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# On Integer Geometry

• Oleg Karpenkov
Chapter
• 2.2k Downloads
Part of the Algorithms and Computation in Mathematics book series (AACIM, volume 26)

## Abstract

In many questions, the geometric approach gives an intuitive visualization that leads to a better understanding of a problem and sometimes even to its solution. This chapter is entirely dedicated to notions, definitions, and basic properties of integer geometry. We start with general definitions of integer geometry, and in particular, define integer lengths, distances, areas of triangles, and indexes of angles. Further we extend the notion of integer area to the case of arbitrary polygons whose vertices have integer coordinates. Then we formulate and prove the famous Pick’s formula that shows how to find areas of polytopes simply by counting points with integer coordinates contained in them. Finally we formulate one theorem in the spirit of Pick’s theorem: it is the so-called twelve-point theorem.

## Keywords

Integer Geometry Integer Area Integer Length Integer Coordinates Integer Triangles
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

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W. Fulton, Introduction to Toric Varieties. Annals of Mathematics Studies, vol. 131 (Princeton University Press, Princeton, 1993). The William H. Roever Lectures in Geometry
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A.G. Khovanskiĭ, Newton polytopes, curves on toric surfaces, and inversion of Weil’s theorem. Russ. Math. Surv. 52(6), 1251–1279 (1997). Russian version: Usp. Mat. Nauk 52(6), 113–142 (1997)
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B. Poonen, F. Rodriguez-Villegas, Lattice polygons and the number 12. Am. Math. Mon. 107(3), 238–250 (2000)
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D. Repovsh, M. Skopenkov, M. Tsentsel, An elementary proof of the twelve lattice point theorem. Math. Notes - Ross. Akad. 77(1–2), 108–111 (2005). Russian version: Mat. Zametki 77(1), 117–120 (2005)

## Copyright information

© Springer-Verlag Berlin Heidelberg 2013

## Authors and Affiliations

• Oleg Karpenkov
• 1
1. 1.Dept. of Mathematical SciencesUniversity of LiverpoolLiverpoolUK