Butterfly Curve

The Butterfly Curve is a special planar curve consisting of elegant functions, first described in 1989 by the American mathematician Temple H. Fay in Mathematics Magazine. Its polar-coordinate form is commonly used, though it can also be defined by two parametric functions in Cartesian coordinates.

In Praise of Useless Things

The Butterfly Curve has no functional role or practical application in modern technology or theory (it is frequently confused with the Lorenz Attractor, to which it is totally unrelated). It serves as an example used when explaining polar coordinate systems and demonstrating how they work in tandem with trigonometric functions. A resemblance created entirely by the human mind. What use is this to us? Honestly, I do not think it looks much like a butterfly (anyway). The mathematician who drew it is also the designer of the Chrysanthemum Curve. Paul Bourke’s codes over here constitute a typical example of them. Fay’s butterfly article begins with some expressions regarding aesthetics. What caught my attention while examining this curve was encountering the term “transcendental” curve.

Transcendental functions

Wikipedia says:

In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable that can be written using only the basic operations of addition, subtraction, multiplication, and division (without the need of taking limits). This is in contrast to an algebraic function.

What is meant here is a type of function different from polynomials, which can be written using the operations of classical ordinary algebra (the four basic operations, finite powers, root extraction, etc.). Transcendental functions cannot be expressed solely through algebraic operations and involve limits, infinite series, exponential growth, etc. The literal meaning of “transcend” is to “go beyond or surpass”. Our butterfly features Euler’s number, and its trigonometric functions oscillate infinitely. While algebraic functions generally draw basic and static curves, transcendental curves tend to produce more lively, harder-to-predict, organic forms.

What the Kant?

The word transcendental goes, of course, beyond curves and polynomials. From here, we transition to the transcendental concept in philosophy, which shares a common root meaning. (I have even started wondering if I have ADHD, given how I jump from topic to topic.) It stems from the Latin verb transcendere (to go beyond). It is used to mean going beyond ordinary or simple rules. Further, allowing my ADHD to run free, I would like to touch on the lessons from Immanuel Kant that I am still trying to grasp. According to Kant, our mind has templates such as space, time, and causality. We perceive the external world through these. Kant named the mental domain that outlines the limits of knowledge, stating that “Objects conform to the rules of our mind (time and space) as transcendental. It is the foundational structure “behind/beyond” simple perception and the empirical world. In his (reverse) Copernican revolution, he places human beings at the center and positions objects and phenomena around them.

Therefore, this meaningless function is self-evident and serves no practical purpose whatsoever. Yet, when plotted on paper, what appears in our mind resembles a butterfly figure. In reality, no such butterfly exists there. A butterfly costume is draped by our mind over the path (phenomenon) traversed by a parametric function over time.

References

Temple H. Fay, “The Butterfly Curve” American Mathematical Monthly 96, No. 5, May 1989.
Temple H. Fay, “A Study in Step Size” Mathematics Magazine 70, No 2, April 1997