The Penrose triangle, also known as the Penrose tribar, the impossible tribar,[1] or the impossible triangle,[2] is a triangular
impossible object, an
optical illusion consisting of an object which can be depicted in a perspective drawing, but cannot exist as a solid object. It was first created by the Swedish artist
Oscar Reutersvärd in 1934.[3] Independently from Reutersvärd, the triangle was devised and popularized in the 1950s by psychiatrist
Lionel Penrose and his son, prominent Nobel Prize-winning mathematician Sir
Roger Penrose, who described it as "impossibility in its purest form".[4] It is featured prominently in the works of artist
M. C. Escher, whose earlier depictions of impossible objects partly inspired it.
Description
The tribar/triangle appears to be a
solid object, made of three straight beams of square cross-section which meet pairwise at right angles at the vertices of the
triangle they form. The beams may be broken, forming cubes or cuboids.
This combination of properties cannot be realized by any three-dimensional object in ordinary
Euclidean space. Such an object can exist in certain Euclidean
3-manifolds.[5] There also exist three-dimensional solid shapes each of which, when viewed from a certain angle, appears the same as the 2-dimensional depiction of the Penrose triangle on this page (such as – for example – the adjacent image depicting a sculpture in
Perth,
Australia). The term "Penrose Triangle" can refer to the 2-dimensional depiction or the impossible object itself.
If a line is traced around the Penrose triangle, a 4-loop
Möbius strip is formed.[6]
Depictions
M.C. Escher's
lithographWaterfall (1961) depicts a watercourse that flows in a zigzag along the long sides of two elongated Penrose triangles, so that it ends up two stories higher than it began. The resulting waterfall, forming the short sides of both triangles, drives a
water wheel. Escher points out that in order to keep the wheel turning, some water must occasionally be added to compensate for
evaporation.
Sculptures
Impossible triangle sculpture as an
optical illusion, East Perth, Western Australia
Impossible Triangle sculpture, Gotschuchen, Austria
Real Penrose Triangle, Stainless Steel, by W.A.Stanggaßinger, Wasserburg am Inn, Germany. This type of impossible triangle was first created in 1969 by the Soviet kinetic artist
Vyacheslav Koleichuk.[7]
^Pappas, Theoni (1989). "The Impossible Tribar". The Joy of Mathematics: Discovering Mathematics All Around You. San Carlos, California: Wide World Publ./Tetra. p. 13.
^Ernst, Bruno (1986). "Escher's impossible figure prints in a new context". In
Coxeter, H. S. M.; Emmer, M.;
Penrose, R.; Teuber, M. L. (eds.). M. C. Escher Art and Science: Proceedings of the International Congress on M. C. Escher, Rome, Italy, 26–28 March, 1985. North-Holland. pp. 125–134. See in particular p. 131.
^Francis, George K. (1988). "Chapter 4: The impossible tribar". A Topological Picturebook. Springer. pp. 65–76.
doi:
10.1007/978-0-387-68120-7_4.
ISBN0-387-96426-6. See in particular p. 68, where Francis attributes this observation to
John Stillwell.