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First it's useful to understand why e.g. a Koch curve has fractional dimension. The answer informally is that it has "too much area" to be just a line, but it's

First it's useful to understand why e.g. a Koch curve has fractional dimension. The answer informally is that it has "too much area" to be just a line, but it's "too much a line" to be a 2d figure. Just notice that its length (1D measure) is infinite (just like a square has "infinite length"), so its dimension must be >1D, but also its area) 2D measure is zero (just like a line has 0 area), so it must also be <2D. Then if you want to make this intuition precise, you need to look into the box-counting dimension, and then the Hausdorf dimension.

westoncb 7y ago
That makes sense to me. There was a specific thing I was looking for in my original question which was to understand how related vs. arbitrary the shared term 'dimension' is. I think I found an answer after browsing a few Wikipedia articles—but happy to be corrected if it doesn't sound right: The usual notion of dimension that people think of is topological only; fractal dimension (i.e. Hausdorf dimension) is also concerned with metric properties. These metric properties are what creates the need/possibility for fractional measures. In that case, the scenario I was considering with 'fractional axes/directions' is not implied by the fractal dimension concept: fractal dimension is dealing with new subject matter, not just an extension of the old concept to deal with real numbers.