figureeight

Figureeight refers to several mathematical and symbolic concepts. In geometry and calculus, it most often denotes a plane curve shaped like a sideways eight, a type of lemniscate. There are multiple varieties, including Bernoulli's lemniscate and the lemniscate of Gerono.

Bernoulli's lemniscate has polar equation r^2 = a^2 cos 2θ; in Cartesian coordinates it can be written as (x^2 + y^2)^2 = a^2 (x^2 − y^2). The lemniscate of Gerono is another form, with a common parametrization x = a cos t, y = a sin t cos t, producing a figureeight-shaped curve.

In knot theory, the figureeight knot, designated 4_1, is the simplest nontrivial knot with crossing number four. It is a hyperbolic knot, and its complement in three-dimensional space supports a complete hyperbolic structure of finite volume. The knot is amphichiral, meaning it is equivalent to its mirror image.

In typography and mathematics, the sideways eight is also used as the infinity symbol, ∞, indicating boundlessness. The term figureeight thus spans geometric curves, knot theory, and symbolic representations, all connected by the characteristic eight-shaped form.