfall appears to run downhill before reaching the top of the waterfall....

), which appeared in 1961.

In their original article the Penroses noted that "each part of the structure is acceptable as representing a flight of steps but the connexions are such that the picture, as a whole, is inconsistent: the steps continually descend in a clockwise direction."

The Shepard tone

Shepard tone

A Shepard tone, named after Roger Shepard, is a sound consisting of a superposition of sine waves separated by octaves. When played with the base pitch of the tone moving upward or downward, it is referred to as the Shepard scale. This creates the auditory illusion of a tone that continually...

, developed in the 1960s, is a similar illusion in terms of sound

Sound

Sound is a mechanical wave that is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations.-Propagation of...

.

History of discovery

At an Escher conference in Rome in 1985, Roger Penrose said that he had been greatly inspired by Escher's work when he and his father discovered both the tri-bar structure and the continuous steps, although Escher at the time had not yet drawn any impossible figures and was not aware of their existence. Roger Penrose had been introduced to Escher's work at the International Congress of Mathematicians in Amsterdam in 1954. He was "absolutely spellbound" by Escher's work, and on his journey back to England he decided to produce something "impossible" on his own. After experimenting with various designs of bars overlying each other he finally arrived at the impossible triangle. Roger showed his drawings to his father, who immediately produced several variants, including the impossible flight of stairs. They wanted to publish their findings but didn't know in what field the subject belonged. Because Lionel Penrose knew the editor of British Journal of Psychology and convinced him to publish their short manuscript, the finding was finally presented as a psychological subject. After the publication in 1958 the Penroses sent a copy of the article to Escher as a token of their esteem.

While the Penroses credited Escher in their article, Escher himself noted in a letter to his son in January 1960 that he was:
Escher was captivated by the endless stairs and subsequently wrote a letter to the Penroses in April 1960:
The staircase design had been discovered previously by the Swedish artist Oscar Reutersvärd

Oscar Reutersvärd

Oscar Reutersvärd , "the father of the impossible figure", was an artist who pioneered the art of impossible objects. These are images such as what was later renamed the Penrose triangle that appear to depict solid objects, which, however, are impossible to construct in reality...

, but neither Penrose nor Escher were aware of his designs. Inspired by a radio programme on Mozart's method of composition — described as "creative automatism", i.e. each creative idea written down inspired a new idea — Reutersvärd started to draw a series of impossible objects on a journey from Stockholm to Paris in 1950 in the same "unconscious, automatic" way. He did not realise that his figure was a continuous flight of stairs while drawing, but the process enabled him to trace his increasingly complex designs step by step. When M. C. Escher's Ascending and Descending was sent to Reutersvärd in 1961, he was impressed but didn't like the irregularities of the stairs (2×15+2×9). Throughout the 1960s, Reutersvärd sent several letters to Escher to express his admiration for his work, but the Dutch artist failed to respond. Roger Penrose only discovered Reutersvärd's work in 1984.