[lit-ideas] Re: How long is the coast of Britain?
- From: "Richard Henninge" <RichardHenninge@xxxxxxxxxxx>
- To: <lit-ideas@xxxxxxxxxxxxx>
- Date: Thu, 21 Oct 2010 06:31:40 +0200
From Wikipedia:
For many shapes that are often considered in mathematics, physics and other
disciplines, the Hausdorff dimension is an integer. However, sets with
non-integer Hausdorff dimension are important and prevalent. Benoît
Mandelbrot, a popularizer of fractals, advocates that most shapes found in
nature are fractals with non-integer dimension, explaining that "[c]louds
are not spheres, mountains are not cones, coastlines are not circles, and
bark is not smooth, nor does lightning travel in a straight line." [1]
Hold that thought: Mandelbrot (as a heuristic, note that his name is
literally almond-bread and that he gave himself a supernumerary middle
initial B. that stands for nothing--or should that be B without the
vestigial period, and not initial, since initial of nothing?), who is a
"popularizer" (like few mathematicians) of fractals, his word, "advocates,"
like the sophists in Ancient Athens with their beautiful set speeches, paid
to be suasive (ask yourself why the Wiki-writer couldn't bring herself [a
nod to Walter's pronominally feminine world in which everybody is she] to
have him "argue" like everybody else, "that most shapes found--"Eureka," I
found it, in Greece!--in nature are fractals, while "explaining" that
"clouds are not spheres," mountains cones, coastlines circles, bark smooth,
nor lightning straight-lined. Well, d'oh. Wouldn't it be pretty to think so,
though, like the things dreamt of in our philosophy, our ideal geometrical
forms? But neither-neither are they obedient to any other quadratically
expressed function discovered, or better, shaped out of airy nothing by
giving it a name--just a more complexly chaotic-seeming name. In every chunk
of the real "separable" world (the ideal mathematical equivalent of which,
the topologist's "separable metric space," can be conceived of as a sort of
finer, uniform tofu) one could carve out, so to speak, scrimshaw of
intricate fractal forms, but our delight in such forms is that we (with our
lightning robotic assistant, or personal extension, the semiconductor
processor) can generate forms that--dare we say so--beggar nature's own
forms in their complexity, but a complexity that we can explain--flatten
out, unpleat--into its every nook and cranny--we can even write out the code
of it, give you a copy, you could even do it yourself, by the numbers, as
many times as you want. See if your vaunted natural world can do that, with
such ease and grace! Henry Ford meets fields of daffodils and begins pumping
them out into the new animated world in vast arrays of numberless smileys.
An early paper by Benoit Mandelbrot entitled How Long Is the Coast of
Britain? Statistical Self-Similarity and Fractional Dimension and subsequent
work by other authors have claimed that the Hausdorff dimension of many
coastlines can be estimated. Their results have varied from 1.02 for the
coastline of South Africa to 1.25 for the west coast of Great Britain.
However, 'fractal dimensions' of coastlines and many other natural phenomena
are largely heuristic
make that totally heuristic, essentially, quintessentially heuristic
and cannot be regarded rigorously as a Hausdorff dimension. It is based on
scaling properties of coastlines at a large range of scales, but which does
not however include all arbitrarily small scales, where measurements would
depend on atomic and sub-atomic structures, and are not well defined.
The best prophylaxis to brake these hubristic heuristic tendencies is
Wittgenstein's checker: The world is everything that is the case, and that
of which we cannot speak, of that we must remain silent. There will be no
end to the coast of Britain if we start "finding" programmable patterns in
it. Fractals may alter the length of that coastline, as people change
people-based impressions, definitions, programs, sentences, but they are not
about to make a new
coastline in the world about which we can say, lo, it is the case that this
coastline is fractal, without being accusable of making a metaphor, giving
an illustration, speaking in parables, and being approximate, and human, all
too human.
Richard Henninge
University of Mainz
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