Every fact

137 facts, every one sourced

Every fact on this site in one list, each linked to its source. Our build stores the source text and checks that the key words and figures in each fact really appear in it; see the sources page.

1 Your brain builds a 3D model of the world from the flat image on the back of your eye. Michael Bach
2 With an impossible object, every corner makes sense on its own. The trouble is that the corners combine viewpoints that can’t all be true at once. Michael Bach
3 At the start of the 18th century the philosopher George Berkeley argued that we can’t see depth directly, because the image of anything on the retina is flat, like a painting. Britannica
4 Britannica notes that modern research does not back Berkeley’s idea that reasoning is the key to seeing depth; today’s theories mix inborn abilities with things we learn. Britannica
5 Psychologists also study how we know a cube has six sides when only three can be seen at once. Britannica
6 Even seeing a wire cube in a flat line drawing is an interpretation: many shapes could cast those same lines, but the brain picks a cube as the most likely answer. Michael Bach
7 Stare at a Necker cube and it suddenly flips between two orientations, typically every 3 to 5 seconds. Michael Bach
8 You can’t hold one version for long: after 10 seconds or so it flips again. Blinking can make it flip. Michael Bach
9 L. A. Necker was a Swiss mineralogist who came across the flip by chance while looking at a drawing of a cube-like crystal. He described it in 1832. Michael Bach
10 Michael Bach points out that an impossible figure is a different thing from the Necker cube’s flipping, though he finds his impossible version even less stable. Michael Bach
11 The “missing corner” cube can be seen three ways: a cube with a corner cut out, a small cube stuck on the front, or a cube sitting in the corner of a room. Michael Bach
12 Without stereo depth or clear lighting, all three readings are geometrically possible. Michael Bach
13 Cognitive scientists ask why we keep seeing an impossible figure as a real object even after we know it can’t exist. Illusions Index
14 One standard answer is that parts of the visual system are “cognitively impenetrable”: knowing better doesn’t change what they output. Illusions Index
15 Philosophers are interested because impossible figures seem to give us experiences with contradictory content. Illusions Index
16 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
17 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
18 Mathematicians have studied impossible figures to model them for uses such as computer vision. Illusions Index
19 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
20 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
21 The Penrose triangle is also called the tribar, the impossible tribar or the impossible triangle. MathWorld
22 Lionel and Roger Penrose published it in the British Journal of Psychology, volume 49, in February 1958. Escher in The Palace
23 The article was called “Impossible objects: a special type of visual illusion”. Escher in The Palace
24 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
25 Oscar Reutersvärd had drawn the shape earlier, though the Penroses didn’t know. Escher in The Palace
26 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
27 MathWorld notes that Escher both inspired the triangle and later made it famous. MathWorld
28 The Illusions Index says Reutersvärd created it first and the Penroses discovered it independently and made it popular. Illusions Index
29 Lionel Penrose (1898–1972) was a British psychiatrist, geneticist and mathematician; his son Roger was born in 1931. Illusions Index
30 Roger Penrose was born on 8 August 1931 in Colchester, Essex. Britannica
31 He later shared the 2020 Nobel Prize for Physics for his work on black holes. Britannica
32 He also discovered Penrose tiling, a set of shapes that covers a flat surface without a repeating pattern. Britannica
33 He was knighted in 1994. Britannica
34 There are impossible figures with four sides and more, too. Illusions Index
35 The idea stretches to many-sided barred shapes, including the “tribox”. MathWorld
36 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
37 Monument Valley, a 2014 game from Ustwo Games, is built around impossible figures. Illusions Index
38 Oscar Reutersvärd was born in Stockholm in 1915. Escher in The Palace
39 He died on 2 February 2002. Escher in The Palace
40 The Escher museum calls him the archetypal father of the impossible figure. Escher in The Palace
41 The Illusions Index calls him the ‘father of the impossible figure’. Illusions Index
42 In 1934, aged only 18, he drew an impossible triangle while sketching cubes in perspective. Escher in The Palace
43 MathWorld says he drew it in 1934 during a “long lecture”. MathWorld
44 The Illusions Index says it is anecdotally, but widely, reported that he made it in 1934 while doodling in a Latin class. Illusions Index
