Rhombic tilings of [formula omitted]-Ovals, [formula omitted]-cyclic difference sets, and related topics

Abstract

Each fixed integer n has associated with it ⌊n2⌋ rhombs: ρ1,ρ2,…,ρ⌊n2⌋, where, for each 1≤h≤⌊n2⌋, rhomb ρh is a parallelogram with all sides of unit length and with smaller face angle equal to h×πn radians.An Oval is a centro-symmetric convex polygon all of whose sides are of unit length, and each of whose turning angles equals ℓ×πn for some positive integer ℓ. A (n,k)-Oval is an Oval with 2k sides tiled with rhombs ρ1,ρ2,…,ρ⌊n2⌋; it is defined by its Turning Angle Index Sequence, a k-composition of n. For any fixed pair (n,k) we count and generate all (n,k)-Ovals up to translations and rotations, and, using multipliers, we count and generate all (n,k)-Ovals up to congruency. For odd n if a (n,k)-Oval contains a fixed number λ of each type of rhomb ρ1,ρ2,…,ρ⌊n2⌋ then it is called a magic (n,k,λ)-Oval. We prove that a magic (n,k,λ)-Oval is equivalent to a (n,k,λ)-Cyclic Difference Set. For even n we prove a similar result. Using tables of Cyclic Difference Sets we find all magic (n,k,λ)-Ovals up to congruency for n≤40.Many related topics including lists of (n,k)-Ovals, partitions of the regular 2n-gon into Ovals, Cyclic Difference Families, partitions of triangle numbers, u-equivalence of (n,k)-Ovals, etc., are also considered.