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Fractals Fractals Everywhere

I remember years ago I was really into those beautiful mathematical shapes called fractals.
If you study these things even casually, a gentleman called Benoit Mandlebrot will invariably turn up in your Google searches.

Mandlebrot was the Sterling Professor Emeritus of Mathematical Sciences at Yale University before he retired but he will be most remembered for coining the term fractal and bringing their beautiful shapes and insights to the masses.

In his 1982 seminal book “The Fractal Geometry of Nature,” he described these mathematical objects which were virtually ignored by mainstream scientists who famously called them monstrous and pathological.

Why am I talking about this guy? Well primarily because he died recently and I thought it would be a good chance to potentially introduce him and his ideas to a few people.

So, what are fractals?

Most people would recognize them as those complex beautiful computer generated shapes you see on t-shirts or geek-conventions.

Fractals are essentially geometric shapes whether natural or created from equations that have an attribute called self-similarity. That means that tiny pieces of the shape looks like the whole thing. At any scale then from far away to zoomed in incredibly far, you see similar shapes repeating themselves over and over.

A tree is a classic example. Look at its biggest branch and it looks like the whole tree. Look at a smaller branch on that branch and it looks similar as well. Look at a twig on that branch it is similar as well albeit simpler. Every November, once I get over being bummed about all the beautiful fall foliage being gone, I start to appreciate many of the bare trees and how beautifully fractal they can be.

It turns out that these shapes, once they can be mathematically described and manipulated, can be used to describe and understand many of the complex shapes found in nature. This includes clouds, mountains, and coastlines, the branching of nerves and arteries and a plethora of  other natural and biological shapes.

Mandlebrot said these shapes were once considered unmeasurable but now could “be approached in rigorous and vigorous quantitative fashion.”

Another key attribute of fractals is the idea of fractal dimension which is not as boring as it sounds, really.
To illustrate this think of a coastline. Mandlebrot did this and it pretty much led to his whole epiphany on fractals.
He asked himself, how long is the coast of Great Britain? Easy question right? No it’s not. Just look it up. To this day there is likely to be various estimates that seem to be more divergent than they ought to be.

Think about it. The measured length of a coastline depends on the size of your ruler or how detailed the map your using is.  A small map will yield one estimate but zoom into the map and more nooks and crannies of the coast reveal themselves and the coast gets longer. Zoom in some more and the same thing happens. Mandlebrot said the coastline could be considered infinite when looked at this way. Which number is correct then?

Well, the shape of a coast has fractal properties. Tiny pieces of it look like much longer stretches of coast. That shape isn’t a one-dimensional line and it’s not a two-dimensional surface like a table, it’s somewhere in between. It can be said to have fractional dimensions. 

Draw a line on a piece of paper. That’s one dimensional. Now draw a line so squiggly that it fills the entire piece of paper. That line has now become two-dimenional because it fills the entire 2-d surface of a piece of paper. Fractals then, like coastlines, can be calculated to have fractal dimensions of say 1.3 (In Great Britain’s case) between a 1-dimensional line and a 2-dimensional surface. This number applies to the coastline at all scales just like its self-similarity applies at all scales

Once you determine the fractal dimension of a coastline, you can then calculate a meaningful length for the coast based on that dimension.

Fractal shapes were known and examined mathematically before Mandlebrot even going back to the 17th century but Mandlebrot is widely considered the one to define the science of fractal mathematics. He also discovered the Mandelbrot set which is my favorite fractal of all time. Look it up if you haven’t seen it. If you’re too lazy, just looks below.

Understanding fractals doesn’t just help with coastlines or pretty pictures. Mandlebrot’s and other’s insights have helped with…

  • describing network architectures like the internet
  • diagnosing some diseases
  • Predicting size and timing of earthquakes
  • financial markets
  • file compression algorithms
  • how mammalian brains fold as they grow
  • how galaxies cluster
  • geology, medicine, history, cosmology and engineering

And the list goes on and on.

Yale College Dean Mary Miller said.

“He revolutionized geometry and made it possible to think about measurements and visualization of forms through an entirely new kind of geometry, Visualization of fractals is fundamental to college and university study now.”

 

Heinz-Otto Peitgen, a professor of mathematics and biomedical sciences at the University of
Bremen said:

“…if we talk about impact inside mathematics, and applications in the sciences,he is one
of the most important figures of the last 50 years.”

So I hope I’ve piqued your interest in fractals and I’ve enticed you to explore this topic further. There’s plenty online to keep you interested for hours. Enjoy

2 comments to Fractals Fractals Everywhere

  • Jsug

    Fractal art was one of the topics that got me really interested in computer programming, eventually leading to my degree in computer science. I used to play around a lot with a public domain DOS program, called Fractint. It had iterators for several different fractal sets, and gave you a mind-numbing array of tweaks to the inputs and outputs, so you could generate a limitless array of stunning scenes.

    On a related topic, Jonathan Coulton (a geek rock star) wrote a song a couple years ago, “Mandelbrot Set”, which is all about Benoit and his discoveries. It was inspired by his reading of “The Fractal Geometry…” in high school.

  • Mark

    If you’re interested in fractals then you shouldn’t miss it’s cousin the ‘strange attractor’.

    As just for it’s sheer beauty, the organic attractor: http://petervandernoord.nl/blog/2010/11/the-strange-beauty-of-the-gumowski-mira-attractor/

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