Ref #2: What’s a Fractal

Draw your own fractals with my fractal drawing software, TWM Fractals.  With the new “Animate” feature, you can watch as your fractals iterate before your very eyes!  (It’s quite entertaining.)  Click the link to download.  (It requires java JRE.  I’m not really sure which version, but as long as it’s not super old, it should work.)

In geometry, a fractal is the infinite iteration of a recursively defined figure.  That is, it is a figure whose sides are defined recursively and iterated to infinity.  A simple example of a fractal it Koch’s Snowflake .  Koch’s Snowflake is a geometric fractal based around an equilateral triangle.

The algorithm for turning such a triangle into a fractal is as follows:  subdivide each side into four equal parts such that, in the middle of each side, a triangle protrudes that is similar to the original, only missing one side. It will look like this:

Once this is completed, the figure is said to have undergone one iteration.  Now we repeat the process for each side, include those newly formed sides:


and so on to infinity…

Once the figure has been iterated to infinity, it is considered a fractal.  This means that every part of its perimeter has the exact same structure (while some parts are smaller and others larger).  Fractals are often considered to be fraction-dimensional figures.  This is because, in any integer dimension, an infinite sum of infinitesimal parts (that is, 0over0) is an integral, which always has a finite solution.  But in the case of a fractal, we have an infinite sum of infinitesimal parts (still 0over0) that has an infinite solution.  This means that the perimeter of Koch’s flake has an infinite length (as each iteration increases the length by a factor of 4/3).  This is because the order of infinity that describes the number of sides is higher than the order of inverse infinity that describes the length of each side.  It is often thought of as a paradox that a figure, such as a fractal, can have a finite area but infinite perimeter and an infinite perimeter made only of infinitesimals.