Some different mathematical notions of dimension are: correlation dimension, embedding dimension, topological dimension, Hausdorff dimension, fractal dimension, self-similarity dimension, box-counting dimension, capacity dimension, information dimension. (see scientific space)
The self-similarity dimension can be thought of in terms of a photocopy reduction. At what settings and with how many copies can we reproduce the entire structure? If we take the Koch (snowflake) curve, set the reduction to 1/3 and make four copies, we can paste them back together to make the Koch curve. If we repeat this same operation, we have reduced the original by 1/3 x 1/3 = 1/9 and produced 4 x 4 = 16 copies, continuing to 1/27, 64 etc. which we can generalize as a reduction factor of 1/3 to the kth power corresponding to 4 to the kth power pieces. The self-similarity dimension of the Koch curve is log4/log3 = 1.2619.
for an irregular curve Eg coastline of britain, we can determine the power law of the relation between the unit of measurement and the obtained length (this is a log/log graph which will be a straight line) taking the slope of that line and adding 1 will give the self-similarity dimension (approx 1.36 for British coastline, which means it is rougher than Koch curve)
According to Benoit Mandelbrot fractal forms have dimensions that are not integers. Their dimensional ambiguities arise from some of their unusual aspects. For instance, the Peano curve is a line that can be shown to fill a plane. We would usually assign 1 dimension to a line and two dimensions to a plane, but if this line can fill a plane whose area is calculated in two dimensions, it would seem that it cannot be a one-dimensional line (even if infinitely long).