The History of Fractals

Fractals are seen everywhere but is a fairly recent branch of mathematics.
Although the concept of a fractal has been around for a long time, Benoit Mandelbrot was the first person to define it in 1975.

The first hint of fractals appeared in the seventeenth century when Liebniz explored self similarity but only with a straight line.  The next reference came in 1872 when a mathematician produced the first graph resembling what we call fractals.  The man who produced this graph used an abstract and analytic definition which Helge Van Koch hated.  So Koch used a geometric definition which produces the Koch Curve or Koch Snowflake.

Shortly after, Waclaw Sierpinski created his famous triangle and carpet.  Over time, many others continued exploring this topic but without computers it was difficult to create a picture showing the true beauty of fractals.  Benoit Mandlebrot came up with the term "Fractal" which is from the Latin term "Fractus" meaning broken or fractured. He is considered the father of Fractals.

One type of fractals are the Mendlebrot Fractals named after Benoit Mandlebrot.  In this set, they basically look at how the complex functions behave when repeated multiple times.  This particular fractal has lots of heads and legs called "brots"

The form of the function is f(z) = z^2 + c but it involves a lot of complex numbers with a, b, and i. Remember i. is the square root of -1.  For each c, they begin by placing 0 in for z and two things happen.  Either it gets larger and larger moving away from zero or it stays close to zero and never going very far.

A second type of fractals are the Julia fractals discovered by Gaston Julia around 1915.  He investigated the equation f(z) = z^4 + z^3/(z-1) + z^2/(z^3 + 4z^2 + 5) + c which has lead to several commonly used functions such as zn+1= c sin (zn), zn+1 = c i. cos(zn), and two others.  Julia fractals are mapped from each pixel to a rectangular region of the complex plane.  It starts at the pixel.  If it goes to infinity, the area is white but if it does not, then its black.

There is a relationship between Mandlebrot fractals and Julia fractals.  If the c is inside a Mandlebrot set, it is connected and if they are from outside the set, they are disconnected and appear as dust.

The third type of fractal is the Newton fractal which is based on Newton's Method - used to find roots of functions.  It appears as xn+1 = xn - f(xn)/f'(xn).  The xn converges rapidly to f(xn).  If you use colors to represent the type of root such as red if it converges to -1, green if it goes to the complex root on the right side or blue if it goes to the complex root on the left side and black if it doesn't converge at all.  The shade of the color is determined by how quickly it converges.

So now you know the three main types of fractals and the mathematics associated with them.  I'll be delving into the Koch Snowflake another time because it is one that can easily be incorporated into the classroom.  Let me know what you think, I'd love to hear. Have a great day.