Editing Strange loop (section)

== Examples ==
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Hofstadter points to [[Johann Sebastian Bach|Bach]]'s ''Canon per Tonos'', [[M. C. Escher]]'s drawings ''[[Waterfall (M. C. Escher)|Waterfall]]'', ''[[Drawing Hands]]'', ''[[Ascending and Descending]]'', and the [[liar paradox]] as examples that illustrate the idea of strange loops, which is expressed fully in the proof of [[Gödel]]'s [[Gödel's incompleteness theorems|incompleteness theorem]].

The "[[chicken or the egg]]" paradox is perhaps the best-known strange loop problem.

The "[[ouroboros]]", which depicts a dragon eating its own tail, is perhaps one of the most ancient and universal symbolic representations of the reflexive loop concept.

A [[Shepard tone]] is another illustrative example of a strange loop. Named after [[Roger Shepard]], it is a [[sound]] consisting of a superposition of tones separated by [[octave]]s. When played with the base [[Pitch (music)|pitch]] of the tone moving upwards or downwards, it is referred to as the ''Shepard scale''. This creates the [[auditory illusion]] of a tone that continually ascends or descends in pitch, yet which ultimately seems to get no higher or lower. In a similar way a sound with seemingly ever increasing tempo can be constructed, as was demonstrated by [[Jean-Claude Risset]].
{{listen|filename=DescenteInfinie.ogg|title=A Shepard–Risset glissando|description=|format=[[Ogg]]}}

Visual illusions depicting strange loops include the [[Penrose stairs]] and the [[Barberpole illusion]].

A [[quine (computing)|quine]] in software programming is a program that produces a new version of itself without any input from the outside. A similar concept is [[metamorphic code]].

[[Intransitive dice#Efron's dice|Efron's dice]] are four dice that are [[intransitivity|intransitive]] under gambler's preference. I.e., the dice are ordered {{nowrap|A > B > C > D > A}}, where {{nowrap|''x'' > ''y''}} means "a gambler prefers ''x'' to ''y''".

Individual preferences are always transitive, excluding preferences when given explicit rules such as in Efron's dice or [[rock-paper-scissors]]; however, aggregate preferences of a group may be intransitive.  This can result in a [[Condorcet paradox]] wherein following a path from one candidate across a series of majority preferences may return to the original candidate, leaving no clear preference by the group.  In this case, some candidate beats an opponent, who in turn beats another opponent, and so forth, until a candidate is reached who beats the original candidate.

The liar paradox and [[Russell's paradox]] also involve strange loops, as does [[René Magritte]]'s painting ''[[The Treachery of Images]]''.

The mathematical phenomenon of [[polysemy]] has been observed to be a strange loop.  At the denotational level, the term refers to situations where a single entity can be seen to ''mean'' more than one mathematical object. See Tanenbaum (1999).

''[[The Stonecutter]]'' is an old Japanese [[fairy tale]] with a story that explains social and natural hierarchies as a strange loop.

A strange loop can be found by traversing the links in the “See also” sections of the respective [[English Wikipedia]] articles. For instance: This article->[[Mise en abyme]]->[[Recursion]]->this article.<ref>Wikipedia contributors. (2024, December 14). [[Strange loop]]. In ''Wikipedia, The Free Encyclopedia''. Retrieved 10:34, December 25, 2024, from https://en.wikipedia.org/w/index.php?title=Strange_loop&oldid=1263113776</ref>{{Circular reference|date=December 2024}}