Penrose triangle: Difference between revisions
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{{Short description|Impossible object}} | {{Short description|Impossible object}} | ||
{{Redirect-distinguish|Impossible triangle|Impossible trinity}} | |||
{{CS1 config|mode=cs1}} | {{CS1 config|mode=cs1}} | ||
[[Image:Penrose-dreieck.svg|thumb|upright=1|Penrose triangle]] | [[Image:Penrose-dreieck.svg|thumb|upright=1|Penrose triangle]] | ||
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| contribution-url = https://www.maplesoft.com/mapleconference/resources/54_Zeng_IsometricEmbedding_slides.pdf | | contribution-url = https://www.maplesoft.com/mapleconference/resources/54_Zeng_IsometricEmbedding_slides.pdf | ||
| doi = 10.1007/978-3-030-81698-8_29 | | doi = 10.1007/978-3-030-81698-8_29 | ||
| isbn = | | isbn = 978-3-030-81698-8 | ||
| pages = 438–457 | | pages = 438–457 | ||
| publisher = Springer International Publishing | | publisher = Springer International Publishing | ||
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If a line is traced around the Penrose triangle, a 4-loop [[Möbius strip]] is formed.{{r|gardner}} | If a line is traced around the Penrose triangle, a 4-loop [[Möbius strip]] is formed.{{r|gardner}} | ||
== Creation of the Penrose triangle from partial figures == | |||
[[File:Penrosedreieck Entstehung.svg|thumb|Creation of the Penrose triangle (right) from two real perceptible partial figures]] | |||
If you move the left part of the figure parallel to the right until its upper horizontal edge coincides with the upper horizontal edge of the middle part of the figure, the Penrose triangle (right) is created by overlapping the two parts. | |||
The first two partial views of the Penrose triangle are individually perceptible, whereas the resulting tribar represents an impossible figure.<ref>[http://www.mathe-werkstatt.de/allgem/unmoegl.htm Unmögliche Figuren (Escher)] in ''mathe-werkstatt.de'' (based on the creation of Escher's figures)</ref> | |||
==Depictions== | ==Depictions== | ||
[[File:Penrosetrianglemodel.jpg|thumb|A 3D-printed version of the Reutersvärd Triangle illusion]] | [[File:Penrosetrianglemodel.jpg|thumb|A 3D-printed version of the Reutersvärd Triangle illusion]] | ||
[[M.C. Escher]]'s [[lithograph]] ''[[Waterfall (M. C. Escher)|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 points out that in order to keep the wheel turning, some water must occasionally be added to compensate for [[evaporation]]. A third Penrose triangle lies between the other two, formed by two segments of waterway and a support tower.<ref>{{cite book|title=M. C. Escher: The Graphic Work|publisher=Taschen|year=2000|isbn= | [[M.C. Escher]]'s [[lithograph]] ''[[Waterfall (M. C. Escher)|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 points out that in order to keep the wheel turning, some water must occasionally be added to compensate for [[evaporation]]. A third Penrose triangle lies between the other two, formed by two segments of waterway and a support tower.<ref>{{cite book |title=M. C. Escher: The Graphic Work |location=Cologne |publisher=Taschen |year=2000 |isbn=3-8228-5864-1 |page=16}}</ref> | ||
===Sculptures=== | ===Sculptures=== | ||
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| year = 1986}} See in particular p. 131.</ref> | | year = 1986}} See in particular p. 131.</ref> | ||
<ref name=fedorov>{{Cite journal|last=Федоров|first=Ю.|date=1972|title=Невозможное-Возможно|journal=Техника Молодежи|volume=4|pages=20–21|url=http://zhurnalko.net/=nauka-i-tehnika/tehnika-molodezhi/1972-04--num22}}</ref> | <ref name=fedorov>{{Cite journal |last=Федоров |first=Ю. |date=1972 |title=Невозможное-Возможно |journal=Техника Молодежи |language=ru |volume=4 |pages=20–21 |url=http://zhurnalko.net/=nauka-i-tehnika/tehnika-molodezhi/1972-04--num22}}</ref> | ||
<ref name=francis>{{cite book | <ref name=francis>{{cite book | ||
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| volume = 49}}</ref> | | volume = 49}}</ref> | ||
