Saturday, November 24, 2012

The Fractal Dimension

     A line has one dimension. A square has two dimensions. Fractals lie somewhere between the two. Fractals occupy more space than a straight line but they are not a solid surface. Think of a toddler with a crayon scribbling inside a square on a piece of paper. The scribble is all over inside the square to the point where it is approaching the dimension of the square's planer surface but not quite filling it.

Research shows that we enjoy looking at objects that have a fractal dimension. I created a fractal tree to create the fractal dimension.


       Infinity connects gractals to the idea of being measured. The most famous example of how fractal geometry effects measurement is the coastline of England. Euclidean geometry puts a rough outline around England and measures. Fractal geometry includes every nook and cranny of the coastline which nearly triples the length of the coast. Imagine if smaller and smaller units of measure are used the coastline would approach an infinite length. 
     The image on the left shows the coast of England measured with Euclidean geometry. The image on the far right shows how fractal geometry measures the coastline more accurately.     


Here are some pieces of my art based on coastlines.


     Self-similarity is when all parts of an object are the same shape. Man made fractals are usually strictly self-similar meaning that each piece is exactly the same as the other pieces. In nature the pieces are simply self-similar meaning that there are small discrepencies between pieces. A fern leaf is self-similar. The smaller branching leaves look like the entire fern leaf. One of my favorite natural fractals that is an excellent example of self-similarity is the romanesque or broccoflower.


     Scaling means that a small piece of an object looks a lot like the larger object it came from. You can see this in the Mandelbrot Set. The small sets around the edge of the large Mandelbrot looks like the fractal it came from.

     A twig on a branch looks like the branch which in turns looks like the tree. Clouds, ferns and coastlines are all good examples of scaling. Below is my art showing scaling. The small modules of the broccoflower look like the larger modules which looks like the larger portion itself.


     In fractal geometry recursion refers to a loop in which the output of one stage becomes the input for the next. The Koch Curve is an early example of this type of recursion. The curve is begun by drawing a line segment and dividing it into thirds. An equilateral triangle is formed on the central segment and then the bottom of the triangle removed. This is the seed shape or generator. It is the output that now becomes the input. The visual of the Koch Curve will help the explanation.

     The seed shape, or generator, is used to replace every line segment in the original drawing as seen on level 2. This process can continue into infinity with the seed shape replacing every newly created line segment. This type of recursion is iteration.
     Here is iteration applied to my art.