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Below is an animation of the growth of the Mandelbrot set as you iterate towards infinity. The animation shows the continuous growth of the Mandelbrot set as it tends towards infinity – iterations {1-1024}, increments are powers of two.

Animation of the growth of the Mandelbrot set as you iterate towards infinity by Michael James Dean

An Introduction to Fractals

Fractals are not easy to explain mathematically, similar to problems conceptualizing the fourth dimension. Fractals, even though seeming to exist in the 1st dimension (that of a line), are not so simple. In fact, “a curve with fractal dimension very near to 1, say 1.10, behaves quite like an ordinary line, but a curve with fractal dimension 1.9 winds convolutedly through space very nearly like a surface” (Wikipedia – Fractal Dimension).

Read about the mathematics of fractals in Wikipedia’s introduction. Fractals are described as having three main characteristics:

  • Self-similarity: analogous to zooming into an image. With fractals, you notice a repeating pattern.
  • Fractional Dimension: greater than it’s topological dimension (e.g., a line’s topographical dimension is 1). See explanation about fractal dimension above.
  • Measurement: cannot be measured like lines, shapes, and solids. If you set out to measure a fractal curve (the length of the line), the pattern would keep repeating and thus require a little more length, It may help to think about the infinity concept – infinitely smaller.

This rather long Nova video is worth the time if you want to know more about fractals.

History of the Fractal

The history of fractals started in the 17th century with Gottfried Leibniz. According to Wikipedia:

Gottfried Leibniz pondered recursive self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense). In his writings, Leibniz used the term “fractional exponents”, but lamented that “Geometry” did not yet know of them. Indeed, according to various historical accounts, after that point few mathematicians tackled the issues and the work of those who did remained obscured largely because of resistance to such unfamiliar emerging concepts, which were sometimes referred to as mathematical “monsters”.

You can view the Wikipedia page for more. I’m fascinated by the terms  “resistance” and “mathematical monsters” – how do you think fractals might have been more difficult to understand in the 1800s versus today? Let’s view some graphic animations of fractals.

Animated Fractal Mountain

Animated Fractal Mountain by António Miguel de Campos (public domain)

Okay, so what does this all mean in terms of dimension? According to Wikipedia, “The fractal dimension of a curve can be explained intuitively thinking of a fractal line as an object too detailed to be one-dimensional, but too simple to be two-dimensional”. Let’s explore some different ways fractals have been represented.

Cantor sets

Screen shot 2013-02-12 at 10.34.36 PM

The Julia Set

Julia Set

Screen shot 2013-02-12 at 9.48.43 AM

Koch curve

Von_Koch_curve

7 first steps of the building of the von Koch curve

Sierpiński’s Triangle

Animated construction of Sierpinski Triangle

Animated construction of Sierpinski Triangle by Dino

The term fractal dimension was published by Mandelbrot in 1975. He investigated “self-similarity” when mapping the coastline of Britain. According to Wikipedia, “Mandelbrot solidified hundreds of years of thought and mathematical development in coining the word “fractal” and illustrated his mathematical definition with striking computer-constructed visualizations…Currently, fractal studies are essentially exclusively computer-based.

Britain-fractal-coastline-combined

Britain Fractal Coastline Combined by originals made by Avsa mixed by Acadac

Here’s a link to some free Fractal Generator Programs if you want to experiment.

By the way, fractal relationships do not only occur as curves and lines, but have been represented three-dimensionally. What would you suspect the fractal dimension of this object is?

MandelbrotOrbitInfimum

Mandelbrot Orbit Infimum by Paintthread

One thought on “The Fractal Dimension

  1. Pingback: Copyright, Fair Use, and Creative Commons | UGA Edit 2000

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