<ref name=wa>{{cite web|url=https://www.wa.gov.au/government/media-statements/Court%20Coalition%20Government/Unveiling-of-East-Perth-redevelopment%27s-latest-piece-of-public-art-19991105|title=Unveiling of East Perth redevelopment's latest piece of public art|date=5 November 1999|publisher=Government of West Australia|access-date=2025-05-09}}</ref> | <ref name=wa>{{cite web |url=https://www.wa.gov.au/government/media-statements/Court%20Coalition%20Government/Unveiling-of-East-Perth-redevelopment%27s-latest-piece-of-public-art-19991105 |title=Unveiling of East Perth redevelopment's latest piece of public art |date=5 November 1999 |publisher=Government of West Australia |access-date=2025-05-09}}</ref> | ||
}} | }} | ||
==External links== | ==External links== | ||
* [ | * [https://im-possible.info/english/articles/real/real3.html An article about impossible triangle sculpture in Perth] | ||
* [https://web.archive.org/web/20080120055138/http://www.cs.technion.ac.il/~gershon/EscherForReal/ Escher for Real constructions] | * [https://web.archive.org/web/20080120055138/http://www.cs.technion.ac.il/~gershon/EscherForReal/ Escher for Real constructions] | ||
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{{Roger Penrose}} | {{Roger Penrose}} | ||
<!--Categories--> | <!--Categories--> | ||
[[Category:Topology]] | [[Category:Topology]] | ||
Latest revision as of 23:59, 16 September 2025
Template:Short description Template:Redirect-distinguish Template:CS1 config
The Penrose triangle, also known as the Penrose tribar, the impossible tribar,Template:R or the impossible triangle,Template:R is a triangular impossible object, an optical illusion consisting of an object which can be depicted in a perspective drawing. It cannot exist as a solid object in ordinary three-dimensional Euclidean space, although its surface can be embedded isometrically (bent but not stretched) in five-dimensional Euclidean space.[1] It was first created by the Swedish artist Oscar Reutersvärd in 1934.Template:R Independently from Reutersvärd, the triangle was devised and popularized in the 1950s by psychiatrist Lionel Penrose and his son, the mathematician and Nobel Prize laureate Roger Penrose, who described it as "impossibility in its purest form".Template:R 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.Template:R A surface with the same geodesic distances as the depicted surface of the tribar, but without its flat shape and right angles, are to be preserved, can also exist in 5-dimensional Euclidean space, which is the lowest-dimensional Euclidean space within which this surface can be isometrically embedded.[1] 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, such as the sculpture "Impossible Triangle" in East Perth, Australia.Template:R 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.Template:R
Creation of the Penrose triangle from partial figures
If you move the left part of the figure parallel to the right until its upper horizontal edge coincides with the upper horizontal edge of the middle part of the figure, the Penrose triangle (right) is created by overlapping the two parts.
The first two partial views of the Penrose triangle are individually perceptible, whereas the resulting tribar represents an impossible figure.[2]
Depictions
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 points out that in order to keep the wheel turning, some water must occasionally be added to compensate for evaporation. A third Penrose triangle lies between the other two, formed by two segments of waterway and a support tower.[3]
Sculptures
-
"Impossible Triangle", Brian McKay and Ahmad Abas, East Perth, Australia, 1999Template:R
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Impossible Triangle sculpture, Gotschuchen, Austria
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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.Template:R
See also
References
External links
Template:Optical illusions Template:Roger Penrose
- ↑ a b Script error: No such module "citation/CS1".
- ↑ Unmögliche Figuren (Escher) in mathe-werkstatt.de (based on the creation of Escher's figures)
- ↑ Script error: No such module "citation/CS1".