47 research outputs found
Multi-triangulations as complexes of star polygons
Maximal -crossing-free graphs on a planar point set in convex
position, that is, -triangulations, have received attention in recent
literature, with motivation coming from several interpretations of them.
We introduce a new way of looking at -triangulations, namely as complexes
of star polygons. With this tool we give new, direct, proofs of the fundamental
properties of -triangulations, as well as some new results. This
interpretation also opens-up new avenues of research, that we briefly explore
in the last section.Comment: 40 pages, 24 figures; added references, update Section
Martin Gardner and His Influence on Recreational Math
Recreational mathematics is a relatively new field in the world of mathematics. While sometimes overlooked as frivolous since those who practice it need no advanced knowledge of the subject, recreational mathematics is a perfect transition for people to experience the joy in logically establishing a solution. Martin Gardner recognized that this pattern of proving solutions to questions is how mathematics progresses. From his childhood on, Gardner greatly influenced the mathematical world. Although not a mathematician, he inspired many to pursue careers and make advancements in mathematics during his 25-year career with Scientific American. He encouraged novices to expand their knowledge, enlightened professionals of computer science developments, and established his own proofs
Multitriangulations as complexes of star-polygons
International audienceMaximal -crossing-free graphs on a planar point set in convex position, that is, -triangulations, have received attention in recent literature, motivated by several interpretations of them. We introduce a new way of looking at -triangulations, namely as complexes of star polygons. With this tool we give new, direct proofs of the fundamental properties of -triangulations, as well as some new results. This interpretation also opens up new avenues of research that we briefly explore in the last section
Ribbon Tilings and Multidimensional Height Functions
We fix and say a square in the two-dimensional grid indexed by
has color if . A {\it ribbon tile} of order is a
connected polyomino containing exactly one square of each color. We show that
the set of order- ribbon tilings of a simply connected region is in
one-to-one correspondence with a set of {\it height functions} from the
vertices of to satisfying certain difference restrictions.
It is also in one-to-one correspondence with the set of acyclic orientations of
a certain partially oriented graph.
Using these facts, we describe a linear (in the area of ) algorithm for
determining whether can be tiled with ribbon tiles of order and
producing such a tiling when one exists. We also resolve a conjecture of Pak by
showing that any pair of order- ribbon tilings of can be connected by a
sequence of local replacement moves. Some of our results are generalizations of
known results for order-2 ribbon tilings (a.k.a. domino tilings). We also
discuss applications of multidimensional height functions to a broader class of
polyomino tiling problems.Comment: 25 pages, 7 figures. This version has been slightly revised (new
references, a new illustration, and a few cosmetic changes). To appear in
Transactions of the American Mathematical Societ
The Topology of Tile Invariants
In this note we use techniques in the topology of 2-complexes to recast some
tools that have arisen in the study of planar tiling questions. With spherical
pictures we show that the tile counting group associated to a set of tiles
and a set of regions tileable by is isomorphic to a quotient of the second
homology group of a 2-complex built from . In this topological setting we
derive some well-known tile invariants, one of which we apply to the solution
of a tiling question involving modified rectangles.Comment: 25 pages, 24 figure
Geometric Properties and a Combinatorial Analysis of Convex Polygons Constructed of Tridrafters
The aim of this thesis is to show how the use of parity in tandem with the triangular grid as well as a newly introduced and similar method are insufficient to provide proof for why convex regions composed using the full set of shapes known as proper tridrafters have a portion shifted in a fashion known as against-the-grain. These two methods are applied in a combinatorial fashion