Möbiusn.

Brit.

/ˈməːbɪəs/,

/ˈməʊbɪəs/, U.S.

/ˈmoʊbiəs/

Forms: 1900s– Möbius, 1900s– Mobius, 1900s– Moebius. Also with lower-case initial.

Origin: From a proper name. Etymon: proper name Möbius.

Etymology: < the name of August Ferdinand Möbius (1790–1868), German mathematician, who described the one-sided surface in Berichte über die Verhandl. der Königl. Sächsischen Ges. der Wissensch. zu Leipzig (1865) 17 41.The surface was also discovered independently by J. B. Listing: see quot. 1993 for Möbius band n. at sense 1c.

Chiefly Mathematics.

a. Möbius surface n. = Möbius strip n. at sense 1b.

ΘΚΠ

the world > relative properties > number > geometry > surface > [noun] > generated by rotation or twisting

pseudosphere1884

Möbius surface1900

Möbius strip1904

Möbius band1932

Klein bottle1941

1900 Trans. Amer. Math. Soc. 1 39 T will cut out of R a unilateral Moebius surface.

1939 Amer. Jrnl. Math. 61 552 The resulting manifold represents a Möbius surface bounded by C.

1984 Amer. Math. Monthly 91 125 It is seen that all these double points of this Moebius surface fall on a straight line.

b. Möbius strip n. (also †Möbius' strip) a surface having only one side and one edge, formed by twisting one end of a rectangular strip through 180 degrees and joining it to the other end. Also in extended use.

ΘΚΠ

the world > relative properties > number > geometry > surface > [noun] > generated by rotation or twisting

pseudosphere1884

Möbius surface1900

Möbius strip1904

Möbius band1932

Klein bottle1941

1900 Trans. Amer. Math. Soc. 1 39 An algebraic surface containing as a part the unilateral paper-strip of Moebius.]

1904 E. R. Hedrick tr. E. Goursat Course in Math. Anal. I. 546 Möbius' strip.

1941 R. Courant & H. E. Robbins What is Math.? v. 260 If the Moebius strip is cut along this [center] line..we find that it remains in one piece.

1960 F. Land Lang. Math. xi. 171 We have in the Moebius strip the strange object which consists of one continuous surface bounded by one continuous curve.

1965 H. Eves Surv. Geom. II. xv. 357 Show that a Klein bottle can, by one cut, be converted into a disc and a Möbius strip.

1970 New Yorker 28 Nov. 58/3 Barbara, taking a pair of scissors..turned the clipping on fragrant Christmas gifts into a Möbius strip.

1991 D. Rieff Los Angeles i. ii. 49 There is the nagging feeling that to enter the freeway is to move onto some enormous Möbius Strip, a series of contours that turn back on themselves.

c. Möbius band n. = Möbius strip n. at sense 1b.

ΘΚΠ

the world > relative properties > number > geometry > surface > [noun] > generated by rotation or twisting

pseudosphere1884

Möbius surface1900

Möbius strip1904

Möbius band1932

Klein bottle1941

1932 Ann. Math. 33 683 A locally Euclidean Möbius band.

1950 Astounding Sci. Fiction Dec. 75/2 ‘The Möbius band’, Turpelo said, ‘has unusual properties because it has a singularity. The Klein bottle, with two singularities, manages to be inside of itself.’

1974 Times 10 Sept. 14/8 If the curtain at Covent Garden three weeks hence goes up on yet another ring, circle, hoop, saucer, dish or Möbius band I shall be heard to blow my nose in an ominous falling figure..which..betokens Doom.

1993 Nature 10 June 506/1 The catchy title Möbius and his Band is not only misleading but also wrong, because the essay on topology by Norman Biggs shows that the Möbius band is not really his, having been described a little earlier by J. B. Listing, a pioneer of topology and pupil of Gauss.

2. Möbius transformation n. a conformal transformation which maps a complex number z into ^{az + b}/ _{cz + d}, where ad ≠ bc.

ΘΚΠ

the world > relative properties > number > arithmetic or algebraic operations > transformation > [noun] > correspondence > preserving relations or elements

inclusion1870

orthomorphosis1885

isomorphism1892

identity1910

homoeomorphism1918

homomorphism1935

topological mapping or transformation1939

isometry1941

Möbius transformation1941

injection map(ping)1950

monomorphism1954

bijection1963

surjection1964

1941 Trans. Amer. Math. Soc. 49 380 In minimal coordinates, any direct Moebius transformation may be written as U = ^{au + b}/ _{cu + d}, V = ^{a′v + b′}/ _{c′u + d′}.

1974 Ann. Math. 99 444 Hyperbolic distance is preserved under Möbius transformations.

1990 Glasgow Math. Jrnl. 32 317 Let Γ denote the modular group, consisting of the Möbius transformations z → ^{az + b}/ _{cz + d}..ad − bc = 1.

This entry has been updated (OED Third Edition, September 2002; most recently modified version published online March 2022).

