Each fixed integer nnhas associated with it ⌊n2⌋ rhombs: ρ1,ρ2,…,ρ⌊n2⌋, where, for each 1≤h≤⌊n2⌋, rhomb ρhρh is a parallelogram with all sides of unit length and with smaller face angle equal toh×πn radians.
An Oval is a centro-symmetric convex polygon all of whose sides are of unit length, and each of whose turning angles equalsℓ×πn for some positive integer ℓℓ. A (n,k)(n,k)-Oval is an Oval with 2k2k sides tiled with rhombsρ1,ρ2,…,ρ⌊n2⌋; it is defined by its Turning Angle Index Sequence, a kk-composition of nn. For any fixed pair (n,k)(n,k) we count and generate all (n,k)(n,k)-Ovals up to translations and rotations, and, using multipliers, we count and generate all (n,k)(n,k)-Ovals up to congruency. For odd nn if a (n,k)(n,k)-Oval contains a fixed number λλ of each type of rhombρ1,ρ2,…,ρ⌊n2⌋ then it is called a magic (n,k,λ)(n,k,λ)-Oval. We prove that a magic (n,k,λ)(n,k,λ)-Oval is equivalent to a (n,k,λ)(n,k,λ)-Cyclic Difference Set. For even nn we prove a similar result. Using tables of Cyclic Difference Sets we find all magic (n,k,λ)(n,k,λ)-Ovals up to congruency for n≤40n≤40.
Many related topics including lists of (n,k)(n,k)-Ovals, partitions of the regular 2n2n-gon into Ovals, Cyclic Difference Families, partitions of triangle numbers, uu-equivalence of (n,k)(n,k)-Ovals, etc., are also considered.
McSorley, John and Schoen, Alan. "Rhombic tilings of (n,k)-Ovals, (n,k,λ)-cyclic difference sets, and related topics." Discrete Mathematics 313, No. 1 (Jan 2013): 129-154. doi:10.1016/j.disc.2012.08.021.