A mathematical figure that is self-similar; that is, as you zoom into it, you will see the same pattern repeated over and over again. In theory, one can expand the fractal forever.

Fractals are generated using complex functions, and weren't studied much before computers were around to draw them quickly.

Many things in nature have a fractal appearance; a classic example is the fern.

Famous fractals include the Mandelbrot and Julia sets.

An image that retains its detail and pattern regardless of how deeply one "zooms" into it. The self-similarity of the image is usually due to a recursive mathematical formula which is used to create the image. The Mandelbrot set is one example, but a lesser-known example is Newton's method, which generates Julia set fractals:

Newton's method is a function that is used to estimate the roots of a polynomial of any degree. It is of the form:

x n+1 = xn - d(f(x) / f'(x))

where d is a dampening factor (explained below), f(x) is the polynomial you want the roots of, and f'(x) is the derivative of that polynomial. When running the function for the first time, set xn to a number that is close to the root you want to find (this arbitrary number is called a "guess"). Newton's method won't give exact answers, but will take a certain number of iterations to approach the root: you pick when to stop. d is a number between 0 and 1; using a smaller value makes the function take longer to get to a root, but the function will be more stable.

Images generated by Newton's method are available. In the image, each pixel is used as a guess into Newton's method for whatever the polynomial happens to be. The pixel is colored depending on the root it converged to and the number of iterations it took (darker = more iterations). Start with:

http://www.flash.net/~nav2/z6minus1.gif     which is Newton's Method on the polynomial z^6 - 1. Then:
http://www.flash.net/~nav2/z6minus1_C_0+I.gif     (same function, centered at 0+I), then:
http://www.flash.net/~nav2/z6minus1_Z_2000.gif     (centered at 0+I, zoomed in 2000x), and finally:
http://www.flash.net/~nav2/z6minus1_D_onefourth.gif     (same function, centered at 0, d = 1/4).
Fighting misinformation, one day at a time: The definition is fine, even if only because what fractal means doesn't really have a clear definition. But to be more specific:

I believe Mandelbrot's original definition of a 'fractal' was "an object for which the self-similarity dimension (also called the fractal dimension, basically a simplified special-case of the Hausdorff dimension) exceeds the Euclidean, or embedding dimension". If this sounds overly self-involved, no worries, you can ignore it for now and use the more general definition; Mandelbrot later wanted the term 'fractal' to cover any recursive structures that were not totally self-similar (like the mandelbrot set, as opposed to the utterly self-similar koch curve, as well as the physical examples given). So basically anything that's roughly self-similar on varying scales is fair game.

Like allover said, many of the original 'fractals' evolved from so-called 'pathological functions'... for example, the Weierstrass Function, which is everywhere continous but nowhere differentiable.

That said, all of the fractals that you've seen probably fall into these three basic catagories:

1. Computer-aided Iterative functions on the complex plane (or hypercomplex n-space)
i.e. Mandelbrot, Quaternion animations
2. Reduction/Replacement-based self-similar
i.e. Koch Snowflake
3. Natural Fractal Systems
i.e. Trees

1. Computer-aided Iterative functions on the complex plane (or hypercomplex n-space)

Computer generated fractals aren't really created with any method that outright dismisses Euclidean geometry, if approached from a numerical perspective. Most of the colorful 'fractals' that you see are generated using an iterative formula on the complex plane, which is actually a fairly simple arithmetical process:

Essentially, you take each point on the plane (to the limit of screen resolution), apply a fixed formula to that point, apply the formula to the result, and repeat as many times as is necessary to see if the final result is diverging (always getting bigger), or converging (approaching zero). Points can be assigned different colors based on whether they are divergent or convergent (yields a two color picture), or for a more psychedelic approach, a spectrum of colors can be used that corresponds to the rate of divergence. That's all there is to Mandelbrot set and Julia set fractals, which is what you see the most of. The formulas, zoom and coloring techniques are what varies. I believe both of the linked nodes give the details of some of the formulas used.

