X-Google-Language: ENGLISH,ASCII-7-bit X-Google-Thread: f996b,1c2897ed56918deb X-Google-Attributes: gidf996b,public X-Google-ArrivalTime: 2001-09-18 08:09:41 PST Path: archiver1.google.com!newsfeed.google.com!newsfeed.stanford.edu!news.tele.dk!small.news.tele.dk!195.54.122.107!newsfeed1.bredband.com!bredband!newsfeed1.telenordia.se!algonet!newsfeed1.funet.fi!newsfeeds.funet.fi!nntp.inet.fi!inet.fi!levitin.saunalahti.fi!uutiset.saunalahti.fi!not-for-mail From: Ilmari Karonen Newsgroups: alt.ascii-art Subject: Re: Mike Throll 5 Date: 18 Sep 2001 15:09:16 GMT Organization: (dis)Order of the Holy Spoon (or whatever) Lines: 54 Message-ID: <1000823107.6971@itz.pp.sci.fi> References: <3ba5bac8.2311763@News.CIS.DFN.DE> <9o5cf9$m7f$1@news.fas.harvard.edu> <3ba710ac.2496292@News.CIS.DFN.DE> Reply-To: Ilmari Karonen NNTP-Posting-Host: simpukka.saunalahti.fi X-Trace: tron.sci.fi 1000825756 22255 195.74.0.20 (18 Sep 2001 15:09:16 GMT) X-Complaints-To: newsmaster@saunalahti.fi NNTP-Posting-Date: 18 Sep 2001 15:09:16 GMT User-Agent: postit.pl 0.05 Xref: archiver1.google.com alt.ascii-art:7609 In article <3ba710ac.2496292@News.CIS.DFN.DE>, Timofei Shatrov wrote: >On 17 Sep 2001 17:38:49 GMT, uncle monty tried to >confuse everyone with this message: > >>Given an initial separation of the plane RxR into a subset A and not-A, >>and an iterative process which at each step assigns membership in A to >>each point outside A whose circular neighbourhood of radius 1 has a >>greater intersection with A than with not-A, and removes each point >>previously within A whose circular neighbourhood of radius 1 has a greater >>intersection with not-A than with A, is it possible to choose initial A >>such that its area will increase by a factor of 1000 or more during the >>process? >> >>Tell me, does the blot have to be simply connected in the beginning? >I think yes. It also has to be finite. Obviously it must have a well-defined finite area, for the question to make sense. Other than that, finiteness makes little difference -- see below. >>Do we need the final limiting area to be more than 1000 times the >>initial area, or can some intermediate area be 1000 times? > >I think all finite figures will eventually die. You need to make it 1000 times >larger only at one turn. Yes, all finite figures must eventually vanish. It's easy to see that all convex figures will vanish, and that if a figure vanishes, all its sub-figures must also vanish. As any finite figure may surrounded by a convex hull, the proof follows. The use of the word "finite" above is slightly vague, but things are simplified by the fact that any figure with a finite area must after one iteration be a subset of a circle with a finite diameter. Proof by contradiction left as an exercise for the reader. This offers no direct help with the original problem, of course. >>I imagine it is some >>impossible fractal shape to begin with. > >No way. It's not very complicated. Just try to invent blot that can increase and >then magic word "asymptotics". Ah, I think I got it, but I'm not sure. To verify my idea, I'd need a formula for the radius, after one iteration, of a solid circle with an original radius of R. This involves trigonometry, and I'm lousy at it. -- Ilmari Karonen - http://www.sci.fi/~iltzu/ "Control: cmsg newgroup sci.math.tasteless" -- Red Drag Diva in the monastery