Penrose triangle

The Penrose triangle

Impossible Triangle sculpture, East Perth, Australia. The structure is actually disjointed, and was photographed from one of the two spots that it was designed to be seen from.More pictures from other angles.

The Penrose triangle, also known as the tribar, is an impossible object. It was first created by the Swedish artist Oscar Reutersvärd in 1934. The mathematician Roger Penrose independently devised and popularised 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.

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.

This combination of properties cannot be realized by any 3-dimensional object. Nevertheless, there do exist 3-dimensional solid shapes each of which, when viewed from a certain angle, has the appearance of possessing all the qualities mentioned in the above paragraph. That is, it appears the same as the purple, green, and yellow 2-dimensional depiction of the Penrose triangle on this page.

M.C. Escher's lithograph Waterfall 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.

The concept can be extended to other polygons, making, for example the "Penrose square," but the visual effect is not as striking.

There exists some terminological confusion over whether "Penrose triangle" refers to the 2-dimensional depiction of an impossible 3-dimensional object, or that impossible object itself. (Philosophically, it is unclear what "that impossible object" refers to, other than a set of conditions that cannot be satisfied.)

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

Other Penrose polygons

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

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  Penrose rectangle

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  Penrose pentagon

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  Penrose hexagon

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  Penrose octagon

External links

• Escher for Real constructions
• The impossible triangle made possible
• Three-dimensional model of a Penrose triangle for SketchUp

Credits

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