Lemniscate

History and examples

Lemniscate of Booth

Main article: Hippopede

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. The name "lemniscate of Booth" for this curve dates to its study by the 19th-century mathematician James Booth.{{citation

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

Main article: 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.{{citation

Lemniscate of Gerono

Main article: 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). 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.

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,
• 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.

References

References

1. {{cite Dictionary.com. lemniscate
2. {{L&S. lemniscatus. ref
3. Erickson, Martin J.. (2011). "Beautiful Mathematics". [[Mathematical Association of America]].
4. {{OEtymD. lemniscus
5. {{L&S. lemniscus. ref
6. {{LSJ. lhmni/skos. λημνίσκος. ref.
7. {{LSJ. i(ppope/dh. ἱπποπέδη. shortref.
8. Köller, Jürgen. "Acht-Kurve".
9. Basset, Alfred Barnard. (1901). "An elementary treatise on cubic and quartic curves". Deighton, Bell.
10. Chandrasekhar, S. (2003). "Newton's Principia for the common reader". Oxford University Press.
11. (2005). "Aesthetics and architectural composition : proceedings of the Dresden International Symposium of Architecture 2004". Pro Literatur.
12. Darling, David. (2004). "The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes". John Wiley & Sons.