Recurrent Theme of Pick's Theorem

TL;DR

This paper reviews and extends the derivations of Pick's theorem, connecting it to number theory and lattice geometry, and introduces elementary methods for constructing primitive cells and deriving the formula.

Contribution

It provides new variants of Pick's theorem derivations and demonstrates elementary constructions for primitive lattice cells using Euclidean algorithm.

Findings

Multiple derivations of Pick's theorem are presented.

Elementary methods for constructing primitive cells are introduced.

The Euclidean algorithm is shown to generate infinite primitive lattice cells.

Abstract

We review and possibly add some new variant to the existing derivations of the formula for the area of Jordan lattice polygons drawn on two-dimensional lattices. The formula is known as Pick's theorem and is related to the number theory elementary result-Bezout lemma. It is pointed out that Euclidean algorithm can be easily used in construction of infinite number of distinct primitive cells for any two-dimensional lattice. Pick's formula itself can also be obtained in an elementary "cut and re-assemble" finite process.