Escher’s three impossible buildings
Belvedere, Ascending and Descending and Waterfall, described in words.
Words only. We don’t show Escher’s prints on this site. The M.C. Escher Foundation’s own pages show each one, and we link to them below. The drawings elsewhere on this site are our own simple versions of the figures he used, not copies of his prints.
The artist
M.C. Escher was born on 17 June 1898 in Leeuwarden, Netherlands, and died on 27 March 1972 in Laren. Britannica
He made some 450 lithographs, woodcuts and wood engravings, and about 2,000 drawings and sketches. Britannica He is sometimes called the “father of modern tessellations”. Britannica
He did not excel at mathematics at school. Britannica Britannica: his mature style, after 1937, combined meticulous realism with puzzling optical illusions. Britannica
Three impossible figures, three prints
Belvedere, Waterfall and Ascending and Descending are based on three impossible figures: a cube, a triangle and a staircase. Escher in The Palace
Of all his impossible buildings, Escher himself used the term for only three: Belvedere, Ascending and Descending and Waterfall. Escher in The Palace
Meeting the Penroses
The International Congress of Mathematicians opened in Amsterdam on 2 September 1954. Escher in The Palace Escher’s solo exhibition at the Stedelijk Museum ran from 27 August to 26 September 1954. Escher in The Palace
There Roger Penrose and the geometer Donald Coxeter first came across Escher’s work. Escher in The Palace It was after seeing Escher’s Relativity at the exhibition that Penrose and his father published their article. 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
Lionel Penrose sent Escher their Christmas puzzle page, “Puzzles for Christmas”. Escher in The Palace
Bach traces a chain of inspiration from the Penroses’ 1958 figure to Escher’s 1960 Ascending and Descending, and a letter exchange between them. Michael Bach
Belvedere (1958)
Belvedere is a lithograph from May 1958. mcescher.com
‘Belvedere’ comes from the Latin for ‘beautiful view’: a tower or gallery added to a building for the view. Escher in The Palace
Of his three famous impossible prints, Escher invented the figure for Belvedere himself: an impossible cube. Escher in The Palace In Belvedere, a drawing of a Necker cube lies on the floor; the man on the bench holds an impossible cube. Escher in The Palace
The people in Belvedere wear clothes from the late Middle Ages. Escher in The Palace The museum thinks the landscape is almost certainly Italian, perhaps the Abruzzo or Sicily. Escher in The Palace
See Belvedere on the Escher in The Palace site
Ascending and Descending (1960)
Ascending and Descending is a lithograph from March 1960. mcescher.com Escher finished Ascending and Descending on 18 March 1960. Escher in The Palace
The Penrose article sparked the idea, and he started work in late 1959. Escher in The Palace In April 1960 he wrote to the Penroses to thank them for their invention. Escher in The Palace
In a letter to his son Arthur, Escher described the staircase as a sealed ring, “like a snake eating its own tail”. Escher in The Palace In his own book, Escher imagined the people on the stairs as monks of some unknown sect, climbing for a few hours a day as a ritual duty. Escher in The Palace Two “recalcitrant individuals” refuse to take part. Escher in The Palace
See Ascending and Descending on the Escher in The Palace site
Waterfall (1961)
Waterfall is a lithograph from October 1961. mcescher.com Waterfall’s water channel is built from three Penrose triple bars combined. Escher in The Palace
Water always flows to the lowest point, but in Waterfall the lowest, furthest point is also the highest and nearest. The water falls and the cycle starts again: perpetual motion. Escher in The Palace
He first sketched the building without water; the flowing water came later, to show the absurdity even better. Escher in The Palace Escher’s early Waterfall sketches also try out the impossible cube he had used for Belvedere. Escher in The Palace
A former curator called the second look at Waterfall “the fascinating OMG moment”: the water is flowing upwards. Escher in The Palace The museum suggests Waterfall is perhaps his most famous print of all. Escher in The Palace
In a letter dated 30 November 1961 Escher wrote that the brand-new Waterfall was based on an idea from “the two Penroses”, adding that they mentioned his name in their article. Escher in The Palace
See Waterfall on the Escher in The Palace site
His secret
Escher said in a 1963 lecture that to show something impossible you must first deceive yourself, then your audience, so a casual viewer would not even notice. Escher in The Palace
Researchers describe impossible structures as having odd geometry that isn’t obvious “pre-attentively”: within a single fixation of the eye, about 120 milliseconds. J. Imaging, 2020 A 2020 study modelling the human retina argued that Escher’s secret was to place the clashing parts just outside the area the eye can check for 3D sense in one fixation. J. Imaging, 2020
Built for real
In 2011 a German design student known as mcwolles built a wooden copy of Waterfall’s water path, which seems to run uphill. Michael Bach Bach explains one of its tricks: the shape only looks impossible from one single viewing angle. It is made of two Penrose triangles. Michael Bach
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