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I have a Ph.D. in computational and theoretical chemistry with advanced but field-oriented knowledge of mathematics.

I am fascinated by fractals, but I am unable to understand them from the formal point of view. To my level of understanding, they look like a graphical rendering of an ill-conditioned iterative problem, where small variations of the initial condition lead to huge changes in the final result, but that's just what I got out of it with my current knowledge.

How would you explain fractals (such as the Mandelbrot set) to a layperson with basic mathematics knowledge from high school, and how would you instead explain it to someone which has more math training, but not formal.

This question is collateral to a post on the Mandelbrot set I did on my blog some time ago. If you have any comments on what I was doing with my tinkering of the parameters (to get some keywords for further exploration), it's greatly appreciated. I would like to explain it better to my readers, but I am unable to do it. Thanks

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You may want to be a little more specific. There are many different types of fractals and not all have nice iterative definitions like the Mandelbrot set. Have you read the wikipedia article? Have you looked into the Hausdorff dimension? Sensitivity to initial conditions is really characteristic of chaos. While chaos theory and fractals overlap, neither is a subset of the other. – Eric O. Korman Aug 1 '10 at 19:04
That's ok, this means I need a 30.000 feet aerial view on the topic in the answer :) – Stefano Borini Aug 1 '10 at 19:09
I would just recommend to read the book Chaos and Fractals - Peitgen, Jürgens & Saupe since it is packed with beautiful illustrations but it is also very easy to read as a beginner. – anon Aug 1 '10 at 19:54
Diagrams are the best way to explain fractals – Casebash Aug 2 '10 at 10:23
I second Muad's answer. I was a high school student at the height of the fractals/chaos craze, and was for some time more interested in them than any other piece of mathematics. The book by Peitgen, Jurgens and Saupe was like a gift from the gods (although, strictly speaking, I bought it myself, which says something given my limited allowance at the time). It discusses the pretty pictures, explicit algorithms for creating them, and (a good bit of) the real mathematics behind them, all seamlessly and beautifully written. – Pete L. Clark Aug 3 '10 at 1:40
up vote 19 down vote accepted

For a "high-level" explanation, I would say this: Fractals are surprisingly complex patterns that result from the repeated application of relatively simple operations/rules.

One of the easiest to visualize examples is the Koch snowflake, constructed by adding smaller triangles to each face of the figure at each iteration:

Koch snowflake

A more real-world example is the fern leaf. The DNA in a single plant cell encodes enough information to describe the structure of an entire leaf (and the entire plant, for that matter) without explicitly describing the location of each cell. Instead, the cells grow according to a set of simple rules that result in the self-similar appearance of the fern, even at smaller and smaller levels:

Barnsley fern

For a more complex mathematical explanation that still remains tied to the real world, have a look at the basic Ricker model of population growth and the resulting bifurcation diagrams:

bifurcation diagram

The x-axis on this graph is population growth rate and the y-axis is population density. Although it looks complex, All it takes is a handful of iterations of the basic formula on a hand calculator to see how the results can oscillate between seemingly random population levels.

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+1 for the fern, I didn't know that!!!! – BBischof Aug 1 '10 at 20:00

That's easy. I would buy them some of this stuff:

(Then I would tell them to stare at it for awhile.) It's called Romanesco broccoli, and it grows like this for the same reasons, as explained in e.James' answer, that ferns do.

Alternately, I would tell them to check out Indra's Pearls by Mumford, Series, and Wright.

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Hey, I liked your previous picture; it looked more real and naturally occurring. :-) – ShreevatsaR Aug 1 '10 at 22:40
Well, this one has much better contrast. I thought it was more suited to staring at. – Qiaochu Yuan Aug 1 '10 at 22:42
Now I'm hungry. How does fractal broccoli taste? – e.James Aug 1 '10 at 22:43
I would say "just like non-fractal broccoli"... except, who has ever seen non-fractal broccoli?! – MathematicalOrchid May 4 '12 at 12:08

There are a lot of examples that are not hard to define rigorously, e.g. the Cantor set. This can be defined rigorously simply by taking a suitable intersection of nested intervals when the middle thirds are successively removed. It is easy to make all sorts of variants, constructing "Cantor-like" sets when you replace "middle third" with "middle fourth" or even with a construction where the ratio of the interval removed changes with the iteration. (This is important because you can then get a set of positive measure.) Leaving aside the remark about measure, all this uses nothing more than elementary mathematics.

One way to think of fractals is as of subsets of the phase space of certain dynamical systems. For instance, consider the Smale horseshoe. This is a two-dimensional map that can be visualized geometrically; this is explained in the Wikipedia article. Then the collection of points that stay in the square when you iterate this procedure both forwards and backwards is a fractal set, basically a two-dimensional Cantor set, which is very interesting from the perspective of dynamical systems (it's chaotic---one consequence of this is that knowing a point to arbitrary precision will not tell you what its orbit looks like later on with much precision). So much of the interesting behavior of dynamical systems is on fractals. You can define a map of the interval into itself such that the points in the interval that stay in the interval no matter how many times you iterate the map is just the ordinary Cantor set, as well.

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The link explains, in simple terms, the process of creating several different fractal types including Mandelbrot, Julia, Newton, and Orbit Traps.

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