„Penrose-Treppe“ – Versionsunterschied

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[[File:Impossible staircase.svg|thumb|Penrose stairs.]]
[[Datei:Impossible staircase.svg|mini|Penrose-Treppe]]
<gallery class="float-right">
Penrosetreppe links.svg|Linke Teiltreppe
Penrosetreppe rechts.svg|Rechte Teiltreppe
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Die '''Penrose-Treppe''', auch die '''unmögliche Treppe''' genannt, ist eine sogenannte [[unmögliche Figur]], die von dem britischen Mathematiker [[Lionel Penrose]] und seinem Sohn [[Roger Penrose]] im Jahre 1958 entdeckt und veröffentlicht wurde. Es ist eine Variation des [[Penrose-Dreieck]]s und ist eine zweidimensionale Darstellung einer dreidimensionalen Treppe mit geschlossenem Innenraum, die in sich selbst zurückläuft, so dass eine Illusion erzeugt wird, dass sie unendlich hinauf bzw. hinunter führt. Damit ist sie physikalisch unmöglich und nur eine Wahrnehmungstäuschung.
The '''Penrose stairs''' or '''Penrose steps''', also dubbed the '''impossible staircase''', is an [[impossible object]] created by [[Lionel Penrose]] and his son [[Roger Penrose]].<ref name="Penrose 1958">{{harvnb|Penrose|Penrose|1958| pp=31–33}}</ref> A variation on the [[Penrose triangle]], it is a two-dimensional depiction of a staircase in which the stairs make four 90-degree turns as they ascend or descend yet form a continuous loop, so that a person could climb them forever and never get any higher. This is clearly impossible in three dimensions.


Bei geeigneter Trennung der Treppe entstehen jedoch einzeln real wahrnehmbare Teile, wie ein Vergleich der Bilder zeigt.
The "continuous staircase" was first presented in an article that the Penroses wrote in 1959, based on the so called "triangle of Penrose" published by Roger Penrose in the ''British Journal of Psychology'' in 1958. [[M. C. Escher]] then discovered the Penrose stairs in the following year and made his now famous lithography ''Klimmen en dalen'' (''[[Ascending and Descending]]'') in March 1960. Penrose and Escher were informed of each other's work that same year.<ref>{{harvnb|Hallyn|2000| p=172}}</ref>
Escher developed the theme further in his print ''Waterval'' (''[[Waterfall (M. C. Escher)|Waterfall]]''), which appeared in 1961.


In dem Artikel ''Impossible objects: A special type of visual illusion'' von 1958 fand man erstmals Abbildungen dieser Treppe und auch anderer unmöglicher Figuren, z.&nbsp;B. des Penrose-Dreiecks. Eine von Roger Penrose an den niederländischen Grafiker [[M.&nbsp;C. Escher]] geschickte Kopie des Artikels inspirierte diesen dazu, mehrere perspektivische Holzschnitte mit unmöglichen Objekten anzufertigen.<ref>[http://www.optical-illusion-pictures.com/paradox.html Paradox Illusions (englisch)]</ref>
In their original article the Penroses noted that "each part of the structure is acceptable as representing a flight of steps but the connexions are such that the picture, as a whole, is inconsistent: the steps continually descend in a clockwise direction."<ref>{{harvnb|Ernst|1992| p=72}}</ref>


== Rezeption ==
The [[Shepard tone]], developed in the 1960s, is a similar illusion in terms of [[sound]].<ref name="Deutsch-2010">{{harvnb|Deutsch|2010}}</ref><ref name="IllusionWorks" />
* In dem Film [[Inception|''Inception'' (2010)]] sieht man einige Protagonisten auf einer Penrose-Treppe laufen.


