Pick's theorem

Pick's Theorem, named after the Austrian mathematician Georg Alexander Pick, describes a fundamental property of simple grid polygons. These are polygons whose vertices have integer coordinates all. ( Think of a polygon, which is painted on graph paper, with vertices only in the intersections of the grid )

Statement of the theorem

A is the area of ​​the polygon, I is the number of lattice points inside the polygon, and R, does the number of lattice points on the boundary of the polygon:

In the adjacent example and. The area of this polygon is thus grid square units.

Pick's Theorem can be used to prove Euler's polyhedron formula.

If not only simple polygons, but also those with "holes ", it must be replaced by " " the summand "-1", the Euler characteristic of the polygon.

Idea of ​​proof

  • The theorem is additive: If one merges two polygons with integer vertices which intersect in a common line to a polygon with integer vertices, then the real surfaces and the surfaces add up according to the formula in the set. For the boundary points in the interior of the track to be interior points, and the endpoints of the segment are two boundary points.
  • The sentence can be verified directly for axis-parallel rectangles.
  • Because of the additivity of the set then also applies to right-angled triangles with short sides parallel to the axis, as it is here at half rectangles.
  • It is also applicable for Trapeze with three axis-parallel sides ( right triangle plus rectangle). Looking to either side of the given polygon trapezoid bounded by this side, two vertical lines through the endpoints and a distant but firm selected horizontal line, then the given surface can be represented as a signed sum of these trapezoids. From the additivity then the assertion follows.
  • Alternatively, the last step can also prove that the theorem for arbitrary triangles considered by supplemented by right triangles to form a rectangle. Then the theorem follows by induction, since one with more than three vertices can decompose by a running entirely inside the polygon diagonal into two simple polygons with fewer corners of each simple polygon.

Conclusions

An interesting result of the set of pick is that a planar triangle with vertices integer that does not contain lattice points other than these vertices, the surface 1 /2. Are and two such triangles, thus forming the affine transformation that transforms in the grid (meaning only the grid points ) from to itself.

Generalization

Pick's Theorem is generalized by Ehrhart polynomials to three and more dimensions. To put it simply: For one-dimensional polytope of the volume is considered a scaled by a factor of copy; for large overlaps in the first approximation grid points.

A simple formula which links the number of lattice points of a higher-dimensional polytope to which the volume is not available. To have about the three-dimensional case, the simplices of the four points (0, 0, 0 ), (1, 0, 0 ), ( 0, 1, 0) and (1, 1, r) to be clamped, respectively, the volume r / 6, but contained except the vertices no further integer point.

Source

  • Georg Alexander Pick: Geometric to the theory of numbers. (Processing one held in the German Mathematical Society in Prague lecture. ) In: Proceedings of the German scientific- medici American Association for Bohemia " lotus " in Prague 19 (1899 ), pp. 311-319.
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