Self-similarity

     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

     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.

Recursion

     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.

The Importance of Components

     At this point in my research I knew I wanted to create fractal art but was unsure how to proceed. I could not do the math or find fractal software I was comfortable with. The work of Ron Eglash gave me the key. Ron has identified five key components that define what a fractal is. These components are self-similarity, scaling, recursion, infinity and the fractal dimension. Once I understood these components I was able to identify fractals all around me. Suddenly I had a a wealth of fractals to draw by hand.

     This is a pencil rendering of a slice of cabbage which grows in a fractal manner.

      The next five blogs will describe each of the components so that you can better understand them yourself.

The Importance of Fractal Geometry

We are all familiar with Euclidean geometry. This means we understand circles, square, rectangle... shapes that are easily defined using conventional mathematics. Fractal geometry addresses those shapes that are not so conventional. Lets talk about coastlines. These are not regular shapes. In the past the coast line of England (for example) has been measured by outlining the coast and then assigning an amount to that outline. To be truly accurate one needs to measure ALL parts of the coast line. Than means every nook and cranny, every irregular twist and turn. The conventional measurement lists the length of England's coastline at about 1,000 miles. Include all the nooks and crannies and the actual coast is closer to 3.000 miles. Fractal geometry can be much more precise.
The image above shows conventional measurement on the left and fractal measuring on the right.

Benoit

So Koch and Sierpinski were experimenting with ideas that turned out to be fractal in nature. They did not know this. No one knew this because the concept of fractals and fractal geometry had not been conceived of. These concepts were brought together and refined by Benoit Mandelbrot who put these concepts together to form what we now consider as fractals and fractal geometry. By using these concepts Benoit was able to create the Mandelbrot Set. The Mandelbrot Set uses the ideas of infinity and zero mass. He is the father of fractals and fractal geometry.

The Koch Snowflake

      The Koch Curve is created by using a seed shape to replace each straight line in the figure, which is a line. This seed shape is also referred to as the first generation, but I think to envision it as a seed makes its function clearer. You plant a seed and it grows. That is how it works with a fractal.
     The Koch Snowflake uses the same seed shape as the Koch Curve. It is two sides of a triangle in the center of a straight line of which it is a third.
       Instead of beginning with a straight line one begins with an equilateral triangle. Each side of the triangle is replaced with the seed shape, two sides of a triangle in the center of a straight line or which it is a third. Then each line segment  of the snowflake is replaced with the seed shape.

     A picture is worth a thousands words. Above is the progression of the creation of a Koch Snowflake. It begins with an equilateral triangle. Each straight line of the triangle is replaced with the seed shape. Then each new line is replaced with the seed shape. And on and on into infinity. (A fractal component.)