A strange loop
arises when, by moving up or down through a hierarchical system, one finds oneself back where one started.
Strange loops may involve self-reference
. The concept of a strange loop was proposed and extensively discussed by Douglas Hofstadter
in Gödel, Escher, Bach
, and is further elaborated in Hofstadter's book I Am a Strange Loop
, which appeared in 2007.
A tangled hierarchy is a hierarchical system in which a strange loop appears.
A strange loop is a hierarchy of levels, each of which may consist of objects, processes, or virtually anything else (such is the generality of the notion). Each level is linked to at least one other by some type of relationship. A strange loop hierarchy, however, is "tangled" (what Hofstadter refers to as a "heterarchy"), in that there is no well defined highest or lowest level. The levels are organized such that moving through them eventually returns one to one's starting point, i.e., the original level. Examples of strange loops that Hofstadter offers include many of the works of M. C. Escher
, the information flow network between DNA
through protein synthesis
and DNA replication
, and self-referential Gödelian statements
in formal systems
In I Am a Strange Loop, Hofstadter defines strange loops as follows:
And yet when I say "strange loop", I have something else in mind — a less concrete, more elusive notion. What I mean by "strange loop" is — here goes a first stab, anyway — not a physical circuit but an abstract loop in which, in the series of stages that constitute the cycling-around, there is a shift from one level of abstraction (or structure) to another, which feels like an upwards movement in a hierarchy, and yet somehow the successive "upward" shifts turn out to give rise to a closed cycle. That is, despite one's sense of departing ever further from one's origin, one winds up, to one's shock, exactly where one had started out. In short, a strange loop is a paradoxical level-crossing feedback loop.
- Quines in software programing- wherein a program produces a new version of itself without any input from the outside. Metamorphic code.
- 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).
Strange Loops in Popular Culture
- The use of a Shepard tone as the sound effect for the Infinite Staircase in Super Mario 64.
- The use of a Shepard tone in the song "Echoes" by Pink Floyd, from the album Meddle.
- In one level of the game Psychonauts, Raz shrinks in size to walk around the surface of a board game which contains a building with a window through which the original room with the board game can be seen. The player cannot enter or exit the room through this window, however, so it may not be considered a true Strange Loop.
- The novelty song "I'm My Own Grandpa" (sometimes rendered as "I'm My Own Grandpaw") by Dwight Latham and Moe Jaffe, performed by Lonzo and Oscar in 1948. The song is sung by a man who, through an unlikely (but legal) combination of marriages, becomes stepfather to his own stepmother — and thus becomes his own (step- + step- (or step- squared)) grandfather.
- The myriad time-travel paradoxes of classic Science Fiction can be perceived as creative versions of Strange Loops that fail by self-cancelling feedback. The two base forms of this are the Predestination paradox and the Ontological paradox
- Usually, these stories are variations on a core trope: a time-traveler alters events in the past in a way that precludes the initiation of the whole process in the first place - either by rendering time-travel itself impossible, or by somehow rendering his own existence null and void. Archetypal examples would include almost any of the many “Shoot Your Grandfather” stories (cf, "The Men Who Murdered Mohammed" by Alfred Bester).
- A quintessential (and recognized classic) SF Time-Travel Strange Loop story is “All You Zombies—” by Robert A. Heinlein. In this short story, Heinlein creates an Anti-Paradox (a self-generating/self-supporting/self-dependent tautological sequence) that exactly meets Douglas Hofstadter's criteria for a Strange Loop as quoted in Definitions (above) - specifically in regard to the criteria ...
- "... despite one's sense of departing ever further from one's origin, one winds up, to one's shock, exactly where one had started out. In short, a strange loop is a paradoxical level-crossing feedback loop.”
- The recursive loop in “All You Zombies—” occurs by way of an alteration of the past that leads to the very future in which same alteration becomes both necessary and inevitable. Specifically: a young hermaphrodite in female mode (as surgically adjusted right after birth so as to render her female reproductive organs dominant and fertile) becomes her own mother (supplying the egg and the womb) after being impregnated by her older time-travelling male version (after giving birth she/he is surgically adjusted into fertile and active male mode). This time-looped self-encounter/courtship/impregnation is stage-managed by her/his/their much older, male self who brings the younger male version back in time to meet (and court, seduce and impregnate) the female iteration. This all to ensure his/her/their eternal, self-generating closed-loop existence.
- nota bene: during the story, the song "I'm My Own Grandpa" cited in Examples (above) is played on the jukebox in the bar managed by the narrator - who is the "final edition" of the main character.
- P.J. Tanenbaum, "Simultaneous intersection representation of pairs of graphs," Journal of Graph Theory 32 (1999) 171-190. ISSN 1097-0118