automorphicadj.
/ˌɔːtə(ʊ)ˈmɔːfɪk/, U.S. /ˌɔdəˈmɔrfɪk/, /ˌɔdoʊˈmɔrfɪk/, /ˌɑdəˈmɔrfɪk/, /ˌɑdoʊˈmɔrfɪk/ 1. Mathematics. Of a function: invariant with respect to a group of linear transformations of a certain kind. Of a transformation: †leaving a certain function invariant (obsolete).In modern use, if a function f(x) is automorphic with respect to a group G, then f(Tn(x)) = f(x) for every element Tn of G. This use originated with F. Klein (see etymology).
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1859 A. Cayley in Proc. Royal Soc. 9 101 The question of the automorphic linear transformation of the function x2 + y2 + z2, that is the transformation by linear substitutions, of this function into a function x′2 + y′2 + z′2 of the same form, is in effect solved by some formulæ of Euler's for the transformation of coordinates.
1892 W. Burnside in Proc. London Math. Soc. 23 52 I have used the phrase ‘automorphic function’, as introduced by Professor Klein, to denote generally any function which is unchanged by the substitutions of a discontinuous group, whatever be the nature of the group.
1922 Bull. Amer. Math. Soc. 28 360 On returning to the corresponding series in x, we should then have automorphic functions, homogeneous of degree 0 in x0, x1, x2, which converge.
1972 M. Kline Math. Thought xxix. 726 Automorphic functions are generalizations of the circular, hyperbolic, elliptic, and other functions of elementary analysis.
2006 A. Ash & R. Gross Fearless Symmetry xxi. 240 An automorphic representation is indeed a kind of group representation, but with a totally different source and target from a Galois representation.
2. Characterized by automorphism (automorphism n. 2); formed on oneself.
ΘΚΠ
the mind > mental capacity > consciousness > subjectivity, relation to self > ascription of self to other > [adjective]
automorphic1872
1872 H. Spencer in Contemp. Rev. Dec. 2 The conception which anyone frames of another's mind, is inevitably more or less after the pattern of his own mind—is automorphic.
1930 Jrnl. Amer. Inst. Criminal Law & Criminol. 21 273 It is this tendency to automorphic interpretation..which sometimes makes futile the most laborious and sincere attempt to be objective and fair.
1975 New Literary Hist. 7 45 Unfortunately, ideas as to who the best readers are tend to be controversial and automorphic.
2009 A. M. Colman Dict. Psychol. (ed. 3) 71/1 Automorphic perception [is]..a tendency to perceive others as physically more similar to oneself than they actually are... Amedeo Modigliani..depicted..most of his subjects with elongated features resembling his own.
3. Crystallography. = idiomorphic adj. rare.
ΘΚΠ
the world > the earth > minerals > mineral structure or appearance > [adjective] > crystalline > specific crystal
idiomorphic1887
automorphic1888
hypidiomorphic1888
euhedral1906
idioblastic1908
peloidal1933
the world > matter > chemistry > crystallography (general) > crystal (general) > structures and forms > [adjective] > idiomorphic
idiomorphic1887
automorphic1888
euhedral1906
1888 F. H. Hatch in J. J. H. Teall Brit. Petrogr. 435 The mineral constituents of rocks are said to be idiomorphic in so far as they are bounded by faces peculiar to the species. The word is synonymous with Rohrbach's ‘automorphic’.
1906 J. P. Iddings Rock Minerals ii. 64 Euhedral, well faced, completely bounded by crystal planes, automorphic, idiomorphic.
2009 A. E. Glikin Polymineral-Metasomatic Crystallogenesis i. 44 Both pseudomorphic and automorphic replacements of the solid part of the crystals are accompanied by the same type of inclusion replacement.
4. Mathematics. Designating an n-digit number which when raised to a power gives a number whose last n digits are the same as the original number. Cf. circular number n. at circular adj. and n. Compounds 1.E.g. 5 is automorphic because 52 is 25, 53 is 125, 54 is 625, etc.; 76 is automorphic because 762 is 5776, 763 is 438,976, etc.
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1939 W. W. R. Ball & H. S. M. Coxeter Math. Recreations & Ess. (ed. 11) ii. 60 In the scale of six, the last seven digits of an automorphic number must be either 1350213 or 4205344.
1959 Math. Gaz. 43 270 Numbers ending in 0 or 1 are of course automorphic.
1995 Daily Tel. 1 Nov. 20/5 Last week's bonus number 5, though, is a treasure house of useless facts. It is the smallest automorphic number (when multiplied by itself any number of times, the answer always ends in 5; the number 6 is similar).
Derivatives
ˌautoˈmorphically adv. in an automorphic way; (Mathematics) by an automorphic function; as regards automorphism.
ΘΚΠ
the mind > mental capacity > consciousness > subjectivity, relation to self > ascription of self to other > [adverb]
automorphically1872
1872 H. Spencer in Contemp. Rev. Dec. 3 He interprets them automorphically—carrying with him not simply his own faculties..but also the modes of thinking in which he was brought up.
1896 Proc. Amer. Acad. Arts & Sci. 1895–6 31 181 An exactly similar theory exists for the group of linear substitutions which transform automorphically an alternate bilinear form with cogredient variables.
1971 D. Bures Abelian Subalgebras of von Neumann Algebras xv. 118 Hence N ∩H = H and ℰ is automorphically normal.
1992 Social Psychol. Q. 55 302/1 Actors 1 and 5 are automorphically equivalent because both are chosen by one other actor.
This entry has been updated (OED Third Edition, June 2011; most recently modified version published online December 2021).
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adj.1859