From: Gerry Myerson Subject: Re: "Pick's Theorem" (?) Date: Fri, 15 Dec 2000 09:27:20 +1000 Newsgroups: sci.math Summary: [missing] In article <91bdc8$eoo$1@nnrp1.deja.com>, brak729@my-deja.com wrote: > A theorem that I have run into a few times (I believe it's called > Pick's Theorem or something) is this: > > If a polygon has all of its vertices on lattice points, its area is > I + B/2 - 1 > Where I = the number of points found inside the polygon and B = > the number of points found on the boundary of the polygon. Does > anyone know any proofs of this? Any other information would be > nice as well, for example, who discovered this fact. This came up a couple of years ago & Bill Dubuque posted a nice reply which I append. After Bill's reply there's a bibliography which I picked (no pun intended) up somewhere that has some overlap with Bill's but also some other items. Gerry Myerson (gerry@mpce.mq.edu.au) ****************************************** From: Bill Dubuque Newsgroups: k12.ed.math,sci.math Subject: Pick's Theorem: Area of Lattice Polygon [was: Geoboards - Pic's Theorem] Date: Wed, 03 Jun 1998 00:25:19 GMT John Van't Land wrote to k12.ed.math on 31 May 1998: | | When I first started teaching math over 25 years ago, I remember using | geoboards and learning about Pic's Theorem or something like that. | The theorem said that on a geoboard, to find the area of any enclosed | region (it need not be a quadrilateral) you add the number of nails | which the rubber band touches, subtract the number of untouched nails in | the interior, and add one (or something like this--I don't remember the | exact way it was done). | | Does anyone know what I'm trying to describe. Was it Pic's Theorem, | Pik's Theorem, Piques' Theorem...? It's known as Pick's Theorem (G. Pick, 1900; Jbuch 31, 215). I have appended below some references to expository papers. A lattice point is a point with integral coordinates, and a lattice polygon is one whose vertices are lattice points. Pick's Theorem says that the area of a simple lattice polygon P is given by I + B/2 - 1, where I is the number of lattice points in the Interior of P and B is the number of lattice points on the Boundary of P. Pick's Theorem is equivalent to Euler's formula and closely connected to Farey series; it may be generalized to non-simple polygons and to higher dimensions, see the papers cited below. To help remember the correct formula, you can check it on easy cases (unit square, small rectangles, etc) or, better, you can view how it arises from additivity of area. One can view Pick's formula as weighting each interior point by 1, and each boundary point by 1/2, except that two boundary points are omitted. Now suppose we are adjoining two polygons along an edge as in the diagram below. Let's check that Pick's formula gives the same result for the union as it does for the sum of the parts (and thus it gives an additive formula for area, as required). 1/2 1/2 1/2 1/2 ... - @ @ - ... ... - @ @ - ... / \ / \ / \ / \ 0 @ @ 0 @ 0 | | . 1/2 @ @ 1/2 . . @ 1 . . | + | . => . . . 1/2 @ @ 1/2 . . @ 1 . | | 0 @ @ 0 @ 0 \ / \ / \ / \ / ... - @ @ - ... ... - @ @ - ... 1/2 1/2 1/2 1/2 The edge endpoints we choose as the two omitted boundary points. The inside points on the edge were each weighted 1/2 + 1/2 on the left, but are weighted 1 on the right since they become interior. All other points stay interior or stay boundary points, so their weight remains the same on both sides. So Pick's formula is additive. -Bill Dubuque Bruckheimer, Maxim; Arcavi, Abraham. Farey series and Pick's area theorem. Math. Intelligencer 17 (1995), no. 4, 64--67. MR 96h:01019 Grunbaum, Branko; Shephard, G. C. Pick's theorem. Amer. Math. Monthly 100 (1993), no. 2, 150--161. MR 94j:52012 Morelli, Robert. Pick's theorem and the Todd class of a toric variety. Adv. Math. 100 (1993), no. 2, 183--231. MR 94j:14048 Varberg, Dale E. Pick's theorem revisited. Amer. Math. Monthly 92 (1985), no. 8, 584--587. MR 87a:52015 Liu, Andy C. F. Lattice points and Pick's theorem. Math. Mag. 52 (1979), no. 4, 232--235. MR 82d:10042 Haigh, Gordon. A "natural" approach to Pick's theorem. Math. Gaz. 64 (1980), no. 429, 173--180. MR 82b:51001 DeTemple, Duane; Robertson, Jack M. The equivalence of Euler's and Pick's theorems. Math. Teacher 67 (1974), no. 3, 222--226. MR 56 #2854 Gaskell, R. W.; Klamkin, M. S.; Watson, P. Triangulations and Pick's theorem. Math. Mag. 49 (1976), no. 1, 35--37. MR 53 #3881 ---------------------------- message approved for posting by k12.ed.math moderator k12.ed.math is a moderated newsgroup. charter for the newsgroup at www.wenet.net/~cking/sheila/charter.html submissions: post to k12.ed.math or e-mail to k12math@sd28.bc.ca D. Ren et al. (1993). A fast Pick-type approximation for areas of H-polygons. _Amer. Math. Monthly_ 100(7), 669--673. Ian Stewart. _Another Fine Math You've Got Me Into_. Freeman, 1992. (see ch.5) H.S.M. Coxeter. _Introduction to Geometry_. Wiley, 1961. Hugo Steinhaus. _Mathematical Snapshots_. Bruckheimer, Maxim; Arcavi, Abraham. Farey series and Pick's area theorem. Math. Intelligencer 17 (1995), no. 4, 64--67. MR 96h:01019 Grunbaum, Branko; Shephard, G. C. Pick's theorem. Amer. Math. Monthly 100 (1993), no. 2, 150--161. MR 94j:52012 Morelli, Robert. Pick's theorem and the Todd class of a toric variety. Adv. Math. 100 (1993), no. 2, 183--231. MR 94j:14048 Varberg, Dale E. Pick's theorem revisited. Amer. Math. Monthly 92 (1985), no. 8, 584--587. MR 87a:52015 Liu, Andy C. F. Lattice points and Pick's theorem. Math. Mag. 52 (1979), no. 4, 232--235. MR 82d:10042 Haigh, Gordon. A "natural" approach to Pick's theorem. Math. Gaz. 64 (1980), no. 429, 173--180. MR 82b:51001 DeTemple, Duane; Robertson, Jack M. The equivalence of Euler's and Pick's theorems. Math. Teacher 67 (1974), no. 3, 222--226. MR 56 #2854 Gaskell, R. W.; Klamkin, M. S.; Watson, P. Triangulations and Pick's theorem. Math. Mag. 49 (1976), no. 1, 35--37. MR 53 #3881