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From: wpt@fisher.UUCP (Bill Thurston)
Newsgroups: net.math
Subject: Re: Re: Strange Shapes
Message-ID: <429@fisher.UUCP>
Date: Wed, 28-Nov-84 18:07:51 EST
Article-I.D.: fisher.429
Posted: Wed Nov 28 18:07:51 1984
Date-Received: Thu, 29-Nov-84 05:50:26 EST
References: <3489@ecsvax.UUCP> <121@talcott.UUCP> <6067@brl-tgr.ARPA>
Organization: Princeton Univ. Statistics
Lines: 35

>> Painting an object means putting a coat of paint *of uniform thickness* on
>> the object.  All objects with finite volume and infinite surface area have
>> arbitrarily small cracks and crevices where a coat of paint "does not fit".

>"On" an object in practice means "within cohesive distance" of the object.
>There are always tiny cracks in practice but they are covered by paint.
>This means that it is easy to paint a snowflake; just immerse it.

>And there you have the difference between a physicist and a mathematician.

>This is not as facetious as it sounds.  If you admit conventional concepts
>of infinity, set theory, and the like, then you support such paradoxes as
>the Banach-Tarski dismantling of a sphere into a few congruent parts that
>can be reassembled into a smaller sphere, and so forth.  There is more to
>reality than that, since in practice no real sphere can be so rearranged.

There is a huge difference between the kind of infinity in a snowflake curve,
which serves as a good model for many things in nature, and the transfinite
processes used to derive the Banach-Tarski paradox.  Certainly, no actual
physical model of a cylinder with a snowflake cross-section can exist, 
because of quantization phenomena at small scales. Even if it existed, there
it could be painted by dipping it in a bucket of paint, but the paint would
obscure the fine details.  Nonetheless, there is a real interpretation to
the idea that such a cylinder would have infinite surface area, and would
require infinitely much paint -- in units of (volume of paint / thickness of
the coat).  The physical interpretation is that as the thickness of the
coat goes down, and as the accuracy of the model correspondingly increases,
this ratio goes to infinity: in fact, it obeys a very regular scaling law,
increasing as a certain inverse power of the thickness (which depends on
the fractional dimension of the snowflake curve, in a certain sense.)

The Banach-Tarski paradox is red herring in this discussion.

		Bill Thurston, Mathematics Dept, Princeton University
		{princeton , allegra }!fisher!wpt