Download Stars and Fareys: A Screen Size Romance

The Ohio State University, August 19-21
Stars and Fareys: A Screen Size Romance
Qing Qing Wu
Matthew G Jacobson
([email protected])
([email protected])
Texas State University - San Marcos
[Mentor:Nathaniel Dean]
Abstract of Report Talk: The screen size of the graph K1,n−1 (i.e. a star of order n) is the
smallest number such that the k x k integer lattice supports a drawing of K1,n−1 , where each
vertex is a lattice point and the edges are drawn as non-overlapping line segments. We find
that the screen size of a star is related to the number of visible points on an integer lattice (i.e.
the screen). In particular, given a screen of odd size s = 2k + 1, k ∈ N, if K1,n−1 is centered
) + 9 vertices, where F (i), called
in the middle of the screen, then it can have at most 8F ( s−1
2
the Farey number of i, is the length of the Farey sequence for i.
However, the center of the screen is not necessarily the point that will fit the largest possible
star. Thus, we derive a 2-dimensional generalization of Farey numbers that gives the exact
number of visible points from any point of an integer lattice by looking at the number of
co-prime pairs in a rectangle and applying the sieve principle. We then generalize Euler’s
totient function in order to investigate the asymptotic properties and provide bounds for our
new function.
[Joint work with Charles S. Berahas and Jian Shen]
Received: July 31, 2011