Butterfly Curve (transcendental)
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The butterfly curve is a transcendental
plane curve In mathematics, a plane curve is a curve in a plane that may be a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane c ...
discovered by Temple H. Fay of
University of Southern Mississippi The University of Southern Mississippi (Southern Miss or USM) is a Public university, public research university with its main campus in Hattiesburg, Mississippi. It is accredited by the Southern Association of Colleges and Schools to award bac ...
in 1989. __TOC__


Equation

The curve is given by the following
parametric equation In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point (mathematics), point, as Function (mathematics), functions of one or several variable (mathematics), variables called parameters. In the case ...
s: :x = \sin t \!\left(e^ - 2\cos 4t - \sin^5\!\Big(\Big)\right) :y = \cos t \!\left(e^ - 2\cos 4t - \sin^5\!\Big(\Big)\right) :0 \le t \le 12\pi or by the following
polar equation In mathematics, the polar coordinate system specifies a given point in a plane by using a distance and an angle as its two coordinates. These are *the point's distance from a reference point called the ''pole'', and *the point's direction from ...
: :r = e^ - 2\cos \left(4\theta\right) + \sin^5\left(\frac\right) The term has been added for purely aesthetic reasons, to make the butterfly appear fuller and more pleasing to the eye.


Developments

In 2006, two mathematicians using
Mathematica Wolfram (previously known as Mathematica and Wolfram Mathematica) is a software system with built-in libraries for several areas of technical computing that allows machine learning, statistics, symbolic computation, data manipulation, network ...
analyzed the function, and found variants where leaves, flowers or other insects became apparent. New developments regarding such curves are still under research by mathematicians.


See also

*
Butterfly curve (algebraic) In mathematics, the algebraic butterfly curve is a plane algebraic curve of degree six, given by the equation :x^6 + y^6 = x^2. The butterfly curve has a single singularity with delta invariant three, which means it is a curve of genus sev ...


References


External links


Butterfly Curve plotted in WolframAlpha
Plane curves {{geometry-stub