RE: [math-fun] vaguely remembered carnival game
Hello! Let D be the largest such disk. If D is covered, then in particular its boundary C must be covered. To cover a circular subarc of length 2*pi*r/4 requires a disk of radius at least r*sin(pi/4). Therefore r is at least sqrt(2). Equality is easy to see; hence the answer to your question is yes. The problem involving five unit disks gives rise to Neville's constant. See Proc. London Math. Soc. 14 (1915) 308--326 for his original paper. See my book "Mathematical Constants", pp. 484--488 for the latest developments. Best wishes, Steve Finch http://pauillac.inria.fr/algo/bsolve/
From: "R. William Gosper" <rwg@tc.spnet.com> Reply-To: math-fun <math-fun@mailman.xmission.com> To: math-fun@mailman.xmission.com Subject: [math-fun] vaguely remembered carnival game Date: Thu, 15 Apr 2004 01:03:30 -0700 (PDT)
Is r=sqrt 2 the largest disk coverable with four unit disks? --rwg
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The problem involving five unit disks gives rise to Neville's constant. ~ .6093828640807096546777101695986736315096819515809997683472565, a root of 1296*x^8+2112*x^7-3480*x^6+1360*x^5+1665*x^4-1776*x^3+22*x^2-800*x+625 . See Proc. London Math. Soc. 14 (1915) 308--326 for his original paper. See my book "Mathematical Constants", pp. 484--488 for the latest developments. Best wishes, or for the impecunious, http://www.mathsoft.com/mathresources/constants/geometryconstant/article/0,,...
participants (2)
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R. William Gosper -
Steven Finch