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W109940877 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." @default.
W109940877 created "2016-06-24" @default.
W109940877 creator A5009550882 @default.
W109940877 creator A5058496835 @default.
W109940877 date "2013-01-01" @default.
W109940877 modified "2023-09-29" @default.
W109940877 title "Rhombic tilings of <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML altimg=si497.gif display=inline overflow=scroll><mml:mrow><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>-Ovals, <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML altimg=si498.gif display=inline overflow=scroll><mml:mrow><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>λ</mml:mi><mml:mo…" @default.
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W109940877 doi "https://doi.org/10.1016/j.disc.2012.08.021" @default.
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