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<title>Strange loop</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Strange loop</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="Strange_loop_(disambiguation)" class="mw-redirect mw-disambig" title="Strange loop (disambiguation)">Strange loop (disambiguation)</a>.</div>

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<p>A <b>strange loop</b> is a <a href="Cycle_(graph_theory)" title="Cycle (graph theory)">cyclic</a> structure that goes through several levels in a <a href="Hierarchical" class="mw-redirect" title="Hierarchical">hierarchical</a> system. It arises when, by moving only upwards or downwards through the system, one finds oneself back where one started.

Strange loops may involve <a href="Self-reference" title="Self-reference">self-reference</a> and <a href="Paradox" title="Paradox">paradox</a>. The concept of a strange loop was proposed and extensively discussed by <a href="Douglas_Hofstadter" title="Douglas Hofstadter">Douglas Hofstadter</a> in <i><a href="G%C3%B6del%2C_Escher%2C_Bach" title="Gödel, Escher, Bach">Gödel, Escher, Bach</a></i>, and is further elaborated in Hofstadter's book <i><a href="I_Am_a_Strange_Loop" title="I Am a Strange Loop">I Am a Strange Loop</a></i>, published in 2007.

</p><p>A <b>tangled hierarchy</b> is a <a href="Hierarchy" title="Hierarchy">hierarchical</a> consciousness system in which a strange loop appears.

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<div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2></div>

<p>A strange loop is a hierarchy of levels, each of which is linked to at least one other by some type of relationship. A strange loop hierarchy is "tangled" (Hofstadter refers to this as a "<a href="Heterarchy" title="Heterarchy">heterarchy</a>"), in that there is no well defined highest or lowest level; moving through the levels, one eventually returns to the starting point, i.e., the original level. Examples of strange loops that Hofstadter offers include: many of the works of <a href="M._C._Escher" title="M. C. Escher">M. C. Escher</a>, the <i>Canon 5. a 2</i> from J.S. Bach's <a href="Musical_Offering" class="mw-redirect" title="Musical Offering">Musical Offering</a>, the information flow network between <a href="DNA" title="DNA">DNA</a> and <a href="Enzymes" class="mw-redirect" title="Enzymes">enzymes</a> through <a href="Protein_synthesis" class="mw-redirect" title="Protein synthesis">protein synthesis</a> and <a href="DNA_replication" title="DNA replication">DNA replication</a>, and <a href="Self-reference" title="Self-reference">self-referential</a> <a href="G%C3%B6del's_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödelian statements</a> in <a href="Formal_system" title="Formal system">formal systems</a>.

</p><p>In <i><a href="I_Am_a_Strange_Loop" title="I Am a Strange Loop">I Am a Strange Loop</a></i>, Hofstadter defines strange loops as follows:

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<blockquote><p>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 an 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 <a href="Feedback_loop" class="mw-redirect" title="Feedback loop">feedback loop</a>. (pp. 101–102)</p></blockquote>

<div class="mw-heading mw-heading2"><h2 id="In_cognitive_science">In cognitive science</h2></div>

