Relationship between two objects with the same fractal dimension.

Question

I've been doing some research into fractal dimension (namely Minkwoski-Bouligand dimension) for my university dissertation and have found two seemingly unrelated objects that have the same Minkowski-Bouligand dimension. I have tried to draw parallels between these objects but I can't seem to draw a conclusion. I am aware that we see fractal dimension as a measure of 'roughness' and therefore these objects are equally as 'rough' but I don't really know what this shows. Any help/guidance is greatly appreciated here. Thank You.

Answer 1

You cannot generally conclude, simply because two sets have the same fractal dimension, that they are concretely related in some other way. Consider the following set, for example:

This set consists of two copies of itself - both scaled by the factor $1/2$ and one rotated relative to the whole. If you know just a bit about similarity dimension, then you probably know that this means that the set has dimension $$ \frac{\log(2)}{\log\left(\frac{1}{1/2}\right)} = 1. $$ But the set certainly doesn't look anything like a line segment, which also has dimension 1. More concretely, this set is totally disconnected while a line segment is connected.

On the other hand, you can use fractal dimension to conclude that two sets are concretely dissimilar in some sense.