Types of Fractals

Base-Motif Fractals

WHAT ARE THEY?

Take a shape — any shape that is composed of line segments. Call this shape a base. Now take another shape and call it the motif. Substitute every line segment in the base with the motif. Do the same with the resulting figure. Now continue substituting an infinite amount of time. The process you are doing is called generator iteration. What you get at the end is a base-motif fractal.

Three things effect a base-motif fractal:
1. Shape of the base — the most common bases are a line segment, a square, and an equilateral triangle.
2. Shape of the motif — this obviously has the greatest effect on the outcome. There is an infinitely great variety of motifs.
3. Positioning of the motif — if the base is a square or a triangle, you can place the motif facing either inside or outside, which will result in different fractals. For example, by using the following base and motif you can get two different fractals. In the Koch Snowflake, the motif is facing outside, while in the Koch Antisnowflake, it is facing inside:

If you position the motif the same way all the time during iteration, you will get a regular base-motif fractal. If you position it differently in some places, you will get a special type of a base-motif fractal called a sweep. Base-motif fractals are also categorized depending on their fractal dimension. When the dimension is between 1 and 2, we call it a fractal curve. When it is less than 1 it is called a dust and when it is equal to 2 it is called a Peano Curve.

EXAMPLES

Below are three examples of base-motif fractals that use different bases, motifs, and positioning:

Levy Curve

Positioning: there is only one way to position a motif if the base is a line segment.

Positioning: inside

Positioning: outside

USES?

Because base-motif fractals are so perfectly shaped, they cannot be very useful in modeling real-life things. Yet, the can create simple models of coastlines and borderlines. Recently, they were also used to create relatively good models of economy.