Examples
Hofstadter points to Bach's Canon per Tonos, M. C. Escher's drawings 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 incompleteness theorem.
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 octaves. When played with the base 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. See Barber's pole.
A quine in software programming is a program which produces a new version of itself without any input from the outside. A similar concept is metamorphic code.
Efron's dice are four dice which are intransitive under gambler's preference. In other words, the dice are ordered A > B > C > D > A, where x > y means "a gambler prefers x to y".
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.
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.
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