<p>According to Hofstadter, strange loops take form in human consciousness as the complexity of active symbols in the brain inevitably leads to the same kind of self-reference which <a href="Kurt_G%C3%B6del" title="Kurt Gödel">Gödel</a> proved was inherent in any sufficiently complex logical or arithmetical system (that allows for arithmetic by means of the <a href="Peano_axioms" title="Peano axioms">Peano axioms</a>) in his <a href="G%C3%B6del's_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorem</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Gödel showed that mathematics and logic contain strange loops: propositions that not only refer to <a href="Logical_truth" title="Logical truth">mathematical and logical truths</a>, but also to the symbol systems expressing those truths. This leads to the sort of paradoxes seen in statements such as "<a href="This_statement_is_false" class="mw-redirect" title="This statement is false">This statement is false</a>," wherein the sentence's basis of truth is found in referring to itself and its assertion, causing a logical paradox.<sup id="cite_ref-oreilly_2-0" class="reference"><a href="#cite_note-oreilly-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>Hofstadter argues that the psychological self arises out of a similar kind of paradox. The brain is not born with an "I" – the <a href="Ego_(Freudian)" class="mw-redirect" title="Ego (Freudian)">ego</a> emerges only gradually as experience shapes the brain's dense web of active symbols into a tapestry rich and complex enough to begin <a href="Self-reference" title="Self-reference">twisting back upon itself</a>. According to this view, the psychological "I" is a narrative fiction, something created only from intake of symbolic data and the brain's ability to create stories about itself from that data. The consequence is that a self-perspective is a culmination of a unique pattern of symbolic activity in the brain, which suggests that the pattern of symbolic activity that makes identity, that constitutes subjectivity, can be replicated within the brains of others, and likely even in <a href="Artificial_brain" title="Artificial brain">artificial brains</a>.<sup id="cite_ref-oreilly_2-1" class="reference"><a href="#cite_note-oreilly-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Strangeness">Strangeness</h2></div>

<p>The "strangeness" of a strange loop comes from the brain's perception, because the brain categorizes its input in a small number of "symbols" (by which Hofstadter means groups of neurons standing for something in the outside world). So the difference between the video-feedback loop and the brain's strange loops, is that while the former converts light to the same pattern on a screen, the latter categorizes a pattern and outputs its "essence", so that as the brain gets closer and closer to its "essence", it goes further down its strange loop.<sup id="cite_ref-Hofstadter_3-0" class="reference"><a href="#cite_note-Hofstadter-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Downward_causality">Downward causality</h2></div>

<p>Hofstadter thinks that minds appear to determine the world by way of "downward <a href="Causality" title="Causality">causality</a>", which refers to effects being viewed in terms of their underlying causes. Hofstadter says this happens in the proof of <a href="Kurt_G%C3%B6del" title="Kurt Gödel">Gödel</a>'s <a href="G%C3%B6del's_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorem</a>:

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<blockquote><p>Merely from knowing the formula's meaning, one can infer its truth or falsity without any effort to derive it in the old-fashioned way, which requires one to trudge methodically "upwards" from the axioms. This is not just peculiar; it is astonishing. Normally, one cannot merely look at what a mathematical conjecture <i>says</i> and simply appeal to the content of that statement on its own to deduce whether the statement is true or false. (pp. 169–170)</p></blockquote>

<p>Hofstadter claims a similar "flipping around of causality" appears to happen in minds possessing <a href="Self-consciousness" title="Self-consciousness">self-consciousness</a>; the mind perceives itself as the cause of certain feelings.

</p><p>The parallels between downward causality in formal systems and downward causality in brains are explored by Theodor Nenu in 2022,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> together with other aspects of Hofstadter's metaphysics of mind. Nenu also questions the correctness of the above quote by focusing on the sentence which "says about itself" that it is provable (also known as a Henkin-sentence, named after logician <a href="Leon_Henkin" title="Leon Henkin">Leon Henkin</a>). It turns out that under suitable <a href="Metamathematics" title="Metamathematics">meta-mathematical</a> choices (where the <a href="Hilbert-Bernays_provability_conditions" class="mw-redirect" title="Hilbert-Bernays provability conditions">Hilbert-Bernays provability conditions</a> do not obtain), one can construct formally undecidable (or even formally refutable) Henkin-sentences for the arithmetical system under investigation. This system might very well be Hofstadter's <a href="Typographical_Number_Theory" title="Typographical Number Theory">Typographical Number Theory</a> used in <i>Gödel, Escher, Bach</i> or the more familiar <a href="Peano_Arithmetic" class="mw-redirect" title="Peano Arithmetic">Peano Arithmetic</a> or some other sufficiently rich formal arithmetic. Thus, there are examples of sentences "which say about themselves that they are provable", but they don't exhibit the sort of downward causal powers described in the displayed quote.

