A Leap of .... Curiosity..preparing to jump

     If you have seen my work you probably wondered what it has to do with fractals. It has TONS to do with fractals and fractal geometry. I, however, can not do the math, although I understand the ideas behind it.
     I draw fractals found in nature and the human body. I abstract some of the concepts to do tilings and geographical drawings. My favorite, however, is a Surrealist technique.
      The Surrealists developed and perfected a process called 'decalcomania'. This is a word borrowed from potters and their system of decals to decorate pottery: transferring an image using specialty paper.
For the Surrealists it was transferring paint from one surface to another using surface tension. The tension in the paint as the two surfaces are separated causes a branching pattern to occur. This process, at some point, has all five of the components of fractal geometry. There is self-similarity, scaling, recursion, infinity and the fractal dimension.

      Here is a sample of basic decalcomania.

Hele Shaw

I have been thinking about Hele Shaw cells for almost a year. I have tried to create them with limited knowledge. I know that there is a tension between two surfaces that are separated by a small amount of space. There is fractal art available if I can create the effect. Any ideas out there?

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

       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

     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.)

Koch Curve

     So infinity is an important part of being a fractal. This means that parts of the fractal continue to get smaller into infinity or that the length of the fractal grows to infinity. On and on and on... infinitely.
     The Koch Curve is one of the earliest fractal curves and is another example of infinity.

     The curve begins with a straight line that is divided into three equal segments. An equilateral triangle is created over the center segment and then the bottom of that triangle is removed. This is the first generation of a Koch Curve. Being the first generation means that this is the shape, the form, that will be used to create the rest of the curve. This segment will replace every line in the original generation.
Eventually the length of the Koch Curve will become infinite because one needs to measure each segment and add it to the total length of the curve.

The Sierpinski Triangle

     The work of Waclaw Sierpinski, a Polish mathematician, is now associated with fractal geometry. In 1915 he created what is referred to as the Sierpinski Triangle (Mandelbrot,1977). This triangle demonstrates some of the unusual mathematical concepts that are found in fractal geometry.
   To begin, one must draw an equilateral triangle and then bisect each side. Lines are drawn, connecting the midpoints. A new triangle is formed inside the original creating four triangles within the original. All sides are bisected and joined.

This is a Sierpinski Triangle used as a color wheel by one of my students. 

The amazing fractal think about this triangle is that the length of the perimeters of the triangles are approaching an infinite length while the area approaches zero. 

     I first heard about fractals by listening to NPR. One I looked them up I was hooked on their beauty As I continued to research fractals I found out there was a lot to know.
     The first thing to know is that fractals come from fractal geometry whose 'father' is Benoit Mandelbrot. There were former mathematicians that had ideas that helped lead to fractal geometry, but it was Mandelbrot who put it all together in the early 19702..
     Three of the early mathematicians to know about are Sierpinski, Koch and Julia.

What is the void?

     Who ever talks about fractals and how to create them? Generally, a few people who have complicated software talk about creating fractals. How to create a fractal without software or complicated math is a void. No one talks about those things, except me.
    The intent of this site is to talk about what fractals are, how to identify them and most importantly ways to create your own fractals by hand.
     Please join me on this quest to fill the void and talk to me about fractals. They are my passion.