单词	möbius
释义	Möbiusn. Brit. /ˈməːbɪəs/, /ˈməʊbɪəs/, U.S. /ˈmoʊbiəs/ Forms: 1900s– Möbius, 1900s– Mobius, 1900s– Moebius. Also with lower-case initial. Origin: From a proper name. Etymon: proper name Möbius. Etymology: < the name of August Ferdinand Möbius (1790–1868), German mathematician, who described the one-sided surface in Berichte über die Verhandl. der Königl. Sächsischen Ges. der Wissensch. zu Leipzig (1865) 17 41.The surface was also discovered independently by J. B. Listing: see quot. 1993 for Möbius band n. at sense 1c. Chiefly Mathematics. 1. a. Möbius surface n. = Möbius strip n. at sense 1b. ΘΚΠ the world > relative properties > number > geometry > surface > [noun] > generated by rotation or twisting pseudosphere1884 Möbius surface1900 Möbius strip1904 Möbius band1932 Klein bottle1941 1900 Trans. Amer. Math. Soc. 1 39 T will cut out of R a unilateral Moebius surface. 1939 Amer. Jrnl. Math. 61 552 The resulting manifold represents a Möbius surface bounded by C. 1984 Amer. Math. Monthly 91 125 It is seen that all these double points of this Moebius surface fall on a straight line. b. Möbius strip n. (also †Möbius' strip) a surface having only one side and one edge, formed by twisting one end of a rectangular strip through 180 degrees and joining it to the other end. Also in extended use. ΘΚΠ the world > relative properties > number > geometry > surface > [noun] > generated by rotation or twisting pseudosphere1884 Möbius surface1900 Möbius strip1904 Möbius band1932 Klein bottle1941 1900 Trans. Amer. Math. Soc. 1 39 An algebraic surface containing as a part the unilateral paper-strip of Moebius.] 1904 E. R. Hedrick tr. E. Goursat Course in Math. Anal. I. 546 Möbius' strip. 1941 R. Courant & H. E. Robbins What is Math.? v. 260 If the Moebius strip is cut along this [center] line..we find that it remains in one piece. 1960 F. Land Lang. Math. xi. 171 We have in the Moebius strip the strange object which consists of one continuous surface bounded by one continuous curve. 1965 H. Eves Surv. Geom. II. xv. 357 Show that a Klein bottle can, by one cut, be converted into a disc and a Möbius strip. 1970 New Yorker 28 Nov. 58/3 Barbara, taking a pair of scissors..turned the clipping on fragrant Christmas gifts into a Möbius strip. 1991 D. Rieff Los Angeles i. ii. 49 There is the nagging feeling that to enter the freeway is to move onto some enormous Möbius Strip, a series of contours that turn back on themselves. c. Möbius band n. = Möbius strip n. at sense 1b. ΘΚΠ the world > relative properties > number > geometry > surface > [noun] > generated by rotation or twisting pseudosphere1884 Möbius surface1900 Möbius strip1904 Möbius band1932 Klein bottle1941 1932 Ann. Math. 33 683 A locally Euclidean Möbius band. 1950 Astounding Sci. Fiction Dec. 75/2 ‘The Möbius band’, Turpelo said, ‘has unusual properties because it has a singularity. The Klein bottle, with two singularities, manages to be inside of itself.’ 1974 Times 10 Sept. 14/8 If the curtain at Covent Garden three weeks hence goes up on yet another ring, circle, hoop, saucer, dish or Möbius band I shall be heard to blow my nose in an ominous falling figure..which..betokens Doom. 1993 Nature 10 June 506/1 The catchy title Möbius and his Band is not only misleading but also wrong, because the essay on topology by Norman Biggs shows that the Möbius band is not really his, having been described a little earlier by J. B. Listing, a pioneer of topology and pupil of Gauss. 2. Möbius transformation n. a conformal transformation which maps a complex number z into ^{az + b}/ _{cz + d}, where ad ≠ bc. ΘΚΠ the world > relative properties > number > arithmetic or algebraic operations > transformation > [noun] > correspondence > preserving relations or elements inclusion1870 orthomorphosis1885 isomorphism1892 identity1910 homoeomorphism1918 homomorphism1935 topological mapping or transformation1939 isometry1941 Möbius transformation1941 injection map(ping)1950 monomorphism1954 bijection1963 surjection1964 1941 Trans. Amer. Math. Soc. 49 380 In minimal coordinates, any direct Moebius transformation may be written as U = ^{au + b}/ _{cu + d}, V = ^{a′v + b′}/ _{c′u + d′}. 1974 Ann. Math. 99 444 Hyperbolic distance is preserved under Möbius transformations. 1990 Glasgow Math. Jrnl. 32 317 Let Γ denote the modular group, consisting of the Möbius transformations z → ^{az + b}/ _{cz + d}..ad − bc = 1. This entry has been updated (OED Third Edition, September 2002; most recently modified version published online March 2022). < n.1900
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