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<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<p>Hofstadter points to <a href="Johann_Sebastian_Bach" title="Johann Sebastian Bach">Bach</a>'s <i>Canon per Tonos</i>, <a href="M._C._Escher" title="M. C. Escher">M. C. Escher</a>'s drawings <i><a href="Waterfall_(M._C._Escher)" title="Waterfall (M. C. Escher)">Waterfall</a></i>, <i><a href="Drawing_Hands" title="Drawing Hands">Drawing Hands</a></i>, <i><a href="Ascending_and_Descending" title="Ascending and Descending">Ascending and Descending</a></i>, and the <a href="Liar_paradox" title="Liar paradox">liar paradox</a> as examples that illustrate the idea of strange loops, which is expressed fully in the proof of <a href="G%C3%B6del" class="mw-redirect" title="Gödel">Gödel</a>'s <a href="G%C3%B6del's_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorem</a>.

</p><p>The "<a href="Chicken_or_the_egg" title="Chicken or the egg">chicken or the egg</a>" paradox is perhaps the best-known strange loop problem.

</p><p>The "<a href="Ouroboros" title="Ouroboros">ouroboros</a>", which depicts a dragon eating its own tail, is perhaps one of the most ancient and universal symbolic representations of the reflexive loop concept.

</p><p>A <a href="Shepard_tone" title="Shepard tone">Shepard tone</a> is another illustrative example of a strange loop. Named after <a href="Roger_Shepard" title="Roger Shepard">Roger Shepard</a>, it is a <a href="Sound" title="Sound">sound</a> consisting of a superposition of tones separated by <a href="Octave" title="Octave">octaves</a>. When played with the base <a href="Pitch_(music)" title="Pitch (music)">pitch</a> of the tone moving upwards or downwards, it is referred to as the <i>Shepard scale</i>. This creates the <a href="Auditory_illusion" title="Auditory illusion">auditory illusion</a> 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 <a href="Jean-Claude_Risset" title="Jean-Claude Risset">Jean-Claude Risset</a>.

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<p>Visual illusions depicting strange loops include the <a href="Penrose_stairs" title="Penrose stairs">Penrose stairs</a> and the <a href="Barberpole_illusion" title="Barberpole illusion">Barberpole illusion</a>.

</p><p>A <a href="Quine_(computing)" title="Quine (computing)">quine</a> in software programming is a program that produces a new version of itself without any input from the outside. A similar concept is <a href="Metamorphic_code" title="Metamorphic code">metamorphic code</a>.

</p><p><a href="Intransitive_dice#Efron's_dice" title="Intransitive dice">Efron's dice</a> are four dice that are <a href="Intransitivity" title="Intransitivity">intransitive</a> under gambler's preference. I.e., the dice are ordered <span class="nowrap">A > B > C > D > A</span>, where <span class="nowrap"><i>x</i> > <i>y</i></span> means "a gambler prefers <i>x</i> to <i>y</i>".

</p><p>Individual preferences are always transitive, excluding preferences when given explicit rules such as in Efron's dice or <a href="Rock-paper-scissors" class="mw-redirect" title="Rock-paper-scissors">rock-paper-scissors</a>; however, aggregate preferences of a group may be intransitive. This can result in a <a href="Condorcet_paradox" title="Condorcet paradox">Condorcet paradox</a> 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.

</p><p>The liar paradox and <a href="Russell's_paradox" title="Russell's paradox">Russell's paradox</a> also involve strange loops, as does <a href="Ren%C3%A9_Magritte" title="René Magritte">René Magritte</a>'s painting <i><a href="The_Treachery_of_Images" title="The Treachery of Images">The Treachery of Images</a></i>.

</p><p>The mathematical phenomenon of <a href="Polysemy" title="Polysemy">polysemy</a> 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 <i>mean</i> more than one mathematical object. See Tanenbaum (1999).

</p><p><i><a href="The_Stonecutter" title="The Stonecutter">The Stonecutter</a></i> is an old Japanese <a href="Fairy_tale" title="Fairy tale">fairy tale</a> with a story that explains social and natural hierarchies as a strange loop.

