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.

Lightning bolts? Or fractal fog? Or 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.

Fractals are self-similar objects. As one zooms in on a fractal, it can appear endless.

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?

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.

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. But 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.

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

When first introduced, many mathematicians did not consider his work mathematics.

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?

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