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).

# dimension

in fractal