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The Penrose triangle, also known as the Penrose tribar, the impossible tribar, or the impossible triangle, is a triangular
The tribar/triangle appears to be a
M.C. Escher's
File:Perth Impossible Triangle.jpg, Impossible triangle sculpture as an
An article about impossible triangle sculpture in Perth
Escher for Real constructions
{{DEFAULTSORT:Penrose Triangle Topology Impossible objects Triangles 1934 introductions
impossible object
An impossible object (also known as an impossible figure or an undecidable figure) is a type of optical illusion that consists of a two-dimensional figure which is instantly and naturally understood as representing a projection of a three-di ...
, an optical illusion
Within visual perception, an optical illusion (also called a visual illusion) is an illusion caused by the visual system and characterized by a visual perception, percept that arguably appears to differ from reality. Illusions come in a wide v ...
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. Independently from Reutersvärd, the triangle was devised and popularized in the 1950s by psychiatrist Lionel Penrose
Lionel Sharples Penrose, FRS (11 June 1898 – 12 May 1972) was an English psychiatrist, medical geneticist, paediatrician, mathematician and chess theorist, who carried out pioneering work on the genetics of intellectual disability. Penr ...
and his son, prominent Nobel Prize-winning mathematician Sir Roger Penrose
Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, philosopher of science and Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford, an emeritus f ...
, who described it as "impossibility in its purest form". It is featured prominently in the works of artist M. C. Escher
Maurits Cornelis Escher (; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made mathematically inspired woodcuts, lithographs, and mezzotints.
Despite wide popular interest, Escher was for most of his life neglected in t ...
, whose earlier depictions of impossible objects partly inspired it.
Description
The tribar/triangle appears to be a solid
Solid is one of the four fundamental states of matter (the others being liquid, gas, and plasma). The molecules in a solid are closely packed together and contain the least amount of kinetic energy. A solid is characterized by structur ...
object, made of three straight beams of square cross-section which meet pairwise at right angles at the vertices of the triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
. Such an object can exist in certain Euclidean 3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...
s. 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
Perth is the capital and largest city of the Australian state of Western Australia. It is the fourth most populous city in Australia and Oceania, with a population of 2.1 million (80% of the state) living in Greater Perth in 2020. Perth is ...
, Australia
Australia, officially the Commonwealth of Australia, is a sovereign country comprising the mainland of the Australian continent, the island of Tasmania, and numerous smaller islands. With an area of , Australia is the largest country by ...
). 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
In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and A ...
is formed.
Depictions
M.C. Escher's lithograph
Lithography () is a planographic method of printing originally based on the immiscibility of oil and water. The printing is from a stone (lithographic limestone) or a metal plate with a smooth surface. It was invented in 1796 by the German a ...
''Waterfall
A waterfall is a point in a river or stream where water flows over a vertical drop or a series of steep drops. Waterfalls also occur where meltwater drops over the edge of a tabular iceberg or ice shelf.
Waterfalls can be formed in several ...
'' (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
A water wheel is a machine for converting the energy of flowing or falling water into useful forms of power, often in a watermill. A water wheel consists of a wheel (usually constructed from wood or metal), with a number of blades or buckets ...
. Escher points out that in order to keep the wheel turning, some water must occasionally be added to compensate for evaporation
Evaporation is a type of vaporization that occurs on the surface of a liquid as it changes into the gas phase. High concentration of the evaporating substance in the surrounding gas significantly slows down evaporation, such as when h ...
.
Sculptures
optical illusion
Within visual perception, an optical illusion (also called a visual illusion) is an illusion caused by the visual system and characterized by a visual perception, percept that arguably appears to differ from reality. Illusions come in a wide v ...
, East Perth, Western Australia
File:LargeTribarGotschuchenAustria.JPG, Impossible Triangle sculpture, Gotschuchen, Austria
File:Penrose Triangle auf Ecke stehend.jpg, 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.
See also
*Impossible trident
An impossible trident, also known as an impossible fork, blivet, poiuyt, or devil's tuning fork,Brooks Masterton, John M. Kennedy"Building the Devil's Tuning Fork" ''Perception'', 1975, vol. 4, pp. 107-109 is a drawing of an impossible object (un ...
* Shepard elephant
*Three hares
The three hares (or three rabbits) is a circular motif or meme appearing in sacred sites from East Asia, the Middle East and to the churches of Devon, England (as the " Tinners' Rabbits"), and historical synagogues in Europe. It is used as an a ...
* Penrose steps
* Mobius strip
References
External links
An article about impossible triangle sculpture in Perth
Escher for Real constructions
{{DEFAULTSORT:Penrose Triangle Topology Impossible objects Triangles 1934 introductions