Penrose triangle

The Penrose triangle

The Penrose triangle, also known as the Penrose tribar, or the impossible tribar,[1] is a triangular impossible object. It was first created by the Swedish artist Oscar Reutersvärd in 1934. The psychiatrist Lionel Penrose and his mathematician son Roger Penrose independently devised and popularized it in the 1950s, describing it as "impossibility in its purest form". It is featured prominently in the works of artist M. C. Escher, whose earlier depictions of impossible objects partly inspired it.

Impossible object

The tribar 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.

Impossible triangle sculpture as an optical illusion, East Perth, Western Australia

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.[2] 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.

M.C. Escher's lithograph Waterfall (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 helpfully points out that in order to keep the wheel turning some water must occasionally be added to compensate for evaporation.

A 3D-printed version of the Reutersvaard Triangle illusion
Impossible Triangle sculpture, Gotschuchen, Austria

If a line is traced around the Penrose triangle, a 4-loop Möbius strip is formed.[3].

Although the tribar is named one of the impossible objects, there exist many more that fit into the same category. Other impossible objects include the devil's fork, the dancing elephant, and impossible arch.[4]

Other Penrose polygons

While it is possible to construct analogies to the Penrose triangle with other shapes and regular polygons to create a Penrose polygon, the visual effect is not as striking, and as the sides increase, the object seems merely to be warped or twisted.

See also

References

  1. ↑ Pappas, T. "The Impossible Tribar." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 13, 1989.
  2. ↑ Francis, George (1988). A topological picturebook. Springer. ISBN 0-387-96426-6. In the chapter on the Penrose tribar, Francis attributes this observation to John Stillwell.
  3. ↑ Gardner, Martin (1978). Mathematical Games, August 1978 A Möbius band has a finite thickness, and so it is actually a twisted prism. SCIENTIFIC AMERICAN, A DIVISION OF NATURE AMERICA, INC.
  4. ↑ Adamovic, Jan. "BrainDen.com - Impossible Objects". BrainDen.com. Retrieved 2016-02-09.
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