</p><p>A strange loop can be found by traversing the links in the “See also” sections of the respective <a href="English_Wikipedia" title="English Wikipedia">English Wikipedia</a> articles. For instance: This article-><a href="Mise_en_abyme" title="Mise en abyme">Mise en abyme</a>-><a href="Recursion" title="Recursion">Recursion</a>->this article.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

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<ul><li><i><a href="A_Strange_Loop" title="A Strange Loop">A Strange Loop</a></i> – A <a href="Broadway_theatre" title="Broadway theatre">Broadway</a> <a href="Musical_theatre" title="Musical theatre">musical</a> by <a href="Michael_R._Jackson" title="Michael R. Jackson">Michael R. Jackson</a> that takes its title from and references Hofstadter's strange loop</li>

<li><a href="Absurdism" title="Absurdism">Absurdism</a> – Theory that life in general is meaningless</li>

<li><a href="Autopoiesis" title="Autopoiesis">Autopoiesis</a> – System capable of producing itself</li>

<li><a href="Catch-22_(logic)" title="Catch-22 (logic)">Catch-22</a> – Situation in which one cannot avoid a problem because of contradictory constraints</li>

<li><a href="Causal_loop" class="mw-redirect" title="Causal loop">Causal loop</a> – Theoretical paradox resulting from time travel<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li>

<li><a href="Dilemma" title="Dilemma">Dilemma</a> – Problem requiring a choice between equally undesirable alternatives</li>

<li><a href="Euthyphro_dilemma" title="Euthyphro dilemma">Euthyphro dilemma</a> – Ethical problem on the origin of morality posed by Socrates</li>

<li><a href="Grandfather_paradox" class="mw-redirect" title="Grandfather paradox">Grandfather paradox</a> – Theoretical paradox resulting from time travel<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span> – Going back in time to kill one's own grandfather generates a circular contradiction</li>

<li><a href="Hysteron_proteron" title="Hysteron proteron">Hysteron proteron</a> – Figure of speech reversing a natural or rational order</li>

<li><a href="Irony" title="Irony">Irony</a> – Literary and rhetorical device or general attitude towards life</li>

<li><a href="Klein_bottle" title="Klein bottle">Klein bottle</a> – Non-orientable mathematical surface</li>

<li><a href="Mise_en_abyme" title="Mise en abyme">Mise en abyme</a> – Technique of placing a copy of an image within itself, or a story within a story</li>

<li><a href="Meno" title="Meno">Meno</a> – Dialogue by Plato – Paradox: One must already possess any given piece of knowledge, otherwise it could not be recognized when supposedly "discovered"</li>

<li><a href="Metamorphic_code" title="Metamorphic code">Metamorphic code</a> – Type of code used by computer viruses</li>

<li><a href="M%C3%B6bius_strip" title="Möbius strip">Möbius strip</a> – Non-orientable surface with one edge</li>

<li><a href="M%C3%BCnchhausen_trilemma" title="Münchhausen trilemma">Münchhausen trilemma</a> – Thought experiment used to demonstrate the impossibility of proving any truth</li>

<li><a href="Ontological_paradox" class="mw-redirect" title="Ontological paradox">Ontological paradox</a> – Theoretical paradox resulting from time travel<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li>

<li><a href="Optical_feedback" title="Optical feedback">Optical feedback</a></li>

<li><a href="Ouroboros" title="Ouroboros">Ouroboros</a> – Symbolic serpent with its tail in its mouth</li>

<li><a href="Penrose_stairs" title="Penrose stairs">Penrose stairs</a> – Impossible object</li>

<li><a href="Perpetual_motion" title="Perpetual motion">Perpetual motion</a> – Work being continuously done without an external input of energy</li>

<li><a href="Phoenix_(mythology)" title="Phoenix (mythology)">Phoenix (mythology)</a> – Immortal bird that is cyclically reborn</li>

