What is a

Fractal?

Natali Kuzkova

Ph.D. coffee, 5th May 2015

A fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale. If the replication is exactly the same at every scale, it is called a self-similar pattern. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos.

Fractals also includes the idea of a detailed pattern that repeats itself.

The term "fractal" was first used by mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin (frāctus) meaning "broken" or "fractured", and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature.

The Mandelbrot set is the set of values of c in the complex plane for which the orbit of 0 under iteration of the complex quadratic polynomial remains bounded:

Images of the Mandelbrot set display an elaborate boundary that reveals progressively ever-finer recursive detail at increasing magnifications. The "style" of this repeating detail depends on the region of the set being examined. The set's boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies to the entire set, and not just to its parts.

History

Self-similarity illustrated by

image enlargements. This

panel, no magnification.

The same fractal as above,

magnified 6-fold. Same

patterns reappear, making the

exact scale being examined

difficult to determine.

The same fractal as above,

magnified a 100-fold.

The same fractal as above,

magnified a 2000-fold, where

the Mandelbrot set fine detail

resembles the detail at low

magnification.

Mandelbrot set

Fractals should, in addition, to being nowhere differentiable and

able to have a fractal dimension, be generally characterized by

a gestalt of the following features. Self-similarity, which may be

manifested as:

1. Exact self-similarity: identical at all scales; e.g. Koch snowflake: The Koch snowflake (Koch curve, star. or island) is a mathematical curve and one of the earliest fractal curves to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a continuous curve without tangents, constructible from elementary geometry" (original French title: Sur une courbe continue sans

tangente, obtenue par une construction géométrique élémentaire) by the Swedish mathematician Helge von Koch.

Characteristics

Fig.1. The first four iterations of the Koch

snowflake.

Fig.2. The first seven iterations in animation.

Fig.3. Zooming into

the Koch curve.

2. Quasi self-similarity: approximates the same pattern at different

scales; may contain small copies of the entire fractal in distorted

and degenerate forms; e.g., the Mandelbrot set's satellites are

approximations of the entire set, but not exact copies:

Characteristics

Fig.4. Mandelbrot animation

based on a static number of

iterations per pixel.

3. Statistical self-similarity: repeats a pattern stochastically so numerical or statistical measures are preserved across scales; e.g., randomly generated fractals; the well-known example of the coastline of Britain, for which one would not expect to find a segment scaled and repeated as neatly as the repeated unit that defines.

"How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension" is a paper by Mandelbrot, first published in Science in 1967. In this paper Mandelbrot discusses self-similar curves that have Hausdorff dimension between 1 and 2. These curves are examples of fractals.

The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal-like properties of coastlines. More concretely, the length of the coastline depends on the method used to measure it.

Characteristics

Fractals in nature

1. Sea shells.

The nautilus is one of the most famous

examples of a fractal in nature. The

perfect pattern is called a Fibonacci

spiral.

2. Snow flake.

Fractals in nature

3. Lightning.

Lightning’s terrifying power is both

awesome and beautiful. The fractals

created by lightning are fascinatingly

arbitrary and irregular.

4. Fern.

The fern is one of many flora that are fractal;

it’s an especially good example.

Fractals in nature

5. Crystals.

Both chemically-formed crystals and ice and frost

crystals are breathtaking examples of fractals in nature.

6. Trees and leaves.

From the macro view of a leaf to the span of

a tree’s branches, fractals turn up frequently.

8. Shore lines.

7. Mountain ranges.

Both shorelines and mountain ranges are considered

loosely fractal. These particular examples are

beautiful.

This stunningly complex fractal shoreline is

none other than the pan handle of Florida.