Say you have a filament that's 1 µm in diameter, and 1 meter long. You want to fill up a 1m^3 (1m W x 1m H x 1m L) space with these, how many of these can you place in such a space? Over a trillion! And thus, the combined km length of these will also be over a billion km. At such small scales things can become very long when summed up.
Everytime I see people talking about the length of their coastlines ...
Look at how long this edge of my fractal is, Ma!
I understand the infinite coastline concept in theory but in practice would you even want to measure a coastline that way? Even if your goal was “perfect” precision, coastlines are variable. You’d have to take an average across tides and that would require zooming out enough to get a normal sounding answer.
It is obvious when you think that 3d volumes scale cubicly and 1d lengths scale linearly. Adding another 1 meter to a cube of said filaments would increase the total length by a power of 3!
How would this compare with packing the volume with DNA ?