Strange Loop

A strange loop is a phenomenon in which, whenever movement is made upwards or downwards through the levels of some hierarchical system, the system unexpectedly arrives back where it started. Hofstadter (1989) uses the strange loop as a paradigm in which to interpret paradoxes in logic (such as Grelling's paradox, the liar's paradox, and Russell's antinomy) and calls a system in which a strange loop appears a tangled hierarchy.

Canon 5 from Bach's Musical Offering (sometimes known as Bach's endlessly rising canon) is a musical piece that continues to rise in key, modulating through the entire chromatic scale until it ends in the same key in which it began. This is the first example cited by Hofstadter (1989) as a strange loop.

Other examples include the endlessly rising stairs in M. C. Escher 1960 lithograph Ascending and Descending, the endlessly falling waterfall in his 1961 lithograph Waterfall, and the pair of hands drawing each other in his lithograph Drawing Hands (Hofstadter 1989, pp. 10-15). Painter René Magritte's The Treachery of Images provides another example.

Various authors have written works whose titles involve strange loops. For example, Abbie Hoffman humorously attempted to undermine sales of his own book by entitling it Steal This Book, while the band "System of a Down" did the analogous thing with their album Steal This Album!

REFERENCES

Escher, M. C. "Ascending and Descending." Lithograph. 1960.

 http://www.mcescher.com/Gallery/recogn-bmp/LW435.jpg.Escher, M. C. "Waterfall." Lithograph. 1961.

 http://www.mcescher.com/Gallery/recogn-bmp/LW439.jpg.Hoffman, A. Steal this Book, new ed. Four Walls Eight Windows, 2002.

Hofstadter, D. R. Gödel, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, pp. 10-28, 1989.