Editing Penrose triangle

{{Short description|Impossible object}}
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[[Image:Penrose-dreieck.svg|thumb|upright=1|Penrose triangle]]
The '''Penrose triangle''', also known as the '''Penrose tribar''', the '''impossible tribar''',{{r|pappas}} or the '''impossible triangle''',{{r|brorub}} 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.<ref name=zeng>{{cite conference
 | last1 = Zeng | first1 = Zhenbing
 | last2 = Xu | first2 = Yaochen
 | last3 = Yang | first3 = Zhengfeng
 | last4 = Li | first4 = Zhi-bin
 | editor1-last = Corless | editor1-first = Robert M.
 | editor2-last = Gerhard | editor2-first = Jürgen
 | editor3-last = Kotsireas | editor3-first = Ilias S.
 | contribution = An isometric embedding of the impossible triangle into the Euclidean space of lowest dimension
 | contribution-url = https://www.maplesoft.com/mapleconference/resources/54_Zeng_IsometricEmbedding_slides.pdf
 | doi = 10.1007/978-3-030-81698-8_29
 | isbn = 9783030816988
 | pages = 438–457
 | publisher = Springer International Publishing
 | title = Maple in Mathematics Education and Research: 4th Maple Conference, MC 2020, Waterloo, Ontario, Canada, November 2–6, 2020, Revised Selected Papers
 | series = Communications in Computer and Information Science
 | year = 2021| volume = 1414
 }}</ref> It was first created by the Swedish artist [[Oscar Reutersvärd]] in 1934.{{r|ernst}} 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".{{r|penpen}} It is featured prominently in the works of artist [[M. C. Escher]], whose earlier depictions of impossible objects partly inspired it.

==Description==
[[File:Penrose-triangle-4color-rotation.gif|thumb|A rotating Penrose triangle model to show illusion. At the moment of illusion, there appears to be a pair of purple faces (one partially occluded) joined at right angles, but these are actually parallel faces, and the partially occluded face is internal, not external.]]
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-manifold]]s.{{r|francis}} A surface with the same [[geodesic distance]]s 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.<ref name=zeng/> 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.{{r|wa}} 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.{{r|gardner}}

==Depictions==
[[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=9783822858646|page=16}}</ref>

===Sculptures===
{{-}}
<gallery widths="240" heights="240">
File:Perth Impossible Triangle.jpg|"Impossible Triangle", Brian McKay and Ahmad Abas, East Perth, Australia, 1999{{r|wa}}
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]].{{r|fedorov}}
</gallery>

==See also==
*[[Impossible trident]]
*[[Shepard elephant]]
*[[Penrose stairs]]

==References==
{{reflist|refs=

<ref name=brorub>{{cite journal
 | last1 = Brouwer | first1 = James R.
 | last2 = Rubin | first2 = David C.
 | date = June 1979
 | doi = 10.1068/p080349
 | issue = 3
 | journal = [[Perception (journal)|Perception]]
 | pages = 349–350
 | title = A simple design for an impossible triangle
 | volume = 8| pmid = 534162
 | s2cid = 41895719
 }}</ref>

<ref name=ernst>{{cite conference
 | last = Ernst | first = Bruno
 | editor1-last = Coxeter | editor1-first = H. S. M. | editor1-link = Harold Scott MacDonald Coxeter
 | editor2-last = Emmer | editor2-first = M.
 | editor3-last = Penrose | editor3-first = R. | editor3-link = Roger Penrose
 | editor4-last = Teuber | editor4-first = M. L.
 | contribution = Escher's impossible figure prints in a new context
 | pages = 125–134
 | publisher = North-Holland
 | title = M. C. Escher Art and Science: Proceedings of the International Congress on M. C. Escher, Rome, Italy, 26–28 March, 1985
 | 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=francis>{{cite book
 | last = Francis | first = George K.
 | contribution = Chapter 4: The impossible tribar
 | doi = 10.1007/978-0-387-68120-7_4
 | isbn = 0-387-96426-6
 | pages = 65–76
 | publisher = Springer
 | title = A Topological Picturebook
 | year = 1988}} See in particular p. 68, where Francis attributes this observation to [[John Stillwell]].</ref>

<ref name=gardner>{{cite journal
 | last = Gardner | first = Martin
 | date = August 1978
 | issue = 2
 | journal = [[Scientific American]]
 | jstor = 24960346
 | pages = 18–26
 | title = Mathematical Games: A Möbius band has a finite thickness, and so it is actually a twisted prism
 | volume = 239| doi = 10.1038/scientificamerican1278-18
 }}</ref>

<ref name=pappas>{{cite book
 | last = Pappas | first = Theoni | author-link = Theoni Pappas
 | contribution = The Impossible Tribar
 | location = San Carlos, California
 | page = 13
 | publisher = Wide World Publ./Tetra
 | title = The Joy of Mathematics: Discovering Mathematics All Around You
 | year = 1989}}</ref>

<ref name=penpen>{{cite journal
 | last1 = Penrose | first1 = L. S. | author1-link = Lionel Penrose
 | last2 = Penrose | first2 = R. | author2-link = Roger Penrose
 | date = February 1958
 | doi = 10.1111/j.2044-8295.1958.tb00634.x
 | issue = 1
 | journal = [[British Journal of Psychology]]
 | pages = 31–33
 | pmid = 13536303
 | title = Impossible objects: a special type of visual illusion
 | 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>

}}

==External links==
* [http://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]

{{Optical illusions}}
{{Roger Penrose}}

{{DEFAULTSORT:Penrose Triangle}}
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[[Category:Topology]]
[[Category:Impossible objects]]
[[Category:Triangles]]
[[Category:1934 introductions]]
[[Category:Roger Penrose]]