Rhombic tilings of -Ovals, -cyclic difference sets, and related topics

Each fixed integer n has associated with it ⌊ n 2 ⌋ rhombs: ρ 1 , ρ 2 , … , ρ ⌊ n 2 ⌋ , where, for each 1 ≤ h ≤ ⌊ n 2 ⌋ , rhomb ρ h is a parallelogram with all sides of unit length and with smaller face angle equal to h × π n radians. l is a centro-symmetric convex polygon all of whose sides are of unit length, and each of whose turning angles equals ℓ × π n for some positive integer ℓ . A ( n , k ) -Oval is an Oval with 2 k sides tiled with rhombs ρ 1 , ρ 2 , … , ρ ⌊ n 2 ⌋ ; 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 ρ 1 , ρ 2 , … , ρ ⌊ n 2 ⌋ 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 . elated topics including lists of ( n , k ) -Ovals, partitions of the regular 2 n -gon into Ovals, Cyclic Difference Families, partitions of triangle numbers, u -equivalence of ( n , k ) -Ovals, etc., are also considered.