**Fractals** are objects that are self similarity at different scales

A tree is fractal because it has a trunk with branches which have branches which have branches...

Fractal is self-similar at all possible scales and is a mathematical concept vs. **fractal-like** which is self-similarity at multiple scales and is what is found in nature.

Leaf veins, galaxy clusters, tree roots, mountain ranges, and web page links are all fractal-like.

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**Benoit Mandelbrot** coined the term fractal from a Latin root.

Mandelbrot was trying to develop a "theory of roughness" to describe the geometry of the real world. He created fractal geometry.

Mandelbrot looked at measuring coastlines on a map (famously Great Britain). With a shorter and shorter "ruler" (and a higher and higher resolution map) you get a longer and longer measurement of the coastline because the "rulers" can  fit into even more nooks and crannies of the rough/rugged coastline.

So the question about how long the coastline is depends on the length of the ruler you use to measure it since it' has self-similar roughness/ruggedness at different scales and is not smooth.

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The [**Koch Curve**](http://en.wikipedia.org/wiki/Koch_snowflake) was invented/discovered in the  early 20th-century by Helge von Koch and is a simple fractal.

Start with 1 segment, a straight line, and iterate a simple rule. The rule is:

At each time interval, take each segment, remove the middle 3rd and replace it with 2 segments that are 1/3 the length of the original segment and joined at the ends.

[KochCurve.nlogo](http://s3.amazonaws.com/complexityexplorer/IntroToComplexity/KochCurve.nlogo)

If we imagine an iterated koch curve as a coastline, we can ask, how long is it at each iteration? Each iteration is equivalent to using a smaller ruler.

**Level - Seg Length - Num Segs - Curve Length**  
0 - L - 1 - L  
1 - L/3 - 4 - (4/3)L  
2 - L/9 - 16 - (16/9)L  
3 - L/27 - 64 - (64/27)L  

Here the "Curve Length" is the "coastline length".

So the curve at level n is:  4^n/3^n * L

The growth of our pattern means a 1 meter line segment at level 1 is ~2 billion miles by level 100. So we'd have 2 billion miles of line segments squeezed into a 1 meter long area. 

Fractal-like structures in nature don't get to level 100, only mathematical fractals do, but it's a dramatic demonstration that the fractal geometry is **space filling** which is a very efficient use of space, and it's why we see fractal patterns in nature in places like the circulatory system, root systems and brains.