The Koch Curve 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.
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.