This can be called non-Euclidean, since most shapes that are considered Euclidean are far more regular (circles, squares) ... but there is not necessarily any need to call this a 'new geometry'. The only way I think you could really justify calling this a new geometry would be to take the point of view that associates a formula that acts on the complex plane as a sequence of deformations (stretching, displacement, rotation) of said plane.. using this approach, many (all?) Julia sets can be shown to be Euclidean circles that have been iteratively subjected to geometric transformations. This idea of infinetly perverting the immaculate plane (so to speak) probably is dirty enough so as to be against everything Euclid is taken to stand for.

As mentioned above, iterative function fractals don't have to be 2 dimensional. In fact, the concept can be extended to any dimension that is a power of two (2,4,8,16...), by use of 2^n dimensional algebras, such as quaternions (n=2), octonions (n=3) and so on.

You may say: "Ok, but what good is it to have 4 (or more)dimensional fractals when all we can see is three?" The answer is, of course, that we CAN visualize a four dimensional shape... we just need to turn one of the spatial axes into a time axis. That is, an 'image' of a quaternion-fractal is in fact a video; if you've seen a computer animation of a porous, organic fluid appearing out of nowhere, coalescing and twisting then finally fading from view, you've probably seen a flyby of a multidimensional fractal; a 2d representation of a 3d slice of a 4d object as the 4d object passes at a constant rate through a 3d space projected onto a 2d screen. And if you buy that, I've got a suspension bridge in my backyard, looking to sell it, real cheap. On to something simpler.

2. Reduction/Replacement-based self-similar

Most of these are based on taking a simple line drawing, then replacing some section of that line drawing with a reduced-scale version of the whole. As such, all of these are strictly self similar. I'll let the individual nodes of some of the more commonplace ones speak for themselves:

Cantor Ternary Set (Also called Cantor Set or Cantor Dust)

Sierpinski triangle (Variations: Sierpinski gasket, Sierpinski Sponge(3D))

Koch curve, three of which make a Koch snowflake.

Space-filling curves: The Sierpinski curve and the Peano monster curve
The last item deserves special note: These two are curves that manage to twist around so much that they pass within an arbitrarily small distance of any point in a fixed area. In other words, they are curves that have a non-zero area, and are of dimension 2.

3. Natural Fractal Systems

Basically, anything in nature which exhibits the same kind of scaling features as those shown above. Here are some that I can think of of the top of my head:

a) Trees, other branched plants: each branch with sub-branches is reminiscent of the whole. This recursion is, of course, only carried to a finite level.

b) Clouds: These definetly possess the scaling properties of fratcals... Imagine you had a picture of part of a cloud, with no other frames of reference. Could you tell if that part was a meter across versus 10 meters? Versus 100? Versus a kilometer? This example might come from Gleick.
c) Reflections: Putting reflective objects near each other creates an infinite recursion of reflections, reflections of reflections, and so on. Think of a hallway with mirrored walls as the most simple expression of this phenomenom.

d) Coastlines: "How long is the coast of Norway?" Answer: It depends on how close you look. I don't remember the source of this quotation, but it's quite accurate... the measured length of a coastline, being fractal, will increase as the increment with which you measure decreases. This is because there will always be tiny inlets and peninsulas which are smaller then your ruler's tiniest increment, yet still add to the length. If matter was continous as opposed to discrete, a coastline could technically be infinitely long while still enclosing a finite surface area (like the koch snowflake).

e) Lungs and veins: Two fractal structures with similar purposes: the lungs have developped to enclose the largest possible surface area in the smallest possible volume, in order to ensure efficient oxygen absorbtion. I don't remember the exact surface area of the lungs, but it's really big. Honest. Similarly, you want to minimize the volume of blood being pumped while ensuring total distribution: blood vessels are positioned so that no cells are more then a few cell-widths away from fresh blood (think: space-filling curve).

f) Biological systems as a whole: Although DNA is complicated, living things seem hugely more so. This is because DNA is NOT a blueprint, at least not in the strictest sense of the word... there is almost assuredly no 'map' by which the body lays out veins: instead they grow and branch fractally, creating a hugely complicated structure from the very little (implicit) information in the genes. This information-conserving attribute of fractals is now being used in new forms of data compression, another example of developing technologies mirroring naturally developping systems.

