Generalization of Pick's theorem for surface of polyhedra

Abstract

The Pick's theorem is one of the rare gems of elementary mathematics because this is a very innocent sounding hypothesis imply a very surprising conclusion (Bogomolny 1997). Yet the statement of the theorem can be understood by a fifth grader. Call a polygon a lattice polygon if the co-ordinates of its vertices are integers. Pick's theorem asserts that the area of a lattice polygon P is given by A(P) = I(P) + B(P) / 2 - 1 = V(P) - B(P) / 2 - 1 where I(P), B(P) and V(P) are the number of interior lattice points, the number of boundary lattice points and the total number of lattice points of P respectively. It is worth to mention that the I(P) (understand like digital area) is digital mapping standard in USA since decade (Morrison, J. L. 1988 and 1989). Because the Pick's theorem was first published in 1899 therefore our planned presentation had timing its 100 anniversary. Currently it has greater importance than realized heretofore because of the Pick's theorem forms a connection between the old Euclidean and the new digital (discrete) geometry. During this long period lots of proof had been made of Pick's theorem and many trial of its generalization from simple polygons towards complex polygon networks, moreover tried to extend it to the direction of 3D geometrical objects as well. It is also turned out that nowadays the inverse Pick's formulas comes to the front instead of the original ones, consequently of powerful spreading the digital geometry and mapping. Today the question is not the old one: how can we produce traditional area without co-ordinates, using only inside points and boundary points. Just on the contrary: how is it possible to simply determine digital boundary and digital area (namely the number of boundary points and inside points) using known co-ordinates of vertices. The inverse formulas are: B(P)=ΣGCD (AX, AY, AZ) (1D Pick's theorem) and I(P)=A(P)-B(P)/2+1 (2D Pick's theorem) where GCD is the Great Common Divisor of the co-ordinate differences of two-two neighboring vertices. The our main object is not these formulas to present, but we desire to show that the Pick's theorem (after adequate redrafting) indeed valid for every spatial triangle which are determined by three arbitrary points of a 3D lattice. The original planar theorem is only a special case of it. However if it is true then its valid not only for triangles but all irregular polygons also which are lying in space and have its vertices in spatial lattice points. Finally if the extended Pick's theorem is true for all face of a lattice polyhedron then it is true for total surface as well. Consequently we developed so simple and effective algorithms which solve enumeration tasks without the time- and memory-wasting immediate computing. These algorithms make possible that using the vertex-co-ordinate list and the topological description of a convex or non-convex polyhedron (cube, prism, tetrahedron etc.) getting answer many elementary questions. For example, how many vaxels can be found on the complex surface of a polyhedron, how many on its edges or on its individual faces. We succeeded to extend our results also to the surface of non-cornered geometric objects (circle, sphere, cylinder, cone, ellipsoid etc.), but anyway, this have to be object of another presentation.