# Theorem of Pick

Lattice polygon

The set of Pick , named after the Austrian mathematician Georg Alexander Pick , a fundamental property of simple lattice polygons. These are polygons whose corner points all have integer coordinates. (Imagine a polygon, which is drawn on calculation paper, with the corner points only at the intersections of the grid.)

## Statement of the sentence

Let the area of the polygon be , the number of grid points inside the polygon and the number of grid points on the edge of the polygon, then: ${\ displaystyle A}$${\ displaystyle I}$${\ displaystyle R}$

${\ displaystyle A = I + {\ frac {R} {2}} - 1}$

In the example opposite is and . The area of ​​this polygon is thus grid square units. ${\ displaystyle R = 12}$${\ displaystyle I = 40}$${\ displaystyle \ textstyle 40 + {\ frac {12} {2}} - 1 = 45}$

Pick's theorem can be used to prove Euler's polyhedron formula and, conversely, Pick's theorem follows from Euler's polyhedron formula, so that both theorems are equivalent.

If one considers not only simple polygons, but also those with “holes”, the summand “ ” must be replaced by “ ” in the above formula , whereby the Euler characteristic of the polygon is. One thus obtains: ${\ displaystyle -1}$${\ displaystyle - \ chi (P)}$${\ displaystyle \ chi (P)}$${\ displaystyle P}$

${\ displaystyle A = I + {\ frac {R} {2}} - \ chi (P)}$

## Proof idea

• The theorem is additive: If you merge two polygons with integer corners that intersect in a common segment to form a polygon with integer corners, then the real areas and also the areas add up according to the formula in the sentence. This is because the edge points inside the line become inner points and the end points of the line become two edge points.
• The set can be verified immediately for axially parallel rectangles.
• Because of the additivity, the theorem then also applies to right-angled triangles with axially parallel cathets , since these are half rectangles.
• It also applies to trapezoids with three axially parallel sides (right-angled triangle plus rectangle). If one looks at the trapezoid on each side of the given polygon, which is bounded from this side by two vertical lines by the end points and a distant, but fixed horizontal straight line, the given area can be represented as the signed sum of these trapezoids. The assertion then follows from the additivity.
• As an alternative to the last step, one can also prove that the theorem applies to any triangles by adding right triangles to form an axially parallel rectangle. Then follows the theorem by induction, since every simple polygon with more than three corners can be broken down into two simple polygons with fewer corners by a diagonal running right inside the polygon .

## Inferences

An interesting consequence of Pick's theorem is that a plane triangle with integer vertices, which apart from these vertices contains no integer points, has the surface . If and are two such triangles, the affine mapping that converts into maps the grid (only the grid points are meant here ) onto itself. ${\ displaystyle {\ tfrac {1} {2}}}$${\ displaystyle ABC}$${\ displaystyle A'B'C '}$${\ displaystyle ABC}$${\ displaystyle A'B'C '}$

## generalization

Pick's theorem is generalized to three and more dimensions by Ehrhart polynomials . To put it simply: for a -dimensional polytope of the volume one considers a copy scaled by a factor ; for large covers, as a first approximation, grid points. ${\ displaystyle d}$ ${\ displaystyle P}$${\ displaystyle V}$${\ displaystyle k}$${\ displaystyle k \ cdot P}$${\ displaystyle k}$${\ displaystyle k \ cdot P}$${\ displaystyle k ^ {d} \ cdot V}$

A simple formula that connects the number of integer points of a higher-dimensional polytope with its volume is not available. So have about three-dimensional case the simplices which of the four points , , and are clamped, each volume , but contain, besides the vertices no further integer point. ${\ displaystyle (0,0,0)}$${\ displaystyle (1,0,0)}$${\ displaystyle (0,1,0)}$${\ displaystyle (1,1, r)}$${\ displaystyle {\ tfrac {r} {6}}}$

## literature

• Georg Alexander Pick: Geometry for numerology. (Processing of a lecture given in the German mathematical society in Prague.) In: Reports of the meetings of the German natural science and medicine association for Bohemia “Lotos” in Prague 19 (1899), pp. 311–319.