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A catastrophe which can occur for four control factors and one behavior axis. The butterfly catastrophe is the universal unfolding of the singularity f(x)=x^6 of codimension ...

Due to nonlinearities in weather processes, a butterfly flapping its wings in Tahiti can, in theory, produce a tornado in Kansas. This strong dependence of outcomes on very ...

The fractal-like two-dimensional function f(x,y)=((x^2-y^2)sin((x+y)/a))/(x^2+y^2). The function is named for the appearance of a butterfly-like pattern centered around the ...

Letting Lk be the linking number of the two components of a ribbon, Tw be the twist, and Wr be the writhe, then Lk(K)=Tw(K)+Wr(K). (Adams 1994, p. 187).

If a plane cuts the sides AB, BC, CD, and DA of a skew quadrilateral ABCD in points P, Q, R, and S, then (AP)/(PB)·(BQ)/(QC)·(CR)/(RD)·(DS)/(SA)=1 both in magnitude and sign ...

The term "Cartesian" is used to refer to anything that derives from René Descartes' conception of geometry (1637), which is based on the representation of points in the plane ...

A special point which usually has some symmetric placement with respect to points on a curve or in a solid. The center of a circle is equidistant from all points on the ...

A circle is the set of points in a plane that are equidistant from a given point O. The distance r from the center is called the radius, and the point O is called the center. ...

The involute of the circle was first studied by Huygens when he was considering clocks without pendula for use on ships at sea. He used the circle involute in his first ...

Two complex numbers x=a+ib and y=c+id are multiplied as follows: xy = (a+ib)(c+id) (1) = ac+ibc+iad-bd (2) = (ac-bd)+i(ad+bc). (3) In component form, ...