Lattice Polygons and the Number 12

@article{Poonen2000LatticePA,
  title={Lattice Polygons and the Number 12},
  author={Bjorn Poonen and Fernando Rodriguez-Villegas},
  journal={The American Mathematical Monthly},
  year={2000},
  volume={107},
  pages={238 - 250}
}
1. PROLOGUE. In this article, we discuss a theorem about polygons in the plane, which involves in an intriguing manner the number 12. The statement of the theorem is completely elementary, but its proofs display a surprisingly rich variety of methods, and at least some of them suggest connections between branches of mathematics that on the surface appear to have little to do with one another. We describe four proofs of the main theorem, but we give full details only for proof 4, which uses… Expand

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References

SHOWING 1-10 OF 28 REFERENCES
Classification of Reflexive Polyhedra in Three Dimensions
We present the last missing details of our algorithm for the classification of reflexive polyhedra in arbitrary dimensions. We also present the results of an application of this algorithm to the caseExpand
Pick's theorem
Some years ago, the Northwest Mathematics Conference was held in Eugene, Oregon. To add a bit of local flavor, a forester was included on the program, and those who attended his session wereExpand
Mathematics and its history
Preface.- The Theorem of Pythagoras.- Greek Geometry.- Greek Number Theory.- Infinity in Greek Mathematics.- Polynomial Equations.- Analytic Geometry.- Projective Geometry.- Calculus.- InfiniteExpand
Lattice vertex polytopes with interior lattice points.
Introduction. In real Euclidean space R of dimension D there is the lattice Z of points with integer coordinates. Unless a different lattice is specified, a lattice point will mean a point of Z D ,Expand
Braids, Links, and Mapping Class Groups.
The central theme of this study is Artin's braid group and the many ways that the notion of a braid has proved to be important in low-dimensional topology. In Chapter 1 the author is concerned withExpand
Topological methods in algebraic geometry
Introduction Chapter 1: Preparatory material 1. Multiplicative sequences 2. Sheaves 3. Fibre bundles 4. Characteristic classes Chapter 2: The cobordism ring 5. Pontrjagin numbers 6. The ringExpand
Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties
We consider families ${\cal F}(\Delta)$ consisting of complex $(n-1)$-dimensional projective algebraic compactifications of $\Delta$-regular affine hypersurfaces $Z_f$ defined by Laurent polynomialsExpand
A Course in Arithmetic
Part 1 Algebraic methods: finite fields p-adic fields Hilbert symbol quadratic forms over Qp, and over Q integral quadratic forms with discriminant +-1. Part 2 Analytic methods: the theorem onExpand
Algebraic Geometry
Introduction to Algebraic Geometry.By Serge Lang. Pp. xi + 260. (Addison–Wesley: Reading, Massachusetts, 1972.)
Lie groups, Lie algebras, and their representations
1 Differentiable and Analytic Manifolds.- 2 Lie Groups and Lie Algebras.- 3 Structure Theory.- 4 Complex Semisimple Lie Algebras And Lie Groups: Structure and Representation.
...
1
2
3
...