Penrose stairs

The Penrose stairs or Penrose steps, also dubbed the impossible staircase, is an impossible object created by Lionel Penrose and his son Roger Penrose. A variation on the Penrose triangle, 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 then discovered the Penrose stairs in the following year and made his now famous lithograph Klimmen en dalen (Ascending and Descending) 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), 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."

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 Penrose tribar structure (i.e., the Penrose triangle) and the continuous steps. Escher, in the 1950s, 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 did not know in what field the subject belonged. Because Lionel Penrose knew the editor of the 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.

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