ES> 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.

Even including a "dimension" parameter D, which spacefills when D=2. So, as shown by Peano, you can specify a 2D point with 1 coordinate. Infinitely often, there are three ways to do it. Of course, if you've a high tolerance for disconinuity, just interlace the digits of x and y to make a single number t. --rwg PS, one can find exact values of z(rational) using hashcoding to detect cycles in the recursive (e.g. Koch) definition of the "curve". Equating these to the Fourier series produced two of the recondite identities in www.tweedledum.com/rwg/idents.htm vaticinate <-> inactivate