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| {{about|figure-eight shaped curves in algebraic geometry|other uses|Lemniscate (disambiguation)}}
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| In [[algebraic geometry]], a '''lemniscate''' may refer to any of several figure-eight or <big>[[∞]]</big>-shaped [[curve]]s.<ref name="lemniscatomy"/><ref name="erickson"/> The word comes from the [[Latin language|Latin]] "lēmniscātus" meaning "decorated with ribbons", which in turn may come from the ancient Greek island of [[Lemnos]] where ribbons were worn as decorations,<ref name="erickson">{{citation|title=Beautiful Mathematics|series=MAA Spectrum|publisher=[[Mathematical Association of America]]|first=Martin J.|last=Erickson|year=2011|isbn=9780883855768|contribution=1.1 Lemniscate|pages=1–3|url=http://books.google.com/books?id=LgeP62-ZxikC&pg=PA1}}.</ref> or alternatively may refer to the [[wool]] from which the ribbons were made.<ref name="lemniscatomy"/>
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| Although the name "lemniscate" dates to the late 17th century, the consideration of curves with a figure eight shape can be traced back to [[Proclus]], a Greek [[Neoplatonist]] philosopher and mathematician who lived in the 5th century AD. Proclus considered the cross-sections of a [[torus]] by a plane parallel to the axis of the torus. As he observed, for most such sections the cross section consists of either one or two ovals; however, when the plane is [[tangent]] to the inner surface of the torus, the cross-section takes on a figure-eight shape, which Proclus called a [[horse]] [[Legcuffs|fetter]] (a device for holding two feet of a horse together). The Greek phrase for a horse fetter became the word [[hippopede]], the name for this figure-eight shaped curve, which is also called the lemniscate of Booth. It may be defined algebraically as the zero set of the [[quartic polynomial]] {{math|(''x''<sup>2</sup> + ''y''<sup>2</sup>)<sup>2</sup> − ''cx''<sup>2</sup> − ''dy''<sup>2</sup>}} when the parameter ''d'' is negative. For positive values of ''d'' one instead obtains an oval-shaped curve, the [[oval of Booth]].<ref name="lemniscatomy">{{citation
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| | last = Schappacher | first = Norbert
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| | contribution = Some milestones of lemniscatomy
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| | location = New York
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| | mr = 1483331
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| | pages = 257–290
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| | publisher = Dekker
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| | series = Lecture Notes in Pure and Applied Mathematics
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| | title = Algebraic Geometry (Ankara, 1995)
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| | volume = 193
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| | year = 1997}}.</ref>
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| In 1694, [[Johann Bernoulli]] studied the zero set of the polynomial equation <math>(x^2 + y^2)^2 = 2a^2 (x^2 - y^2)</math> in connection with a problem of "isochrones" that had been posed earlier by [[Leibniz]]. Bernoulli's brother [[Jacob Bernoulli]] also studied the same curve in the same year, and gave it its name, the lemniscate.<ref name="bos">{{citation
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| | last = Bos | first = H. J. M.
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| | contribution = The lemniscate of Bernoulli
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| | location = Dordrecht
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| | mr = 774250
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| | pages = 3–14
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| | publisher = Reidel
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| | series = Boston Stud. Philos. Sci., XV
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| | title = For Dirk Struik
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| | url = http://books.google.com/books?id=OvK9orJNezwC&pg=PA3
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| | year = 1974}}.</ref> It may also be defined geometrically as the locus of points whose product of distances from two foci equals the square of half the interfocal distance.<ref>{{citation
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| | last1 = Langer | first1 = Joel C.
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| | last2 = Singer | first2 = David A.
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| | doi = 10.1007/s00032-010-0124-5
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| | issue = 2
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| | journal = Milan Journal of Mathematics
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| | mr = 2781856
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| | pages = 643–682
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| | title = Reflections on the lemniscate of Bernoulli: the forty-eight faces of a mathematical gem
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| | volume = 78
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| | year = 2010}}.</ref> It is a special case of the hippopede, with <math>d=-c</math>, and may be formed as a cross-section of a torus whose inner hole and circular cross-sections have the same diameter as each other.<ref name="lemniscatomy"/> The [[lemniscatic elliptic function]]s are analogues of trigonometric functions for the lemniscate of Bernoulli, and the [[Gauss's constant|lemniscate constants]] arise in evaluating the [[arc length]] of this lemniscate.
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| Another lemniscate, the [[lemniscate of Gerono]] or lemniscate of Huygens, is the zero set of the quartic polynomial equation <math>y^2=x^2(a^2-x^2)</math>.<ref>{{citation|title=An elementary treatise on cubic and quartic curves|first=Alfred Barnard|last=Basset|publisher=Deighton, Bell|year=1901|pages=171–172|url=http://books.google.com/books?id=T40LAAAAYAAJ&pg=PA171|contribution=The Lemniscate of Gerono}}.</ref><ref>{{citation|title=Newton's Principia for the common reader|first=S|last=Chandrasekhar|publisher=Oxford University Press|year=2003|isbn=9780198526759|page=133|url=http://books.google.com/books?id=qomP58txKQwC&pg=PA133}}.</ref> [[Viviani's curve]], a three-dimensional curve formed by intersecting a sphere with a cylinder, also has a figure eight shape, and has the lemniscate of Gerono as its planar projection.<ref>{{citation|first1=Luisa Rossi|last1=Costa|first2=Elena|last2=Marchetti|contribution=Mathematical and Historical Investigation on Domes and Vaults|pages=73–80|title=Aesthetics and architectural composition : proceedings of the Dresden International Symposium of Architecture 2004|year=2005|location=Mammendorf|publisher=Pro Literatur|editor1-last=Weber|editor1-first=Ralf|editor2-last=Amann|editor2-first=Matthias Albrecht}}.</ref>
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| Other figure-eight shaped algebraic curves include
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| * The [[Devil's curve]], a curve defined by the quartic equation <math>y^2 (y^2 - a^2) = x^2 (x^2 - b^2)</math> in which one connected component has a figure-eight shape,<ref>{{citation|title=The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes|first=David|last=Darling|publisher=John Wiley & Sons|year=2004|isbn=9780471667001|contribution=devil's curve|pages=91–92|url=http://books.google.com/books?id=HrOxRdtYYaMC&pg=PA91}}.</ref>
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| * [[Watt's curve]], a figure-eight shaped curve formed by a mechanical linkage. Watt's curve is the zero set of the degree-six polynomial equation <math>(x^2+y^2)(x^2+y^2-d^2)^2+4a^2y^2(x^2+y^2-b^2)=0</math> and has the lemniscate of Bernoulli as a special case.
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| ==See also==
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| * [[Analemma]], the figure-eight shaped curve traced by the noontime positions of the sun in the sky over the course of a year
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| * [[Lorenz attractor]], a three-dimensional dynamic system exhibiting a lemniscate shape
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| * [[Polynomial lemniscate]], a level set of the absolute value of a complex polynomial
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| ==References==
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| {{Reflist|30em}}
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| ==External links==
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| * {{springer|title=Lemniscates|id=p/l058130}}
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| [[Category:Mathematical terminology]]
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| [[Category:Curves]]
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