Generalization of Pick's theorem for surface of polyhedra

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 (ÄX, ÄY, ÄZ) (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.

Mighty is geometry; joined with art, resistless. Euripedes Geometry is the art of correct reasoning on incorrect figures. George Polya In this chapter we present some intriguing results, and their delightful proofs, about some of the simplest geometric configurations in the plane. These include figures consisting solely of points and lines, including those constructed from the lattice points in the plane. We will deal with structures such as triangles, quadrilaterals, and circles in later chapters. Pick's theorem Pick's theorem is admired for its elegance and its simplicity; it is a gem of elementary geometry. Although it was first published in 1899, it did not attract much attention until seventy years later when Hugo Steinhaus included it in the first edition of his lovely book Mathematical Snapshots [Steinhaus, 1969]. Georg Alexander Pick (1859–1942) was born in Vienna but lived much of his life in Prague. Pick wrote many mathematical papers in the areas of differential equations, complex analysis, and differential geometry. Sadly, Pick was arrested by the Nazis in 1942 and sent to the concentration camp at Theresienstadt, where he perished. A lattice point in the plane is a point with integer coordinates, and a lattice polygon is a polygon whose vertices are lattice points. A polygon is simple if it has no self-intersections. Pick's theorem gives the area A ( S ) of a simple lattice polygon S in terms of the number i of interior lattice points and the number b of lattice points on the boundary: A ( S ) = i + b /2-1.

where Vi and Vb, respectively, denote the number of lattice points in the interior and on the boundary of P. Observe that Vb includes, in addition to the vertices, any lattice points which occur on the boundary between the vertices. An interesting proof of Pick's theorem is contained in [3]. The proof centers around showing that the area of a so-called triangle is 1/2; a primitive triangle has no lattice points inside or on the boundary except for the (non-collinear) vertices themselves. It is not difficult to convince oneself that any simple polygon P can be decomposed into primitive triangles by appropriately joining up its lattice points with non-intersecting segments. For such a triangulation, Pick's theorem merely gives

Pick's theorem is used to prove that if $P$ is a lattice polygon (that is, the convex hull of a finite set of lattice points in the plane), then every lattice point in the $h$-fold sumset $hP$ is the sum of $h$ lattice points in $P$.

Euclidean geometry and fractions not only make unlikely bedfellows, but would seem to be a sure recipe for boredom. However, in this paper, with the help of a little history, we hope to prove the opposite. Given a (rectangular) lattice, then a lattice polygon is a polygon whose vertices are lattice points. Pick's theorem [1] - the area of a simple lattice polygon is given by ½b + i – 1 where b is the number of lattice points on the boundary of the polygon, and i is the number in its interior – is well known and has been proved many times and in many ways.

Polynomials and spatial Pick-type theorems

Pick's famous area theorem has many generalizations and extensions including relatively recent work by Grunbaum and Shephard [3]. One of the generalizations is due to Hadwiger and Wills who considered nonproper lattice polygons having isolated points and one-dimensional parts. The aim of this note is to give generalizations of Hadwiger—Wills formula for nonproper lattice polyhedra in R 3 and R 4 . The four-dimensional considerations indicate difficulties appearing in a search for an arbitrary-dimensional generalization of this type.

Pick's Theorem relates the area of a simple polygon with vertices at integer lattice points to the number of lattice points in its inside and boundary. We describe a formal proof of this theorem using the HOL Light theorem prover. As sometimes happens for highly geometrical proofs, the formalisation turned out to be more work than initially expected. The difficulties arose mostly from formalising the triangulation process for an arbitrary polygon.

The boundary characteristic and Pick's theorem in the Archimedean planar tilings

The author has extended Pick's theorem for simple closed polygonal regions to unions of simple closed polygonal regions–a topic that is manageable for middle grade students. From sets of data including numbers of boundary points and numbers of interior points, students are guided to discover Pick's theorem. Additionally, with the author's creation of crossing points, Pick's theorem is extended to include areas of other polygonal regions. The article is developed along lines of the 1989 Standards of the NCTM in the use of data tables which lead to the discovery of a formula.

A lattice point in the plane is a point with integer coordinates. A lattice segment is a line segment whose endpoints are lattice points. A lattice polygon is a simple polygon whose vertices are lattice points. We find all convex lattice polygons in the plane up to equivalence with two interior lattice points.

In the Euclidean space ℝ3, denote the set of all points with integer coordinate by ℤ3. For any two-dimensional simple lattice polygon P, we establish the following analogy version of Pick's Theorem, k(I(P) + (1/2)B(P) − 1), where B(P) is the number of lattice points on the boundary of P in ℤ3, I(P) is the number of lattice points in the interior of P in ℤ3, and k is a constant only related to the two-dimensional subspace including P.

In Cantor's original proof of the uncountability of the reals (not the diagonalization argument), he constructs, given any countable sequence of real numbers, a real number not in the sequence. When we apply this argument to a certain standard enumeration of the rationals, the real number we produce will necessarily be irrational. Using some planar geometry, including Pick's theorem on the number of lattice points enclosed within certain polygonal regions, we show that this number is the reciprocal of the golden ratio, whence follows the well-known fact that the golden ratio is irrational.

The Ehrhart polynomial of a lattice polygon $P$ is completely determined by the pair $(b(P),i(P))$ where $b(P)$ equals the number of lattice points on the boundary and $i(P)$ equals the number of interior lattice points. All possible pairs $(b(P),i(P))$ are completely described by a theorem due to Scott. In this note, we describe the shape of the set of pairs $(b(T),i(T))$ for lattice triangles $T$ by finding infinitely many new Scott-type inequalities.

One of the most fruitful results from Minkowski's geometric viewpoint on number theory is his so called 1st Fundamental Theorem. It provides an optimal upper bound for the volume of an o-symmetric convex body whose only interior lattice point is the origin. Minkowski also obtained a discrete analog by proving optimal upper bounds on the number of lattice points in the boundary of such convex bodies. Whereas the volume inequality has been generalized to any number of interior lattice points already by van der Corput in the 1930s, a corresponding result for the discrete case remained to be proven. Our main contribution is a corresponding optimal relation between the number of boundary and interior lattice points of an o-symmetric convex body. The proof relies on a congruence argument and a difference set estimate from additive combinatorics.