Applications of Linear Algebra: Fractals

Instructor: Ross Rueger

By: John Jimenez

Introduction:

Early mathematicians disdained the challenge of describing self

similarity that appeared in nature. It was clear that early mathematical

methods lacked the sophistication to describe natural patterns such as

the shape of a mountains, a coastline, or the growth of a fern.

Here we will explore the morphology of what appeared amorphous

through linear algebra. We will deal exclusively with sets in the

Euclidean space, let us first introduce some definitions.

Self Similarity: A closed and bound subset of the Euclidean

plane is said to be self similar if it can be expressed in the form

Here we have two familiar examples of self similar fractals. The

Sierpinski Triangle, and the Sierpinski Carpet are both generated

through iterative processes with the original set showed above (i)

resulting in the final set (f). Next we will show how to generate

and map fractals with a technique that involves linear algebra.

Here (k) is the number of subsets, each of which is congruent to S

scaled by the same factor . The definition describes the decomposition

of S into non-overlapping congruent sets.

(i)

(i)

(f)

(f)

A similitude with scale factor s is a mapping of where, s, , e, and f are scalars.

Generating a fractal:

Here we start with the Unit Square. We apply 8 similitudes with the appropriate scale factor.

The only translation here is by e units in the x direction and f units in the y direction as well as

scaling by s, and = 0.

Similitudes

Affine Transformation:

An affine transformation is another method of generating and mapping fractals

where a, b, c, d, e, and f are scalars.

Transformations:

• Translations • Scaling • Rotation • Sheer mapping • Squeeze mapping

Transformations:

• Scaling • Rotation • Translation

The fern shown above is generated with an initial point undergoing one of the

four transformations successively with the desired amount of iterations until

the final converging set S or image is obtained. One of the major applications

of affine transformations is data compression. For the Barnsley Fern, Michael

Barnsley developed a matrix of constants that completely encodes the fern.

The storing of numbers requires much less memory versus a pixel-by-pixel

description of the fern.

Anton, Howard. "Applications of Linear Algebra/ Fractals." Elementary Linear Algebra. 10th ed.

Hoboken, NJ: Wiley, 2010. 626-39. Print.

Barnsley, M. F. Fractals Everywhere. Boston: Academic, 1988. Print.

References: