Fractal Basics: Learn It 2

Constructing a Fractal

We can construct other fractals using a similar approach of recursion. To formalize this a bit, we’re going to introduce the idea of initiators and generators.

initiators and generators

An initiator is a starting shape.

 

A generator is an arranged collection of scaled copies of the initiator.

To generate fractals from initiators and generators, we follow a simple rule, the fractal generation rule.

fractal generation rule

At each step, replace every copy of the initiator with a scaled copy of the generator, rotating as necessary.

This process is easiest to understand through an example.

Use the initiator and generator shown to create the iterated fractal.

A straight, horizontal line labeled initiator. And a horizontal line that forms a peak in the middle labeled generator.

 

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 [latex]4[/latex] copies of the initiator. In step [latex]1[/latex], the single line segment in the initiator is replaced with the generator. For step [latex]2[/latex], each of the four line segments of step [latex]1[/latex] is replaced with a scaled copy of the generator:

Step 1, the generator. Next, a scaled copy of generator (smaller copy). Next, a scaled copy replaces each line segment of Step 1. In step 2, the fractal.

 

This process is repeated to form Step [latex]3[/latex]. Again, each line segment is replaced with a scaled copy of the generator.

Step 2, the fractal. Next, a scaled copy of generator. Step 3, a more complicated fractal.

 

Notice that since Step [latex]0[/latex] only had [latex]1[/latex] line segment, Step [latex]1[/latex] only required one copy of Step [latex]0[/latex]. Since Step [latex]1[/latex] had [latex]4[/latex] line segments, Step [latex]2[/latex] required [latex]4[/latex] copies of the generator. Step [latex]2[/latex] then had [latex]16[/latex] line segments, so Step [latex]3[/latex] required [latex]16[/latex] copies of the generator. Step [latex]4[/latex], 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.A Koch curve fractal using a horizontal peaked line.

Use the initiator and generator shown to produce the next two stages.

Initiator is a pentagon. Generator is five pentagons arranged to form a larger pentagon.

Using iteration processes like those above can create a variety of beautiful images evocative of nature. More natural shapes can be created by adding randomness to the steps.

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.

Step 0, an obtuse triangle. Step 1, that triangle divided into four triangles. Step 1 with randomness, The triangle divided into four triangles, but the big triangle is now irregular and no longer a true triangle.

 

We then repeat this process.

Step 1 with randomness from the last image. Next is Step 2 without randomness. Next is Step 2 with randomness.

 

Continuing this process can create mountain-like structures. This landscape[1] was created using fractals, then colored and textured.

A digitally created landscape

Fractals

 

You can view the transcript for “What Is A Fractal (and what are they good for)?” here (opens in new window).

You can view the transcript for “Fractals are typically not self-similar” here (opens in new window).

You can view the transcript for “How fractals can help you understand the universe | BBC Ideas” here (opens in new window).

 

  1. http://en.wikipedia.org/wiki/File:FractalLandscape.jpg ↵