Date:Mon, 28 Feb 2000 09:11:03 -0600 (CST)From:Subhash Suri <suri@cs.wustl.edu>To:David Eppstein <eppstein@ics.uci.edu>Subject:Pick's theorem

Thanks for the reply on triangulation, David. I did a search on Pick's theorem, which landed me on your geometry junkyard, but didn't answer the question, so let me ask you this. Is there an analog of Pick's theorem in 3D? In 2D, the theorem says the area of a simple polygon with b boundary lattice points and i interior lattice points is i + b/2 -1. One can ask the question about the volume of a lattice non-convex polyhedron. Is there a relation between volume and the quantities i and b? I would guess may be not, but haven't seen anything about it... thanks. -Subhash

Date:Mon, 28 Feb 2000 08:44:28 -0800 (PST)From:David Eppstein <eppstein@ics.uci.edu>To:suri@cs.wustl.eduSubject:Pick's theorem

Simple example: consider the tetrahedra formed by points (0,0,0) (1,0,0) (0,1,0) (1,1,k). Volume is k/6, each has four corners at grid points and no other boundary or interior grid points. -- David Eppstein UC Irvine Dept. of Information & Computer Science eppstein@ics.uci.edu http://www.ics.uci.edu/~eppstein/

Date:Mon, 28 Feb 2000 10:50:40 -0600 (CST)From:Subhash Suri <suri@cs.wustl.edu>To:David Eppstein <eppstein@ics.uci.edu>Subject:Re: Pick's theorem

Thanks. You are a great counterexample oracle! May be you should link this to Pick's theorem page... -Subhash On Mon, 28 Feb 2000, David Eppstein wrote: > Simple example: consider the tetrahedra formed by points (0,0,0) (1,0,0) > (0,1,0) (1,1,k). Volume is k/6, each has four corners at grid points > and no other boundary or interior grid points. > -- > David Eppstein UC Irvine Dept. of Information & Computer Science > eppstein@ics.uci.edu http://www.ics.uci.edu/~eppstein/ >

From:Bill Dubuque <wgd@nestle.ai.mit.edu>To:sci.mathSubject:Pick's formula in higher dimensions [was: Russian math olympiad problem on lattice]Date:Wed, 23 Aug 2000 23:08:02 -0700

Hull Loss Incident <x@y.z.com> wrote: > > > |Pick's theorem is that the area of a lattice polygon > > |without holes is #points inside + 1/2(#points on boundary) - 1. > > > also add that the formula can be extended to polygons with multiple > holes and connected components, but not in an obvious way to 3-dim > For a tetrahedron the volume is not determined by the number of > lattice points in the interior, faces, vertices and edges. There are some beautiful higher-dimensional extensions of Pick's formula based upon recent deep work in combinatorial algebraic geometry, in particular around toric varieties, with contributions by Brion, Cappell, Khovanskii, Morelli, Pommersheim, Shaneson, etc. For a readable introduction see Morelli's paper [1], and [2] for an online start. -Bill Dubuque [1] Morelli, Robert. Pick's theorem and the Todd class of a toric variety. Adv. Math. 100 (1993), no. 2, 183--231. MR 94j:14048 [2] http://www.emis.math.ca/EMIS/journals/ERA-AMS/1996-01-001/1996-01-001.html

From:Vladimir Lazic <lazicv@verat.net>To:sci.mathSubject:Re: Pick's Theorem on VolumeDate:Sat, 21 Oct 2000 12:00:42 -0700

A sort of, yes! Generalizations of Pick's theorem were delivered by Reeve (for the 3-dimensional case) and Macdonald (in general). See: 1. I. G. Macdonald, The Volume of a lattice polyhedron, Proc. Camb. Phil. Soc. 59 (1963), 719-726. 2. J. E. Reeve, On the volume of lattice polyhedra, Proc. London Math. Soc. (3), 7 (1957), 378-395. 3. J. E. Reeve, A further note on the volume of lattice polyhedra, J. London Math. Soc. 34 (1959), 57-62. Hamilton Davis <hamiltonious@hotmail.com> wrote in message news:8sr8qt$foc$1@bob.news.rcn.net... > Can a variation of Pick's theorem be applied to the volume of a polyhedron?