== Literatur ==
==History of discovery==
* {{Literatur
At an Escher conference in Rome in 1985, Roger Penrose said that he had been greatly inspired by Escher's work when he and his father discovered both the tri-bar structure and the continuous steps, although Escher at the time had not yet drawn any impossible figures and was not aware of their existence. Roger Penrose had been introduced to Escher's work at the International Congress of Mathematicians in Amsterdam in 1954. He was "absolutely spellbound" by Escher's work, and on his journey back to England he decided to produce something "impossible" on his own. After experimenting with various designs of bars overlying each other he finally arrived at the impossible triangle. Roger showed his drawings to his father, who immediately produced several variants, including the impossible flight of stairs. They wanted to publish their findings but didn't know in what field the subject belonged. Because Lionel Penrose knew the editor of ''British Journal of Psychology'' and convinced him to publish their short manuscript, the finding was finally presented as a psychological subject. After the publication in 1958 the Penroses sent a copy of the article to Escher as a token of their esteem.<ref name="Ernst-71">{{harvnb|Ernst|1992| pp=71–2}}</ref>
|Autor=Diana Deutsch
|Titel=The Paradox of Pitch Circularity
|Sammelwerk=Acoustics Today
|Band=6
|Nummer=3
|Datum=2010-07
|Seiten=8–14
|Online=http://deutsch.ucsd.edu/pdf/Acoustics_Today_2010_Jul.pdf
|Abruf=2012-04-08
|DOI=10.1121/1.3488670}}
* {{Literatur
|Autor=Bruni Ernst
|Titel=The Eye Beguiled: Optical illusions
|Verlag=Benedikt Taschen
|Datum=1992
|ISBN=3-8228-9637-3}}
* {{Literatur
|Hrsg=Fernand Hallyn
|Titel=Metaphor and Analogy in the Sciences
|Verlag=Springer
|Datum=2000
|ISBN=0-7923-6560-7
|Online=http://books.google.com/books?id=iKleTxffj5IC&pg=PA172
|Abruf=2012-04-08}}
* {{Internetquelle
|autor=IllusionWorks
|url=http://psylux.psych.tu-dresden.de/i1/kaw/diverses%20Material/www.illusionworks.com/html/impossible_staircase.html
|titel=Impossible Staircase
|datum=1997
|abruf=2012-04-08}}
* {{Internetquelle
|autor=Ferdinand Lehr
|url=https://ferdinandlehr.de/wp-content/uploads/2023/02/Algorithm_Penrose-Stairs_flehr.pdf
|titel=Studie zur algorithmischen Erzeugung von Penrose-Treppen
|datum=2022
|format=PDF
|abruf=2022-10-23}}


== Weblinks ==
While the Penroses credited Escher in their article, Escher himself noted in a letter to his son in January 1960 that he was:
* [https://ferdinandlehr.de/mathematik/penrose-staircase-calculator/ Browserbasierter Penrose-Treppengenerator]
{{cquote|working on the design of a new picture, which featured a flight of stairs which only ever ascended or descended, depending on how you saw it. [The stairs] form a closed, circular construction, rather like a snake biting its own tail. And yet they can be drawn in correct perspective: each step higher (or lower) than the previous one. [...] I discovered the principle in an article which was sent to me, and in which I myself was named as the maker of various 'impossible objects'. But I was not familiar with the continuous steps of which the author had included a clear, if perfunctory, sketch, although I was employing some of his other examples.<ref name="Ernst-75-78">{{harvnb|Ernst|1992| pp=75, 78}}</ref>}}
* [https://www.exergia.de/english/ideen-projekte/design-art/escher-penrose-stairway/ Animierte 3D-Rotation einer Penrose-Treppe]


== Einzelnachweise ==
Escher was captivated by the endless stairs and subsequently wrote a letter to the Penroses in April 1960:
<references />
{{cquote|A few months ago, a friend of mine sent me a photocopy of your article... Your figures 3 and 4, the 'continuous flight of steps', were entirely new to me, and I was so taken by the idea that they recently inspired me to produce a new picture, which I would like to send to you as a token of my esteem. Should you have published other articles on impossible objects or related topics, or should you know of any such articles, I would be most greatful if you could send me further details.<ref name="Ernst-75-78" />}}


[[Kategorie:Wahrnehmungstäuschung]]
The staircase design had been discovered previously by the Swedish artist [[Oscar Reutersvärd]], but neither Penrose nor Escher were aware of his designs.<ref name="IllusionWorks">{{harvnb|IllusionWorks|1997}}</ref> Inspired by a radio programme on Mozart's method of composition&nbsp;— described as "creative automatism", i.e. each creative idea written down inspired a new idea&nbsp;— Reutersvärd started to draw a series of impossible objects on a journey from Stockholm to Paris in 1950 in the same "unconscious, automatic" way. He did not realise that his figure was a continuous flight of stairs while drawing, but the process enabled him to trace his increasingly complex designs step by step. When M. C. Escher's ''Ascending and Descending'' was sent to Reutersvärd in 1961, he was impressed but didn't like the irregularities of the stairs (2×15+2×9). Throughout the 1960s, Reutersvärd sent several letters to Escher to express his admiration for his work, but the Dutch artist failed to respond.<ref>{{harvnb|Ernst|1992| pp=70–1}}</ref> Roger Penrose only discovered Reutersvärd's work in 1984.<ref name="Ernst-71" />
[[Kategorie:Treppen]]

