Pick's TheoremDate: 2/8/96 at 15:19:3 From: JimLindsay Subject: Pick's theorem Dear Dr. Math, I was wondering what Pick's theorem is? Can you tell me? Thanks so much! Sincerely, Brian Lindsay Date: 2/8/96 at 16:22:33 From: Doctor Sarah Subject: Re: Pick's theorem Hi there - There are some useful Web pages on Pick's theorem. I'll include some excerpts but you'll want to go look up the images yourself. You can find more pages by using Alta Vista http://altavista.digital.com/ to search the Web. (1) http://www.mcs.drexel.edu/~crorres/Archimedes/Stomachion/Pick.html "Pick's theorem provides an elegant formula for the area of a simple lattice polygon: a lattice polygon whose boundary consists of a sequence of connected nonintersecting straight-line segments. The formula is Area = I +B /2 - 1 where I = number of interior lattice points and B = number of boundary lattice points For example, the area of simple lattice polygon in the figure is Area = 31 + 15 /2 - 1 = 37.5 The original result can be found in Georg Pick "Geometrisches zur Zahlenlehre" Sitzungber. Lotos, Naturwissen Zeitschrift Prague, Volume 19 (1899) pages 311-319. Recent proofs and extensions of Pick's theorem can be found in W. W. Funkenbusch "From Euler's Formula to Pick's Formula using an Edge Theorem" The American Mathematical Monthly Volume 81 (1974) pages 647-648 Dale E. Varberg "Pick's Theorem Revisited" The American Mathematical Monthly Volume 92 (1985) pages 584-587 Branko Grunbaum and G. C. Shephard "Pick's Theorem" The American Mathematical Monthly Volume 100 (1993) pages 150-161" (2) http://134.121.112.29/fair_95/gym/hm022.html "Pick's Theorem states that the area of any simple polygon whose vertices are lattice points in the plane is given by this formula: A=Vi + 1/2 Vb - 1 where Vi denotes the number of lattice points in the interior of the polygon and Vb denotes the number of lattice points which occur in the boundary between the vertices. See figure..." There's more good information on methods in this paper. Does this help? -Doctor Sarah, The Math Forum |
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