# Fractals

The French-American polymath, Benoit Mandelbrot, is often referred to as 'the
Father of Fractals'. Mandelbrot introduced the word *fractal* in 1975 and
highlighted the need for fractal mathematics. Information on this page is taken
from *Fractals* by Kenneth Falconer.

## Definition

Something is regarded as a fractal if all or most of the following properties hold in some form.

**Fine structure**- Irregularities at arbitrarily small scales. This is very different from, say, a circle where a small portion of the perimeter sufficiently magnified will appear almost indistinguishable from a straight line.

**Self-similarity**- Made up of smaller scale copies of itself. See the section on self-similarity below.

**Classical mathematics are not applicable**- Traditional Euclidean geometric language doesn't easily describe fractals. Fractals often have infinite length and zero area.

**Recursive definition**- Many fractals are constructed by performing a simple step over and over again.

The definition of a fractal is somewhat analogous to the way biologists define 'life'. Something is held to be alive if it has all or most of the characteristics on a list: ability to grow, ability to reproduce, ability to respond to stimuli in some way, etc.

## Example: Von Koch Curve

Take a straight line (\(E_0\)) and divide it into three equal pieces. Erase the middle piece, and replace this by the other two sides of an equilateral triangle on the same base (\(E_1\)). Repeat with each of these four pieces (\(E_2\)).

Repeated indefinitely, this procedure becomes the **von Koch curve** (\(F\)).

This fractal was introduced and studied by the Swedish mathematician Helge von Koch in 1906. This fractal displays fine structure as a consequence of being self-similar: the curve consists of four parts, each a 1/3 scale copy of the entire curve.

If we take three copies of the von Koch curve and join them corner to corner
in a triangle, we get the **von Koch snowflake**.

## Iteration

Fractals are often drawn on a plane with a collection of points. This
collection of points is referred to as a **set**. A set might be a circle, a von
Koch curve, or the silhouette of a person.

A set can be formed by repeatedly applying a function. Given a function
(e.g. \((x, y) -> (x + 1, y)\)) and some initial point, we discover a new
point. Applying the function to this point gives us a third point. The process
of repeatedly applying the function is known as **iterating** the function and
the points visited are the **iterates** of the initial point. Iterates jump
around and build up a curved, stratified figure called the **attractor** of the
function.

Repeated application of a simple operation may give rise to a complex, fractal
form. A fractal attractor is sometimes called a **strange attractor**.

## Self-Similarity and Templates

Two figures are **similar** if they have the same shape, but not necessarily the
same size. Objects are **congruent** if they have the same shape and size. A
**self-similar** set is one that is made up of several smaller similar copies of
itself.

The Koch curve is self-similar since it comprises 4 suitably placed scale \(1/3\) copies of itself. This may be represented diagrammatically by a template consisting of one large rectangle and 4 smaller ones, each a \(1/3\) copy of the large one.

Put generally, a **template** consists of some simple shape (square, rectangle,
triangle, etc.) and a number of smaller similar copies of the shape positioned
somehow in the plane. Each of the smaller shapes represents a transformation
that scales and repositions the larger shape. The template completely defines
the fractal: given a template, there is essentially just one figure, usually a
fractal, that is made up of smaller scaled copies of itself positioned as
indicated.

There are 8 different similarities that transform a square to a smaller
square. The scaling can be direct or there can be a rotation through 90°,
180°, or 270°, or a mirror reflection in a vertical, horizontal, or one of the
two diagonal lines. We refer to these possibilities as the **orientations** of
the scalings. The orientations of the similarity transformations must be
specified to remove ambiguity.

Templates are simply a diagrammatic way of expressing a family of
transformations or functions. Each of the smaller shapes codes the similarity
transformation which takes the large shape and scales it down to the size and
position of the smaller one. The functions have the advantage that they also
encode the orientations of the transformations. The **Chaos Game** is a method
of constructing templates with functions.

- Take any initial starting point.
- Select one of the functions at random.
- Apply the chosen function to get the next point.
- Go to step 2.

An **affine transformation** is more general than a similarity, in that it
scales by different ratios in different directions and so elongates
objects. For example, affine transformations transform squares into rectangles
or parallelograms, and circles into ellipses. A **self-affine** fractal is one
that is made up of smaller affine copies of itself.

## Example: Sierpinsky Triangle

The **(right angled) Sierpinski triangle** is made up of 3 copies of itself at
scale \(1/2\).

The three functions giving the three similarity transformations that transform the unit square to the bottom left, bottom right, and top left squares respectively, have the formulae:

\begin{equation} (x, y) \rightarrow (\frac{1}{2}x,\frac{1}{2}y) \end{equation} \begin{equation} (x, y) \rightarrow (\frac{1}{2}x + \frac{1}{2},\frac{1}{2}y) \end{equation} \begin{equation} (x, y) \rightarrow (\frac{1}{2}x,\frac{1}{2}y + \frac{1}{2}) \end{equation}The traditional Sierpinski triangle is constructed with equilateral triangles.