The word "fractal" comes from the Latin "fractus" for "broken".

Smooth "smooth" baked bread gets broken into "rough" bread, each getting a "fraction" of the loaf.

After the resurrection:

A fractal is a self-similar object. What does it mean to be self-similar? In general:

• A tree branch looks like a little tree.
• A part of a cloud looks like a cloud.
• A part of a fern looks like a fern.
• (and so on)

A tree branch looks like a little tree.

A part of a cloud looks like a cloud.

A part of a mountain looks like a mountain. To get a lake, just put a flat surface at a given elevation.

Is it a lightning bolt? Is it fractal fog? Is it ocean waves? Related concepts include Brownian/random motion.

A part of a fern looks like a fern. And so on.

The Barnsley fern uses IFS (Iterated Function Systems) to create realistic fractals.

The same techniques are the basis of fractal compression methods.

As one increases the detail, one can descend into the fractal - in what can appear like forever - infinite descent.

In reality, at a certain point, the zooming must stop.

f(z) = z*z + c

Zoom is at the Feigenbaum point at:

(-1.401155189..., 0)

Where else do we see concepts of self-similarity?

Benoit Mandelbrot (1924-2010) discovered and popularized the concept of fractals, and coined the word from the Latin "fractus".

At the time he first introduced his work, many mathematicians did not consider his work mathematics.

Does it matter how things are measured? What about from a Biblical perspective? Explain.

Note: There is a difference between deciding what is right and wrong, communicating those ideas, and announcing/pronouncing and executing judgment on others.

Benoit Mandelbrot's book, The Fractal Geometry of Nature (1983, revisions since), introduced me to the world of fractals.

Others, such as Lauren Carpenter took notice of his work.

Lauren Carpenter, then working for Boeing in Seattle, used the new fractal concepts to develop realistic animated natural scenery for flight simulators and marketing materials.

He left to work at Lucasfilm's computer division to help develop animated landscapes, lave flows, etc.,

including animations for Star Wars, Star Trek, etc.

The Lucasfilm's computer division would become Pixar.

The rocket scene from Toy Story 1 shows animated trees in the background.

Toy Story trivia:

• Leaves on a typical tree in Andy's neighborhood: 10,000
• Trees on Andy's block: 100+
• Leaves on Andy's block: 1.2 million

How might a programmer create hundreds of trees and millions of leaves?

Start with a straight line as the base case.

Divide the line segment in into three parts.

Extend the middle segment. Here are the base and step cases. The fractional dimension in log(4/3).

Now do the same for the step case to get the next step case.

Continue in the same manner.

Continue in the same manner.

Continue in the same manner.

Continue in the same manner.

In the limit, the length of the coastline is infinite, continuous (connected), but nowhere differentiable (smooth), what mathematicians call a "monster curve". Note: A simple recursive procedure can draw the entire curve.

The length of a fractal is infinite. In practice, one stops at a certain point.

The measured length of coastlines has varied depending on measurement scale.

Do you remember the Koch coastline?

What happens if we change the step case, at each step, to add some randomness?

To do so, flip a coin.

Imagine this as a coastline. More real looking, right. You can see the inlets and protruding islands. Simple to generate. That's the beauty of fractals.

Color shading makes the computer-generated coastline look more realistic.

Here are four variations of a simple Koch island.

This one randomization at each step results in a somewhat realistic island outline.

In practice, many parameters can be varied and randomized.

I always liked programming, so here are some attempts at programming fractal trees (from November 2010). I started them after watching a video on Fractals with my son who afterwards started seeing fractal landscapes in his Bionicle movies. These images are in reverse order as I "improve" the original code. The code was written in PostScript and improved over time.

Here is an animated GIF image which put together the concepts from the previous trees (below).

Here are the four seasons. Clockwise: Winter, Spring, Summer, Autumn. I used a seed for the random number generator to get the same tree branch layout each time (with extra random calls where needed depending on the season). Different seeds produce slightly different trees. Here are the trees for each of four seasons.

I created it with a small PostScript program in a few minutes.

Here is the tree after a request to add leaves. The leaves are non-fractal for now.

On request, here is the autumn tree.

Here is the same tree before I added the randomness to the angles and branch lengths.

Possible future improvements:

• Have the autumn leaves fall at random to the position below
• Alter the angle and/or length of each leaf/blossom randomly
• Add icicles hanging from the Winter tree.

The Peano curve was the first space-filling curve to be discovered. Such a curve is called a "monster curve",

Here is a Pythagoras fractal tree.

Fractal antennas, in cell phones, are based on ideas found in the fractal curves.

Nathan Cohen created (and patented) the first fractal antenna in 1988.