<li><a href="Pitch_circularity" title="Pitch circularity">Pitch circularity</a> – Fixed series of tones that appear to ascend or descend endlessly in pitch</li>

<li><a href="Polytely" title="Polytely">Polytely</a> – Problem-solving technique</li>

<li><a href="Predestination_paradox" class="mw-redirect" title="Predestination paradox">Predestination paradox</a> – Theoretical paradox resulting from time travel<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li>

<li><a href="Reflexivity_(social_theory)" title="Reflexivity (social theory)">Reflexivity (social theory)</a> – Circular relationships between cause and effect</li>

<li><a href="Rock_paper_scissors" title="Rock paper scissors">Rock paper scissors</a> – Hand game for two players or more</li>

<li><a href="Shepard_tone" title="Shepard tone">Shepard tone</a> – Auditory illusion</li>

<li><a href="Three_hares" title="Three hares">Three hares</a> – Motif of three hares in threefold rotational symmetry</li>

<li><a href="Tupper's_self-referential_formula" title="Tupper's self-referential formula">Tupper's self-referential formula</a> – Formula that visually represents itself when graphed</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3></div>

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</style><cite id="CITEREFJohnson2007" class="citation journal cs1">Johnson, George (March 2007). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="http://www.scientificamerican.com/article.cfm?id=a-new-journey-into-hofsta">"A New Journey into Hofstadter's Mind"</a></span>. <i>Scientific American</i>. <b>296</b> (3): <span class="nowrap">98–</span>102. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007SciAm.296c..98J">2007SciAm.296c..98J</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fscientificamerican0307-98">10.1038/scientificamerican0307-98</a><span class="reference-accessdate">. Retrieved <span class="nowrap">8 October</span> 2011</span>.</cite></span>

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<li id="cite_note-oreilly-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-oreilly_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-oreilly_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFO'Reilly2010" class="citation journal cs1">O'Reilly, Scott (2010). <a rel="nofollow" class="external text" href="http://www.philosophynow.org/issue78/I_Am_A_Strange_Loop_by_Douglas_Hofstadter">"I Am A Strange Loop by Douglas Hofstadter"</a>. <i>Philosophy Now</i><span class="reference-accessdate">. Retrieved <span class="nowrap">8 October</span> 2011</span>.</cite></span>

</li>

<li id="cite_note-Hofstadter-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hofstadter_3-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFHofstadter2007" class="citation book cs1">Hofstadter, Douglas (2007). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/iamstrangeloop00hofs"><i>I Am A Strange Loop</i></a></span>. Basic Books. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-465-03078-1</bdi>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFNenu2022" class="citation journal cs1">Nenu, Theodor (2022). <a rel="nofollow" class="external text" href="https://philpapers.org/rec/NENDHG">"Douglas Hofstadter's Gödelian Philosophy of Mind"</a>. <i>Journal of Artificial Intelligence and Consciousness</i>. <b>9</b> (2): <span class="nowrap">241–</span>266. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS2705078522500011">10.1142/S2705078522500011</a>.</cite></span>

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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Wikipedia contributors. (2024, December 14). Strange loop. In <i>Wikipedia, The Free Encyclopedia</i>. Retrieved 10:34, December 25, 2024, from <a class="external free external" href="https://en.wikipedia.org/w/index.php?title=Strange_loop&oldid=1263113776">https://en.wikipedia.org/w/index.php?title=Strange_loop&oldid=1263113776</a></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3></div>

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</style><div class="refbegin" style="">

<ul><li><cite id="CITEREFTanenbaum1999" class="citation journal cs1">Tanenbaum, P. J. (October 1999). "Simultaneous intersection representation of pairs of graphs". <i><a href="Journal_of_Graph_Theory" title="Journal of Graph Theory">Journal of Graph Theory</a></i>. <b>32</b> (2): <span class="nowrap">171–</span>190. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F%28SICI%291097-0118%28199910%2932%3A2%3C171%3A%3AAID-JGT7%3E3.0.CO%3B2-N">10.1002/(SICI)1097-0118(199910)32:2<171::AID-JGT7>3.0.CO;2-N</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1097-0118">1097-0118</a>.</cite></li>