45 A later colour lithograph of the first impossible triangle carries the title Opus 1. Escher in The Palace
46 He designed an impossible staircase in 1937. Escher in The Palace
47 His early discoveries stayed hidden for decades: Roger Penrose only found out in 1984 that Reutersvärd had drawn the triangle and the stairs first. Escher in The Palace
48 He always used the same isometric projection, where the lines of perspective stay parallel and never meet at a vanishing point. Escher in The Palace
49 He numbered his reproduced figures under the title ‘perspective japonaise’. Escher in The Palace
50 He drew thousands of them, plain, with no settings or decoration, unlike Escher, who hid his impossibilities in realistic scenes. Escher in The Palace
51 In 1950 a grant took him to Paris to study with the artist Fernand Léger. Escher in The Palace
52 He was professor of art history and art theory at Lund University from 1964 to 1981. Escher in The Palace
53 He also designed mazes, some built outdoors in Sweden. Escher in The Palace
54 Seeing what the Penroses and Escher were doing rekindled his own enthusiasm for impossible figures. Escher in The Palace
55 Sweden honoured him with a stamp in 1982. MathWorld
56 He wasn’t the first to fool the eye: William Hogarth (1697–1774) made an engraving in 1754 full of perspective tricks. Escher in The Palace
57 Giovanni Battista Piranesi hid similar tricks in the walls and stairs of his Carceri d’Invenzione prison scenes. Escher in The Palace
58 Michael Bach notes the first impossible object is often credited to Reutersvärd, but Josef Albers and Hogarth were clearly earlier. Michael Bach
59 Bach lists Josef Albers’s “Rolled Wrongly” from 1931 and Hogarth’s Satire on False Perspective from 1754. Michael Bach
60 The oldest impossible scene Bach knows is the Madonna and Child in the Pericope of Henry II, from around 1025, though he thinks that one was probably not on purpose. Michael Bach
61 René Magritte painted an impossible scene too: “Carte blanche”, 1965. Michael Bach
62 The Penroses’ 1958 article included a drawing and a 3D model of a continuous flight of stairs that seems to lead up, or down, for ever. Escher in The Palace
63 Roger’s father Lionel, an expert in genetics and mathematics, built some models from Roger’s drawings. Escher in The Palace
64 The Penrose stairs appear in Christopher Nolan’s 2010 film Inception. Escher in The Palace
65 Lionel Penrose sent Escher their Christmas puzzle page, “Puzzles for Christmas”. Escher in The Palace
66 The devil’s fork is also called the blivet; in his 1964 paper D. H. Schuster called it a “three-stick clevis”. Michael Bach
67 Other names include the impossible fork, the devil’s pitchfork and the poiuyt. MathWorld
68 The Illusions Index also calls it the impossible trident or the devil’s tuning fork. Illusions Index
69 At one end it seems to have three prongs; at the other, only two. Illusions Index
70 Schuster, an American psychologist, based it on an advertisement he saw in an aviation journal. Illusions Index
71 It was first published in The American Journal of Psychology in 1964. Illusions Index
72 Mad magazine made it a running joke, calling it the “poiuyt”: the third row of a keyboard typed from right to left. MathWorld
73 In Mad, throughout the 1960s, the poiuyt stood for anything absurd or impossible. MathWorld
74 Each prong looks normal; it is working out how they join that goes wrong. MathWorld
75 Michael Bach’s way to draw one: start with six evenly spaced upright lines and give them “hats”, and you get three towers or a three-pronged fork. Michael Bach
76 Cover the top, draw connecting curves at the bottom, and you get the bottom of a square bar bent into a U. Michael Bach
77 Uncover the top and you see the two sensible readings don’t fit together. Michael Bach
78 Bach’s impossible “contraption” also includes three “hexnuts” and a “Penrose frame”. Michael Bach
79 The artist Roger Hayward worked the fork into a picture of Greek columns. MathWorld
80 Of his three famous impossible prints, Escher invented the figure for Belvedere himself: an impossible cube. Escher in The Palace
81 Belvedere, Waterfall and Ascending and Descending are based on three impossible figures: a cube, a triangle and a staircase. Escher in The Palace
82 Escher told the Penroses he had the idea for Belvedere before he read their article. Escher in The Palace
83 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
84 A Necker cube is a cube drawn with all its edges showing, so you can’t tell which face is at the front. It can exist on paper; the man’s cube can’t. Escher in The Palace
85 A 2020 study instead names the Necker cube as the impossible structure in Belvedere. J. Imaging, 2020
86 Escher also made a 1958 wood engraving of the same man with his impossible cube, Man with Cuboid. Escher in The Palace
87 Man with Cuboid is tiny: 64 × 64 mm. EscherMath
