The History of Fractals

The term fractal was first coined by Benoit B. Mandelbrot as a way of describing that, in reality, the dimension of a surface or of an equation was often determined by how it was measured.

For example, if you measure the coastline of Norway using a measuring stick which is 10 kilometres in length, you will arrive at a certain numerical answer. But, if you repeat the same task again - this time using a 1 kilometre long measuring stick - you will obtain a second result, which is fractionally larger than the first. Repeating the measurement again using a 100 metre long stick will yield a still larger measurement, and so on. Each measurement using incrementally smaller resolution will result in a larger measurement than the previous one.

This is because as you reduce the unit of measure, your survey will include smaller and smaller irregularities. So as you sum the length of the irregular boundary using progressively smaller scales you find that L₁ is less than L₂, which is less than L₃ ... Hence we find that L_n-1 is always less than L_n.

This rule is true, even if you reduce your unit of measurement to impossibly small scales, and start measuring the irregularities of every grain of sand as you follow the coastline. We do learn though that the length that we measure does not increase without any bounds. In fact there is another rule which states that L_n is less than twice L₁.

The Mandelbrot Set

The Mandelbrot set is one of the most famous fractal landscapes, and it has featured in many scholarly journals such as Scientific American and Time. We know that Benoit Mandelbrot was not the first to discover this fascinating land - its shores had previously been sighted by mathematicians Robert Brooks and J. Peter Matelski - however Mandelbrot was the first person to make a detailed exploration of the concept.

The Mandelbrot set is an equation lying in the realm of numbers between -2 and +0.5 longitude and -1.25 and +1.25 latitude. This land is dominated by a large fractal sea featuring numerous bays, inlets, and tributaries.

As with most lands, the most interesting places to explore are the areas along the coastline. This is where you will find irregular terrain which fits the description above when we tried to accurately measure the length of the coastline of Norway.

The difference now is that instead of trying to measure the length of the coastline, we are going to recursively examine a relatively simple equation: Zn + 1 = Zn2 + C

In this equation, C is the value of each point (the longitude and latitude of the point) and Z1 is zero for each point (an initial value). Longitude is measured in real numbers, but latitude is measured along an imaginary axis. Therefore all latitudinal values are denoted by the letter i which, to mathematicians, indicates the square root of minus 1 - an "imaginary" number.

Starting at an arbitrary position within the set, using the point coordinates -0.6 and 0.4i as an example, the first value calculated is simply:

In the second step the result of the first calculation becomes the first term of the equation, while the original point coordinates remain as the constant term in the equation.

We will also calculate a distance between the current position, -0.4 - 0.08i and the graph zero point, yielding the result 0.407921561 ... as long as this distance is not equal to or greater than 2.0 I will continue the calculations recursively, producing the third term as:
 

Again, the distance calculated is: 0.178283369 ..., which is still much less than 2.0. Although don't always expect this number to keep getting smaller.

With the recursive formula used for the Mandelbrot set, each point calculated will tend to one of two end points. The first end point after X-number of iterations will be when the distance result exceeds 2.0. After this point, the distance will continue to increase indefinitely. The second end point is reached if, after an arbitrary number of iterations, the result is still less than 2.0, and it can be expected to continue to be less than 2.0. In the demonstration program that we have provided, this arbitrary limit is 100.

When the result reaches one of these two limits, we assign an altitude to the point, which in this program is represented by a colour. In other words, the altitude is a reflection of how many iterations were required to reach that point.

Download the demonstration program and try it for yourself!

To use the demonstration program, press F12 to generate the pattern. Zoom in by clicking and dragging your mouse over an area of interest. Press F12 to generate the pattern again.

To see fractals in action we have a free program for you to download and install.

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