g) Matter as a whole: Galaxies orbiting about the galactic cluster, solar systems orbiting about the galactic center, planets orbiting suns, moons orbiting planets, and (more loosely speaking) electrons 'orbiting' the nucleus, nucleons 'orbiting' the nucleic center, bound quarks 'orbiting' their own centers of force... although this is by no means strict self-similarity, it shows an interesting relationship between the complex plane fractals and our universe... consider particulate matter to be points on the n-dimensional spatial continuum (Where n is however many dimensions space-time is currently supposed to have). This space is subjected to deformation and stretching (by the big bang, and accompanying symmetry-breaking), with the fundamental forces becoming distinct from one another and operating on different levels of matter to create the loosely recursive structures we see today.

See also a variety of good nodes at similarity of fractals to natural objects (Thanks to The Alchemist for this link.)

A few notes on Chaos Theory, intended as a correction (counterpoint?) to aesteve77's node:

'Randomness due to multitudinous variables' is something that is commonly mislabeled as 'Chaos'.. really, we've known for a long time that weather is complicated. Lorenz's principle discovery was that simple systems with only a few controlling variables can exhibit chaotic behavior.. He used a simple system of three differential equations (often seen visually as the 3 dimensional 'Lorenz butterfly') to show that there are systems which exhibit aperiodic behavior even if all initial conditions are exactly determined.. that is, systems whose behavior is deterministic but yet never repeats itself (is chaotic, as opposed to random). He also showed that such systems exhibit a phenomenon known as sensitive dependence on initial conditions, which is usually what gets all the press time:

To explain this idea, consider two identical dynamical systems (with chaotic behavior), being systems whose behaviors depend on the exact same rules. 'System' here is extremely general and refers to almost anything that can be isolated to a resonable degree from 'outside influence'. Now, both of these simple systems have a user-selectable initial condition corresponding to one single number, and the condition of the system at any given point in time is also numerically represented. Say the two numbers inputted into each system differ by only a hundredth of a percent. After a few time units pass, and the dynamical rules begin to act, the relative difference between the systems expands to 1 percent. After slightly more time passes, 10 percent, and soon enough, the systems are entirely out of sync... This happens in all chaotic systems, and since both systems are aperiodic (non-repeating), a observer that happened upon the systems would never guess that both the systems began in the same condition, let alone are based on the same rules of behavior. Therefore, even if we could measure all the initial conditions of a sufficiently complex system (say, the weather) to all the accuracy allowed by Heisenberg Uncertainty, the minute error would render predictions longer then a certain time interval ahead totally, irrevocably inaccurate.

Note that there is a direct, non-ambigous connection between fractals and chaos. To learn more about it, see either of the sources listed below:

This is all off the top of my head, but I wanted to clear up a few vague bits and misconceptions, particularly since I think the reality is so much more interesting. I am not, however, a physics major. I am, however, a high school graduate (what a long, strange trip it's been). For more information, read the 'Chaos Hypertextbook' (http://hypertextbook.com/chaos)... or maybe 'Chaos', by James Gleick. If you're at the U of T, look me up, I'll lend you my copy. :)

This is a metanode for the many discussions of fractals that can be found on E2. If I've forgotten any or if you add a new one please /msg me so that I can add it.

Alphabetically:

• Attractor.
• Benoit Mandelbrot.
• Bifurcation.
• Cantor Set.
• Chaos theory.
• Cantor Ternary Set.
• Continuous function fractals.
• Curlicue Fractal.
• Dragon curve.
• fractal dimension.
• fractal image compression.
• Fractal Music.
• fractal transform.
• fractal tree.
• iterated function system.
• Julia set
• Koch Curve.
• Koch snowflake.
• L-systems.
• Lorenz Attractor.
• Lorenz Equations.
• Mandelbrot.
• Mandelbrot Set.
• Menger sponge.
• Peano Curves
• Sierpinski curve.
• Sierpinski triangle.
• strange attractor.

NB: This includes discussions of fractals themselves rather than general discussions of naturally occuring fractals e.g. fractal design of the dramatic human body.

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