<li><cite id="CITEREFNenu2022" class="citation journal cs1">Nenu, T. (September 2022). <a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS2705078522500011">"Douglas Hofstadter's Gödelian Philosophy of Mind"</a>. <i>Journal of Artificial Intelligence and Consciousness</i>. <b>9</b> (2): <span class="nowrap">241–</span>266. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS2705078522500011">10.1142/S2705078522500011</a></span>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/1983%2F07af725f-5af7-44f5-9d9f-df197218c741">1983/07af725f-5af7-44f5-9d9f-df197218c741</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2705-0793">2705-0793</a>.</cite></li></ul>

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</style><div id="Douglas_Hofstadter340" style="font-size:114%;margin:0 4em"><a href="Douglas_Hofstadter" title="Douglas Hofstadter">Douglas Hofstadter</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Books</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><i><a href="G%C3%B6del%2C_Escher%2C_Bach" title="Gödel, Escher, Bach">Gödel, Escher, Bach</a></i> <span style="font-size: 85%;">(1979)</span></li>

<li><i><a href="Metamagical_Themas" title="Metamagical Themas">Metamagical Themas</a></i> <span style="font-size: 85%;">(1985)</span></li>

<li><i><a href="Fluid_Concepts_and_Creative_Analogies" title="Fluid Concepts and Creative Analogies">Fluid Concepts and Creative Analogies</a></i> <span style="font-size: 85%;">(1995)<sup><small>2</small></sup></span></li>

<li><i><a href="Le_Ton_beau_de_Marot" title="Le Ton beau de Marot">Le Ton beau de Marot</a></i> <span style="font-size: 85%;">(1997)</span></li>

<li><i><a href="I_Am_a_Strange_Loop" title="I Am a Strange Loop">I Am a Strange Loop</a></i> <span style="font-size: 85%;">(2007)</span></li>

<li><i>Surfaces and Essences</i> <span style="font-size: 85%;">(2013)</span></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="3" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Concepts and<br>projects</div></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Ambigram" title="Ambigram">Ambigram</a></li>

<li><a href="BlooP_and_FlooP" title="BlooP and FlooP">BlooP and FlooP</a></li>

<li><a href="Copycat_(software)" title="Copycat (software)">Copycat</a></li>

<li><a href="Hofstadter's_butterfly" title="Hofstadter's butterfly">Hofstadter's butterfly</a></li>

<li><a href="Hofstadter's_law" title="Hofstadter's law">Hofstadter's law</a></li>

<li><a href="Hofstadter_points" title="Hofstadter points">Hofstadter points</a></li>

<li><a href="MU_puzzle" title="MU puzzle">MU puzzle</a></li>

<li><a href="Platonia_dilemma" title="Platonia dilemma">Platonia dilemma</a></li>

<li><a href="Six_nines_in_pi" title="Six nines in pi">Six nines in pi</a></li>

<li><a href="Superrationality" title="Superrationality">Superrationality</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Robert_Hofstadter" title="Robert Hofstadter">Robert Hofstadter</a> (father)</li>

<li><a href="Egbert_B._Gebstadter" title="Egbert B. Gebstadter">Egbert B. Gebstadter</a></li>

<li><a href="Indiana_University_Bloomington" title="Indiana University Bloomington">Indiana University Bloomington</a></li>

<li><i><a href="Victim_of_the_Brain" title="Victim of the Brain">Victim of the Brain</a></i></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="3"><div>

<ul><li><sup><small>1</small></sup> Edited by Hofstadter and <a href="Daniel_C._Dennett" class="mw-redirect" title="Daniel C. Dennett">Daniel C. Dennett</a></li>

<li><sup><small>2</small></sup> By Hofstadter and the Fluid Analogies Research Group</li></ul>

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