TITLE:
Every Tiling of the First Quadrant by Ribbon L n-Ominoes Follows the Rectangular Pattern
AUTHORS:
Viorel Nitica
KEYWORDS:
Polyomino, Replicating Tile, L-Shaped Polyomino, Skewed L-Shaped Polyomino, Local Move Property, Tiling Rectangles, Rectangular Pattern, Tiling First Quadrant
JOURNAL NAME:
Open Journal of Discrete Mathematics,
Vol.5 No.2,
June
9,
2015
ABSTRACT: Let and let be the set of four ribbon L-shaped n-ominoes. We
study tiling problems for regions in a square lattice by . Our main result
shows a remarkable property of this set of tiles: any tiling of the first
quadrant by , n even, reduces to a tiling by and rectangles, each rectangle being covered by
two ribbon L-shaped n-ominoes. An application of our result
is the characterization of all rectangles that can be tiled by , n even: a rectangle can be tiled by , n even, if and only if both of its sides
are even and at least one side is divisible by n. Another application is the existence of the local move property
for an infinite family of sets of tiles: , n even, has the local move property for
the class of rectangular regions with respect to the local moves that
interchange a tiling of an square by n/2
vertical rectangles, with a tiling by n/2
horizontal rectangles, each vertical/horizontal rectangle being covered by two
ribbon L-shaped n-ominoes. We show that none of these results are valid for any
odd n. The rectangular pattern of a
tiling of the first quadrant persists if we add an extra tile to
, n even. A rectangle can be tiled by the
larger set of tiles if and only if it has both sides even. We also show that
our main result implies that a skewed L-shaped n-omino, n even, is not a replicating tile of order k2 for any odd k.