I grasp how fractals can be self-similar and have other weird properties. But I don’t quite get how they can have fractional dimensionality, even though that’s the property they’re named after.
How can a shape have a dimensionality *between*, say, two and three?
In: Mathematics
Actual physics educator here (retired):
A piece of paper is one dimension. But wad it up, and it can only exist that way in three dimensions. Zoom in close enough though, and the paper is still two dimensional. So, you could say the paper exists as a 2.5 dimensional surface. That’s the fractional dimension.
In more practical terms, the surface of a globe is a fractional dimension. Two dimensions that can only exist in three dimensional space. (This also begets “non-euclidean” geometry. A triangle can be drawn from the north pole, to the equator, 1/4 of the way around the equator, then back to the north pole. *Three* *90 degree angles* in one triangle. It’s a fractional dimension, non-euclidean triangle.)
A straight line on a piece of flat paper is one dimensional. A slightly curved line on that two-dimensional surface might be 1.1 dimensional (there are ways to calculate it.) A super squiggly-wiggly line on the flat paper might be 1.9 dimensional. Now wad that paper into a ball, and it gets complicated…
Time dilation due to gravity makes space a fractal dimension somewhere between three and four.
Enjoy!
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