Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.3 4.3bsd-beta 6/6/85; site ucbvax.BERKELEY.EDU Path: utzoo!watmath!clyde!burl!ulysses!ucbvax!space From: REM%IMSSS@SU-AI.ARPA (Robert Elton Maas) Newsgroups: net.space Subject: (none) Message-ID: <8603060912.AA02063@s1-b.arpa> Date: Thu, 6-Mar-86 04:24:25 EST Article-I.D.: s1-b.8603060912.AA02063 Posted: Thu Mar 6 04:24:25 1986 Date-Received: Fri, 7-Mar-86 07:38:45 EST Sender: usenet@ucbvax.BERKELEY.EDU Organization: The ARPA Internet Lines: 53 BH> Date: 26 Feb 86 19:55:07 GMT BH> From: hplabs!amdahl!drivax!holloway@ucbvax.berkeley.edu (Bruce Holloway) BH> Subject: Re: Olber's paradox BH> Another solution (maybe): All stellar objects tend to "clump" into BH> solar systems, galaxies, clusters, ad infinitum. So instead of spreading BH> evenly throughout the sky, we just see light from these collections, the BH> scope of said clumps depending on how far away the object(s) is/are. REM> From: Robert Elton Maas REM> Subject: Many solutions to olber's paradox!! (Keep an open mind!) REM> Of course! ... I should have thought of that myself, having workd REM> with Mandelbrot and Gosper and Farmwald and Moravec on fractal stuff REM> at SU-AI... Indeed, if the large-scale clumping of the Universe has REM> sufficiently small fractal dimension, then even in a static and REM> infinite-time Universe you see only a finite amount of light from any REM> point due to inverse-square diminuation and less than square REM> accumulation of stars. JK> Date: 3 Mar 1986 20:46:46-EST (Monday) JK> From: "Josh Knight" JK> Subject: Clumping doesn't fix Olber's paradox JK> I don't think clumping, no matter what its statistical characteristics JK> can avoid the paradox. Basically, if one extends one's line of sight JK> far enough, one finds it ending up on a star, i.e. the entire surface JK> is covered with star surface. You are wrong. See Mandelbrot "The Fractal Geometry of Nature", page 91. If the Hausdorff dimension of the stars is sufficiently low, you'll still see lots of empty-to-infinity space even with infinite stars. True any cone is sure to hit a star (unity probability), but that star fills only a tiny part of the cone so the light coming down that cone at you isn't very bright. Any infinitesimal (mathematical) ray has less-than-unity probability of striking a star, even with infinite total stars. I suspect you don't understand the principle of clustering of low fractal dimension, maybe don't even understand Hausdorff dimension. Have you ever heard of the Cantor ternary set, where you remove the middle third of a segment, and the middle third of the two remaining segments, etc.? If not, study some topology or analysis book that has this simple example. Once you have that, you can study more complicated examples such as needed for Olber's paradox resolution. Do Cantor's set upward (larger scales) as well as smaller scales for a one-dimensional example of clustering in an infinite universe. (Rubber-stamp Cantor's set offset to the right two units, so there's a one-unit gap between original and copy, i.e. the pattern is CxC where C is Cantor's set and X is a completely empty segment, then rubber-stamp that pattern offset by 6 so the pattern is now CxCxxxCxC, then stamp that offset 2*3**2 = 18 units giving CxCxxxCxCxxxxxxxxxCxCxxxCxC, etc. Alternate direction each time so you extend in both directions to avoid having the original set at either end of the universe.) If you grok that example in fullness, you will be prepared to tackle the 3-dimensional case.