Len:

> One sentence you quoted stumped me and I was wondering if you could
> explain what was meant. I understand the mathematical concept, but
> not with respect to coastlines.
> "the measured length of coastlines tended toward infinity as the base
> unit of measurement went to zero"

This concept goes back to Mandelbrot's observation that Richardson's
coastline data meet the criteria for fractals, and that the "true" length of
a fractal curve is infinite.
It turns out that the logarithm of the measured length of most coastlines is
inversely proportional to the logarithm of the "divider setting" - how far
apart the legs of your divider are set as you step out the length on the map
(the use of the phrase "unit of measurement" in this setting is confusing).
Finer divider settings pick up more twists and turns, and so return a
greater length.
The attached graph is taken from Rudy Rucker's "Mind Tools" book, and it
shows that the correlation is linear on a log-log graph. Since zero is an
infinite distance off the left edge of the logarithmic graph, any upward
trend will take the measured coastal length to infinity as the divider
setting reaches zero. (This is of course entirely true for mathematical
entities like fractals, but reaches a practical limit for real objects like
coasts.)
The graph also plots the Portuguese border that Kathryn alluded to, showing
that it does indeed behave like a fractal over a range of divider settings.
I would presume that other borders will behave more or less the same way if
they are defined by natural features like water courses or watersheds, but
not if (as Kathryn says) they are defined as a series of straight-line
segments (in which case the graph will become horizontal when the divider
setting falls below the length of the shortest line segment).

Grant