# 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", from the Greek λημνίσκος meaning ribbons, ^{[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 when the parameter *d* is negative. For positive values of *d* one instead obtains an oval-shaped curve, the oval of Booth. Its name comes from 19th-century mathematician James Booth, who studied both the lemniscate and the oval.^{[1]}

### 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. It is analytically described as the zero set of the polynomial . 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 , 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 .^{[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 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 and has the lemniscate of Bernoulli as a special case.

## See also

- 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
- Lemniscates as Generalized conics

## References

- 1 2 3 4 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. - 1 2 Erickson, Martin J. (2011), "1.1 Lemniscate",
*Beautiful Mathematics*, MAA Spectrum, Mathematical Association of America, pp. 1–3, ISBN 9780883855768. - ↑ Bos, H. J. M. (1974), "The lemniscate of Bernoulli",
*For Dirk Struik*, Boston Stud. Philos. Sci., XV, Dordrecht: Reidel, pp. 3–14, MR 774250. - ↑ 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. - ↑ 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,
*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.

## External links

- Hazewinkel, Michiel, ed. (2001), "Lemniscates",
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4