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In algebraic geometry, a lemniscate (/lɛmˈnɪskɪt/ or /ˈlɛmnɪsˌkeɪt, -kɪt/)^[1] is any of several figure-eight or $\infty$ -shaped curves.^[2]^[3] The word comes from the Latin lēmniscātus, meaning "decorated with ribbons",^[4] from the Greek λημνίσκος (lēmnískos), meaning "ribbon",^[3]^[5]^[6]^[7] or which alternatively may refer to the wool from which the ribbons were made.^[2]

Curves that have been called a lemniscate include three quartic plane curves: the hippopede or lemniscate of Booth, the lemniscate of Bernoulli, and the lemniscate of Gerono. The hippopede was studied by Proclus (5th century), but the term "lemniscate" was not used until the work of Jacob Bernoulli in the late 17th century.

History and examples

Lemniscate of Booth

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 fetter (a device for holding two feet of a horse together), or "hippopede" in Greek.^[8] The name "lemniscate of Booth" for this curve dates to its study by the 19th-century mathematician James Booth.^[2]

The lemniscate may be defined as an algebraic curve, the zero set of the quartic polynomial $(x^{2}+y^{2})^{2}-cx^{2}-dy^{2}$ when the parameter d is negative (or zero for the special case where the lemniscate becomes a pair of externally tangent circles). For positive values of d one instead obtains the oval of Booth.

Lemniscate of Bernoulli

In 1680, Cassini studied a family of curves, now called the Cassini oval, defined as follows: the locus of all points, the product of whose distances from two fixed points, the curves' foci, is a constant. Under very particular circumstances (when the half-distance between the points is equal to the square root of the constant) this gives rise to a lemniscate.

In 1694, Johann Bernoulli studied the lemniscate case of the Cassini oval, now known as the lemniscate of Bernoulli (shown above), in connection with a problem of "isochrones" that had been posed earlier by Leibniz. Like the hippopede, it is an algebraic curve, the zero set of the polynomial $(x^{2}+y^{2})^{2}-a^{2}(x^{2}-y^{2})$ . Bernoulli's brother Jacob Bernoulli also studied the same curve in the same year, and gave it its name, the lemniscate.^[9] 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.^[10] It is a special case of the hippopede (lemniscate of Booth), with $d=-c$ , 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.^[2] The lemniscatic elliptic functions are analogues of trigonometric functions for the lemniscate of Bernoulli, and the lemniscate constants arise in evaluating the arc length of this lemniscate.

Lemniscate of Gerono

Another lemniscate, the lemniscate of Gerono or lemniscate of Huygens, is the zero set of the quartic polynomial $y^{2}-x^{2}(a^{2}-x^{2})$ .^[12]^[13] 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.^[14]

Others

Other figure-eight shaped algebraic curves include

The Devil's curve, a curve defined by the quartic equation $y^{2}(y^{2}-a^{2})=x^{2}(x^{2}-b^{2})$ in which one connected component has a figure-eight shape,^[15]
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 $(x^{2}+y^{2})(x^{2}+y^{2}-d^{2})^{2}+4a^{2}y^{2}(x^{2}+y^{2}-b^{2})=0$ and has the lemniscate of Bernoulli as a special case.

References

^ "lemniscate". Dictionary.com Unabridged (Online). n.d.
^ ^a ^b ^c ^d Schappacher, Norbert (1997), "Some milestones of lemniscatomy", Algebraic Geometry (Ankara, 1995), Lecture Notes in Pure and Applied Mathematics, vol. 193, New York: Dekker, pp. 257–290, MR 1483331.
^ ^a ^b Erickson, Martin J. (2011), "1.1 Lemniscate", Beautiful Mathematics, MAA Spectrum, Mathematical Association of America, pp. 1–3, ISBN 9780883855768.
^ lemniscatus. Charlton T. Lewis and Charles Short. A Latin Dictionary on Perseus Project.
^ Harper, Douglas. "lemniscus". Online Etymology Dictionary.
^ lemniscus. Charlton T. Lewis and Charles Short. A Latin Dictionary on Perseus Project.
^ λημνίσκος. Liddell, Henry George; Scott, Robert; A Greek–English Lexicon at the Perseus Project.
^ ἱπποπέδη in Liddell and Scott.
^ Bos, H. J. M. (1974), "The lemniscate of Bernoulli", For Dirk Struik, Boston Stud. Philos. Sci., XV, Dordrecht: Reidel, pp. 3–14, ISBN 9789027703934, MR 0774250.
^ Langer, Joel C.; Singer, David A. (2010), "Reflections on the lemniscate of Bernoulli: the forty-eight faces of a mathematical gem", Milan Journal of Mathematics, 78 (2): 643–682, doi:10.1007/s00032-010-0124-5, MR 2781856, S2CID 1448521.
^ Köller, Jürgen. "Acht-Kurve". www.mathematische-basteleien.de. Retrieved 2017-11-26.
^ Basset, Alfred Barnard (1901), "The Lemniscate of Gerono", An elementary treatise on cubic and quartic curves, Deighton, Bell, pp. 171–172.
^ Chandrasekhar, S (2003), Newton's Principia for the common reader, Oxford University Press, p. 133, ISBN 9780198526759.
^ Costa, Luisa Rossi; Marchetti, Elena (2005), "Mathematical and Historical Investigation on Domes and Vaults", in Weber, Ralf; Amann, Matthias Albrecht (eds.), Aesthetics and architectural composition : proceedings of the Dresden International Symposium of Architecture 2004, Mammendorf: Pro Literatur, pp. 73–80.
^ Darling, David (2004), "devil's curve", The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes, John Wiley & Sons, pp. 91–92, ISBN 9780471667001.