88 ‘Belvedere’ comes from the Latin for ‘beautiful view’: a tower or gallery added to a building for the view. Escher in The Palace
89 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
90 The people in Belvedere wear clothes from the late Middle Ages. Escher in The Palace
91 The museum thinks the landscape is almost certainly Italian, perhaps the Abruzzo or Sicily. Escher in The Palace
92 M.C. Escher was born on 17 June 1898 in Leeuwarden, Netherlands, and died on 27 March 1972 in Laren. Britannica
93 He did not excel at mathematics at school. Britannica
94 He made some 450 lithographs, woodcuts and wood engravings, and about 2,000 drawings and sketches. Britannica
95 He is sometimes called the “father of modern tessellations”. Britannica
96 Belvedere is a lithograph from May 1958. mcescher.com
97 Ascending and Descending is a lithograph from March 1960. mcescher.com
98 Waterfall is a lithograph from October 1961. mcescher.com
99 Escher finished Ascending and Descending on 18 March 1960. Escher in The Palace
100 The Penrose article sparked the idea, and he started work in late 1959. Escher in The Palace
101 In April 1960 he wrote to the Penroses to thank them for their invention. Escher in The Palace
102 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
103 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
104 Two “recalcitrant individuals” refuse to take part. Escher in The Palace
105 Waterfall’s water channel is built from three Penrose triple bars combined. Escher in The Palace
106 He first sketched the building without water; the flowing water came later, to show the absurdity even better. Escher in The Palace
107 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
108 A former curator called the second look at Waterfall “the fascinating OMG moment”: the water is flowing upwards. Escher in The Palace
109 The museum suggests Waterfall is perhaps his most famous print of all. Escher in The Palace
110 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
111 Escher’s early Waterfall sketches also try out the impossible cube he had used for Belvedere. Escher in The Palace
112 The International Congress of Mathematicians opened in Amsterdam on 2 September 1954. Escher in The Palace
113 Escher’s solo exhibition at the Stedelijk Museum ran from 27 August to 26 September 1954. Escher in The Palace
114 There Roger Penrose and the geometer Donald Coxeter first came across Escher’s work. Escher in The Palace
115 It was after seeing Escher’s Relativity at the exhibition that Penrose and his father published their article. Escher in The Palace
116 Of all his impossible buildings, Escher himself used the term for only three: Belvedere, Ascending and Descending and Waterfall. Escher in The Palace
117 Britannica: his mature style, after 1937, combined meticulous realism with puzzling optical illusions. Britannica
118 Reutersvärd drew his own impossible version of Escher’s Cubic Space Division, where the front blocks don’t really touch. Escher in The Palace
119 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
120 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
121 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
122 The Impossible Triangle in East Perth, Western Australia, is by the artist Brian McKay and the architect Ahmad Abas. DENMAC
123 It stands 13.5 metres high. DENMAC
124 McKay and Abas submitted their design for the commission in 1997. DENMAC
125 It was completed in November 1999. DENMAC
126 Western Australia’s planning minister unveiled it on 5 November 1999. WA Government, 1999
127 The minister said it can only be fully appreciated by moving around it to watch the shape change and the illusion appear. WA Government, 1999
128 The minister’s statement called it an adaptation of a concept developed by Lionel and Roger Penrose in 1954. WA Government, 1999
129 The makers say it was inspired by the Penrose triangle “first articulated” by Oscar Reutersvärd. DENMAC
130 The local art trail says the triangle was devised by Sir Roger Penrose in the 1950s, and that there are only two spots from which it looks complete. Claisebrook Art Trail
131 The local art trail calls it the 13.5 m landmark at the gateway to East Perth. Claisebrook Art Trail
132 It was part of the East Perth Redevelopment Project, which turned 38 hectares of industrial land into a new suburb. DENMAC
133 Its polished striations reflect sunlight and street light, changing look between sunrise and sunset. DENMAC
134 The WA Public Art Inventory gives its location as the roundabout where Royal Street, Brook Street and East Parade meet. WA Public Art Inventory
135 The Illusions Index puts it in Claisebrook Square, and says it was chosen as the winning entry in a competition. Illusions Index
136 A 4-metre Penrose triangle by the association Treffpunkt Physik stands in Gotschuchen, St. Margareten im Rosental, Austria. Wikimedia Commons
137 Its beams don’t actually meet at the top. Only from one viewing angle do they seem to join. Wikimedia Commons