==See also==
*[[Reversible figure]]

==Notes==
{{reflist}}
==References==
{{refbegin}}
* {{cite journal | ref = harv
| last = Deutsch | first = Diana
| title = The Paradox of Pitch Circularity
| journal = Acoustics Today | month = July | year = 2010 | volume = 6 | issue = 3 | pages = 8–14
| url = http://deutsch.ucsd.edu/pdf/Acoustics_Today_2010_Jul.pdf
| accessdate = March 2011 | doi=10.1121/1.3488670
}}
* {{cite book | ref = harv
| last = Ernst | first = Bruno
| title = The Eye Beguiled: Optical illusions
| publisher = Benedikt Taschen | year = 1992
| isbn = 3-8228-9637-3
}}
* {{cite book | ref = harv
| last = Hallyn | first = Fernand
| title = Metaphor and analogy in the sciences
| publisher = Springer | year = 2000
| isbn = 9780792365600
| url = http://books.google.com/books?id=iKleTxffj5IC&pg=PA172
| accessdate = March 2011
}}
* {{cite web | ref = harv
| title = Impossible Staircase
| url = http://psylux.psych.tu-dresden.de/i1/kaw/diverses%20Material/www.illusionworks.com/html/impossible_staircase.html
| author = IllusionWorks | year = 1997
| accessdate = March 2011
}}
* {{cite journal | ref = harv
| last1 = Penrose | first1 = L. S. | last2 = Penrose | first2 = R.
| title = Impossible objects: A special type of visual illusion
| journal = British Journal of Psychology | year = 1958 | volume = 49 | pages = 31–33
}}
{{refend}}

{{DEFAULTSORT:Penrose Stairs}}
[[Category:Optical illusions]]
[[Category:Stairways]]
[[Category:Impossible objects]]

[[bg:Стълби на Пенроуз]]
[[ca:Escala de Penrose]]
[[de:Penrose-Treppe]]
[[es:Escalera de Penrose]]
[[fr:Escalier de Penrose]]
[[it:Scala di Penrose]]
[[nl:Penrose-trap]]
[[ja:ペンローズの階段]]
[[no:Penrosetrapp]]
[[pt:Escada de Penrose]]
[[ru:Лестница Пенроуза]]
[[sv:Penrose trappa]]
[[uk:Сходи Пенроуза]]
[[zh:彭罗斯阶梯]]

Aktuelle Version vom 25. Oktober 2024, 17:04 Uhr

Penrose-Treppe
  • Linke Teiltreppe
    Linke Teiltreppe
  • Rechte Teiltreppe
    Rechte Teiltreppe

Die Penrose-Treppe, auch die unmögliche Treppe genannt, ist eine sogenannte unmögliche Figur, die von dem britischen Mathematiker Lionel Penrose und seinem Sohn Roger Penrose im Jahre 1958 entdeckt und veröffentlicht wurde. Es ist eine Variation des Penrose-Dreiecks und ist eine zweidimensionale Darstellung einer dreidimensionalen Treppe mit geschlossenem Innenraum, die in sich selbst zurückläuft, so dass eine Illusion erzeugt wird, dass sie unendlich hinauf bzw. hinunter führt. Damit ist sie physikalisch unmöglich und nur eine Wahrnehmungstäuschung.

Bei geeigneter Trennung der Treppe entstehen jedoch einzeln real wahrnehmbare Teile, wie ein Vergleich der Bilder zeigt.

In dem Artikel Impossible objects: A special type of visual illusion von 1958 fand man erstmals Abbildungen dieser Treppe und auch anderer unmöglicher Figuren, z. B. des Penrose-Dreiecks. Eine von Roger Penrose an den niederländischen Grafiker M. C. Escher geschickte Kopie des Artikels inspirierte diesen dazu, mehrere perspektivische Holzschnitte mit unmöglichen Objekten anzufertigen.[1]

Rezeption

[Bearbeiten | Quelltext bearbeiten]
  • In dem Film Inception (2010) sieht man einige Protagonisten auf einer Penrose-Treppe laufen.

Literatur

[Bearbeiten | Quelltext bearbeiten]
[Bearbeiten | Quelltext bearbeiten]

Einzelnachweise

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  1. ↑ Paradox Illusions (englisch)
Abgerufen von „https://de.wikipedia.org/w/index.php?title=Penrose-Treppe&oldid=249744706“