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.
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.
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.