From: Clive Tooth Subject: Re: Just another paradox Date: Fri, 14 Apr 2000 23:13:35 +0100 Newsgroups: sci.math Summary: [missing] denis-feldmann wrote: > Jonathan Hoyle a écrit dans le message : > 38F65D37.567D@kodak.com... > > Apparently I did not communicate myself well. I agree with you that it > > is not paradoxical. It is not at all hard for me to understand that an > > infinite area of two dimensions can be covered by a finite amount of > > three dimensional paint. Letting the paint get infinitely thin (which > > is what surface area is to a 3D object) very easily allows this. > > > > My analogy about space-filling curves was to show how an infinite amount > > of one dimensional length is covered with a finite amount of two > > dimensional area. This too is not a paradox. I was attempting to show > > that when you compare measures of differing dimensions, it is not at all > > surprising to have this kind of result. But perhaps it is just too > > confusing an analogy so I'll withdraw it. > > > > Let me try another analogy: Painting an ifinite amount of surface area > > with a finite amount of paint should not be surprising, since a finite > > amount of surface area can be painted with paint of volume 0. Surface > > area has no thickness, so the volume of the surface area itself is 0. > > For example, both the balls x^2+y^2<=1 and x^2+y^2<1 have precisely the > > same volume, even though the second sphere has its surface removed. > > Thus, the volume of paint needed to cover precisely the sphere x^2+y^2=1 > > (and nothing else) is exactly 0. > > > > Hope that helps. > > This should imply that the opposite situation (an infinite voulme bounded bu > a finite surface) never happens (if it did, *that* would be a real paradox). > It should be obvious, but i don't see any idea for a proof. Help, anyone ! It is, apparently, possible to have an arbitrarily large volume bounded by an arbitrarily small area. Gelbaum and Olmsted mention an example due to Besicovitch [in Besicovitch, On the definition and value of the area of a surface, Quarterly Journal of Mathematics, 16 (1945) 86-102]: For two positive numbers epsilon and M, a surface S in three-dimensional space such that: (a) S is homeomorphic to the surface of a sphere, (b) The surface area of S exists and is less than epsilon, (c) The three-dimensional Lebesgue measure of S exists and is greater than M. The construction involves "tubular connections among faces of cubes". -- Clive Tooth http://www.pisquaredoversix.force9.co.uk/ End of document ============================================================================== From: Jonathan Hoyle Subject: Re: Just another paradox Date: Sat, 15 Apr 2000 19:50:47 -0400 Newsgroups: sci.math >> This should imply that the opposite situation (an infinite voulme >> bounded bu a finite surface) never happens (if it did, *that* would >> be a real paradox). It should be obvious, but i don't see any idea >> for a proof. Help, anyone ! Hmmm...interesting question. I thought it was true that a bounded 3D connected set with positive volume must have a surface area greater than or equal to that of a sphere with the same volume. (Can someone either verify or disprove this assumption?) If the above is true, then there is an upper bound on the volume of any bounded 3D connected object with a given surface area, S. However, if this object has non-measurable volume (e.g. some Banach-Tarski-like constructed object), this may not apply. I am curious as to the resolution of this myself. Thanks in advance. Jonathan