# 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}
}
• Published 1 March 2000
• Mathematics, Computer Science
• The American Mathematical Monthly
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
66 Citations

#### Topics from this paper

Moving Out the Edges of a Lattice Polygon
• W. Castryck
• Mathematics, Computer Science
• Discret. Comput. Geom.
• 2012
The dual operations of taking the interior hull and moving out the edges of a two-dimensional lattice polygon are reviewed and it is shown how the latter operation naturally gives rise to an algorithm for enumerating lattice polygons by their genus. Expand
Lattice multi-polygons
• Mathematics
• 2012
We discuss generalizations of some results on lattice polygons to certain piecewise linear loops which may have a self-intersection but have vertices in the lattice $\mathbb{Z}^2$. We first prove aExpand
On the Twelve-Point Theorem for ℓ-Reflexive Polygons
• D. Dais
• Computer Science
• Electron. J. Comb.
• 2019
The present paper contains a second proof of this fact as well as the description of complementary properties of these polygons and of the invariants of the corresponding toric log del Pezzo surfaces. Expand
Lattice Polygons and the Number 2i + 7
• Computer Science, Mathematics
• Am. Math. Mon.
• 2009
The authors came up with new inequalities: Scott's inequality can be sharpened if one takes into account another invariant, which is de fined by peeling off the skins of the polygons like an onion (see Section 3). Expand
94.09 Lattice polygons and the number 12: an elementary proof
• A. Zorzi
• Mathematics
• The Mathematical Gazette
• 2010
94.09 Lattice polygons and the number 12: an elementary proof Introduction Among the most famous numbers a place can be reserved for the number 12. Some manifestations of 12 are listed in [I, sectionExpand
On Integer Geometry
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 toExpand
On the Twelve-Point Theorem for $\ell$-Reflexive Polygons
• D. Dais
• Mathematics
• The Electronic Journal of Combinatorics
• 2019
It is known that, adding the number of lattice points lying on the boundary of a reflexive polygon and the number of lattice points lying on the boundary of its polar, always yields 12. GeneralisingExpand
Lattice Polygons in the Plane and the Number 12
• Mathematics
• Irish Mathematical Society Bulletin
• 2006
A convex polygon P in R all of whose vertices have integer coordinates is called a convex lattice polygon. If the polygon has n lattice points on its boundary, represented by the vectors p1, . . . ,Expand
A short proof of the twelve-point theorem
• Mathematics
• 2005
We present a short elementary proof of the following twelve-point theorem. Let M be a convex polygon with vertices at lattice points, containing a single lattice point in its interior. Denote by mExpand
Affine dimers from characteristic polygons
Recent work by Forsg̊ard indicates that not every convex lattice polygon arises as the characteristic polygon of an affine dimer or, equivalently, an admissible oriented line arrangement on the torusExpand

#### References

SHOWING 1-10 OF 28 REFERENCES
Classification of Reflexive Polyhedra in Three Dimensions
• Physics, Mathematics
• 1998
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
• Mathematics
• 1993
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.