# Lemniscate

In algebraic geometry, a lemniscate is any of several figure-eight or -shaped curves.[1][2] The word comes from the Latin "lēmniscātus" meaning "decorated with ribbons", which in turn may come from the ancient Greek island of Lemnos where ribbons were worn as decorations,[2] or alternatively may refer to the wool from which the ribbons were made.[1]

## History and examples

### Lemniscate of Booth

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 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 $(x^2 + y^2)^2 - cx^2 - dy^2$ when the parameter d is negative. For positive values of d one instead obtains an oval-shaped curve, the oval of Booth.[1]

Curves of Cassini, including lemniscate

### 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. It is analytically described as the zero set of the polynomial equation $(x^2 + y^2)^2 = 2a^2 (x^2 - y^2)$, 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.[3] 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.[4] It is a special case of the hippopede, 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.[1] 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 equation $y^2=x^2(a^2-x^2)$.[5][6] 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.[7]

### 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,[8]
• 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 $(x^2+y^2)(x^2+y^2-d^2)^2+4a^2y^2(x^2+y^2-b^2)=0$ and has the lemniscate of Bernoulli as a special case.

• Analemma, the figure-eight shaped curve traced by the noontime positions of the sun in the sky over the course of a year
• Lorenz attractor, a three-dimensional dynamic system exhibiting a lemniscate shape
• Polynomial lemniscate, a level set of the absolute value of a complex polynomial

## References

1. ^ a b c d Schappacher, Norbert (1997), "Some milestones of lemniscatomy", Algebraic Geometry (Ankara, 1995), Lecture Notes in Pure and Applied Mathematics 193, New York: Dekker, pp. 257–290, MR 1483331.
2. ^ a b Erickson, Martin J. (2011), "1.1 Lemniscate", Beautiful Mathematics, MAA Spectrum, Mathematical Association of America, pp. 1–3, ISBN 9780883855768.
3. ^ Bos, H. J. M. (1974), "The lemniscate of Bernoulli", For Dirk Struik, Boston Stud. Philos. Sci., XV, Dordrecht: Reidel, pp. 3–14, MR 774250.
4. ^ 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.
5. ^ Basset, Alfred Barnard (1901), "The Lemniscate of Gerono", An elementary treatise on cubic and quartic curves, Deighton, Bell, pp. 171–172.
6. ^ Chandrasekhar, S (2003), Newton's Principia for the common reader, Oxford University Press, p. 133, ISBN 9780198526759.
7. ^ Costa, Luisa Rossi; Marchetti, Elena (2005), "Mathematical and Historical Investigation on Domes and Vaults", in Weber, Ralf; Amann, Matthias Albrecht, Aesthetics and architectural composition : proceedings of the Dresden International Symposium of Architecture 2004, Mammendorf: Pro Literatur, pp. 73–80.
8. ^ 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.