Farey Sequences and Pick’s Theorem

• Rebecca Crump, Mathematics Education, University of Delaware

• Sebastian Cioaba, Department of Mathematical Sciences, University of Delaware

The Farey sequence of order n represents an increasing sequence of reduced fractions between 0 and 1 whose denominators cannot exceed n. The properties of this sequence have been studied by many scholars, namely English geologist John Farey, and significant contributions to the nature of the sequence were previously made by French mathematician Augustin-Louis Cauchy, who proved that for any two consecutive fractions, their cross products differ by one, and for any three consecutive fractions, the middle one equals the mediant of the other two. These properties have important consequences in number theory, particularly for the approximations of irrational numbers by rational numbers. The formula described by Austrian mathematician Georg Pick in Pick’s Theorem states that the area of a polygon whose corners are lattice points, points of integer coordinates in the two-dimensional Cartesian plane, can be calculated as i+(b/2)-1 simply by counting i, the number of lattice points inside the polygon, and b, the number of lattice points on the boundary of the polygon. This remarkably simple result in lattice geometry extends to interesting applications and consequences. In our research, we examined the connections between the described properties for Farey sequences and Pick’s Theorem. By conducting investigations of mathematical articles and papers, analyses of Farey sequence patterns, and observations of general lattice geometry, we will detail connections between certain lattice polygons and fractions represented in the Farey sequence on the Cartesian plane.