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
--- 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
I seem to recall that pretty good solutions to the "Travelling Salesman" problem can be gotten by visiting the cities in the order traced out by certain space-filling curves. I don't know if this works for the sphere or not, but perhaps postal codes could be so chosen this way. On the other hand, phone area codes seem to have been chosen with some sort of "coding theory" algorithm, so that points close to one another would have area codes as different as possible. At 04:08 PM 7/22/2004, Mike Stay 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
Quoting Henry Baker <hbaker1@pipeline.com>:
On the other hand, phone area codes seem to have been chosen with some sort of "coding theory" algorithm, so that points close to one another would have area codes as different as possible.
To understand area codes you need to know about real old-style dial phones. They optimized the time it took to dial the high population area codes. New York was 212; Los Angeles 213. States with more than one area code had codes with a "1" as the central digit. States with only a single code used a zero. Three digit codes ending in zero were avoided as exchange codes for many years, on the assumption that they would be required as area codes eventually. They were used intially as additional area codes, but there were far too few of them, so area codes with any three digits came into use a few years ago.
participants (4)
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