--- Mike Stay <staym@clear.net.nz> wrote:
To specify a point in a two-dimensional space, we use two coordinates. I can't find any reference on how to specify a point in a fractional-dimensional space. Do any funsters know?
http://www.cut-the-knot.org/do_you_know/dimension.shtml gives an example of a fractal with rational Hausdorff dimension, 3/2. Apparently, one can specify three points on the curve with two coordinates. I've no idea how one would approach the Koch curve.
-- Mike Stay staym@clear.net.nz http://www.cs.auckland.ac.nz/~msta039
The number of coordinates needed should depend on the topological dimension, not the Hausdorff dimension. The Koch curve is homeomorphic to a circle, and can be represented as a Fourier series. z(t) = x(t)+ i y(t) = sum(a[n] exp(i n t),n=-inf..inf). A point on the curve is specified by the single coordinate t. The representation is not unique, because you are free to choose the "speed s(t)" at which the curve is traced. Bill Gosper has found explicit expressions for the Fourier coefficients. Gene __________________________________ Do you Yahoo!? Take Yahoo! Mail with you! Get it on your mobile phone. http://mobile.yahoo.com/maildemo