The Penrose triangle, also known as the Penrose tribar, the impossible tribar, or the impossible triangle, is a triangular

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-d ...

, 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 Oscar Reutersvärd (29 November 1915 – 2 February 2002) was a Swedes, Swedish graphic artist, who in 1934 pioneered the art of 3D drawings that may initially appear feasible, yet cannot be physically constructed. He is sometimes described as "th ...

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. Penrose w ...

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 fello ...

, 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 State of matter#Four fundamental states, four fundamental states of matter (the others being liquid, gas, and Plasma (physics), plasma). The molecules in a solid are closely packed together and contain the least amount o ...

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 Edge (geometry), edges and three Vertex (geometry), 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, an ...

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, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

. 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 state, sovereign country comprising the mainland of the Australia (continent), Australian continent, the island of Tasmania, and numerous List of islands of Australia, sma ...

). 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 Augu ...

is formed.

Depictions

M.C. Escher Maurits Cornelis Escher (; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made Mathematics and art, mathematically inspired woodcuts, lithography, lithographs, and mezzotints. Despite wide popular interest, Escher was for ...

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 wa ...

'' (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 humidi ...

Sculptures

File:Perth Impossible Triangle.jpg, Impossible triangle sculpture as an

, 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.

References

External links

An article about impossible triangle sculpture in Perth

Escher for Real constructions
{{DEFAULTSORT:Penrose Triangle Topology Impossible objects Triangles 1934 introductions

Description

Depictions

Sculptures

See also

References

External links