Lattice Polygons in the Plane and the Number 12

Abstract

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, . . . , pn (in anticlockwise order), then we say that the length l(P) is n. The dual (convex) lattice polygon P∨ is by definition the convex hull of the difference vectors qi = pi+1 − pi, where indices throughout the note are considered modulo n. In this note we will give a simple proof of the fact that if P be a convex lattice polygon in R whose only interior lattice point is the origin, then l(P) + l(P∨) = 12. This result has its origin in a correspondence between convex lattice polygons and certain toric varieties, and has several ingenious proofs (see [1] and [2]). These proofs use either Noether’s formula or modular forms to explain the occurrence of the number 12. Our proof observes that if l(P) increases by one then l(P∨) decreases by one, so that their sum is constant. The number 12 then appears as 3 (the smallest possible length) plus 9 (the largest possible length.) Our proof (which grew out of the second authors project, supervised by the first author) also supports the suggestion in [2] that the number 3 can be viewed as a discrete analogue of π in this context.