Fractal Basics
Learning Outcomes
- Define and identify self-similarity in geometric shapes, plants, and geological formations
- Generate a fractal shape given an initiator and a generator
- Scale a geometric object by a specific scaling factor using the scaling dimension relation
- Determine the fractal dimension of a fractal object
Self-similarity
A shape is self-similar when it looks essentially the same from a distance as it does closer up.Iterated Fractals
This self-similar behavior can be replicated through recursion: repeating a process over and over.Example
Suppose that we start with a filled-in triangle. We connect the midpoints of each side and remove the middle triangle. We then repeat this process. If we repeat this process, the shape that emerges is called the Sierpinski gasket. Notice that it exhibits self-similarity—any piece of the gasket will look identical to the whole. In fact, we can say that the Sierpinski gasket contains three copies of itself, each half as tall and wide as the original. Of course, each of those copies also contains three copies of itself.Initiators and Generators
An initiator is a starting shape A generator is an arranged collection of scaled copies of the initiatorFractal Generation Rule
At each step, replace every copy of the initiator with a scaled copy of the generator, rotating as necessaryExample
Use the initiator and generator shown to create the iterated fractal. This tells us to, at each step, replace each line segment with the spiked shape shown in the generator. Notice that the generator itself is made up of 4 copies of the initiator. In step 1, the single line segment in the initiator is replaced with the generator. For step 2, each of the four line segments of step 1 is replaced with a scaled copy of the generator: This process is repeated to form Step 3. Again, each line segment is replaced with a scaled copy of the generator. Notice that since Step 0 only had 1 line segment, Step 1 only required one copy of Step 0. Since Step 1 had 4 line segments, Step 2 required 4 copies of the generator. Step 2 then had 16 line segments, so Step 3 required 16 copies of the generator. Step 4, then, would require [latex]16\cdot4=64[/latex] copies of the generator. The shape resulting from iterating this process is called the Koch curve, named for Helge von Koch who first explored it in 1904.Example
Use the initiator and generator below, however only iterate on the “branches.” Sketch several steps of the iteration. We begin by replacing the initiator with the generator. We then replace each “branch” of Step 1 with a scaled copy of the generator to create Step 2. Step 1, the generator. Step 2, one iteration of the generator.
We can repeat this process to create later steps. Repeating this process can create intricate tree shapes.[footnote]http://www.flickr.com/photos/visualarts/5436068969/[/footnote]
Try It
Use the initiator and generator shown to produce the next two stages.Answer:
Example
Create a variation on the Sierpinski gasket by randomly skewing the corner points each time an iteration is made. Suppose we start with the triangle below. We begin, as before, by removing the middle triangle. We then add in some randomness. We then repeat this process. Continuing this process can create mountain-like structures. This landscape[footnote]http://en.wikipedia.org/wiki/File:FractalLandscape.jpg[/footnote] was created using fractals, then colored and textured.Licenses & Attributions
CC licensed content, Original
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- Self similarity. Authored by: OCLPhase2. License: CC BY: Attribution.
- Iterated tree and twisted gasket. Authored by: OCLPhase2. License: Public Domain: No Known Copyright.