Paper by Erik D. Demaine

Reference:

Kota Chida, Erik Demaine, Martin Demaine, David Eppstein, Adam Hesterberg, Takashi Horiyama, John Iacono, Hiro Ito, Stefan Langerman, Ryuhei Uehara, and Yushi Uno, “Multifold tiles of polyominoes and convex lattice polygons”, in Abstracts from the 23rd Thailand-Japan Conference on Discrete and Computational Geometry, Graphs, and Games (TJCDCGGG 2021), September 3–5, 2021, pages 28–29.

BibTeX

@InProceedings{MultiTiling_TJCDCGGG2021,
  AUTHOR        = {Kota Chida and Erik Demaine and Martin Demaine and David Eppstein and Adam Hesterberg and Takashi Horiyama and John Iacono and Hiro Ito and Stefan Langerman and Ryuhei Uehara and Yushi Uno},
  TITLE         = {Multifold tiles of polyominoes and convex lattice polygons},
  BOOKTITLE     = {Abstracts from the 23rd Thailand-Japan Conference on Discrete and Computational Geometry, Graphs, and Games (TJCDCGGG 2021)},
  bookurl       = {https://www.math.science.cmu.ac.th/tjcdcggg/},
  MONTH         = {September 3--5},
  YEAR          = 2021,
  PAGES         = {28--29},

  comments      = {Described in <A HREF="https://11011110.github.io/blog/2021/09/25/multilayer-tiles.html">a blog post by David Eppstein</A>.},
  paperkind     = {abstract},
  unrefereed    = 1,
  withstudent   = 1,
  papers        = {MultiTiling_TJM},
}

Abstract:

A family of 2-dimensional shapes 𝒯 is a called a tiling if they (rotating and reflecting are allowed) cover the whole plane without gaps or overlaps, and if all shapes belonging to 𝒯 are congruent each other, then the shape is called a tile. We study k-fold tilings which are extended to cover the plane with the thickness of k folds and k-fold tiles which belong to it. Intuitively it means that a family of 2-dimensional shapes covers the plane such that they overlap k times at any point in the plane. Since clearly a (1-fold) tile is a k-fold tile for any positive integer k, the subjects of our research are 2-dimensional shapes with property “not a tile, but a k(≥ 2)-fold tile.” We call a plane shape satisfying this property a nontrivial k-fold tile. In this talk, we clarify some facts as follows: first, we show that for any integer k ≥ 2, there exists a polyomino satisfying a property “not a h-fold tile for any positive integer h < k, but a k-fold tile.” We also find for any integer k ≥ 2, polyominoes with the minimum number of cells among ones that are nontrivial k-fold tiles. Next, we prove that for any integer k = 5 or k ≥ 7, there exists a convex unit-lattice polygon with an area of k that is a nontrivial k-fold tile.

Comments:

Described in a blog post by David Eppstein.

Availability:

The abstract is available in PDF (154k).

See information on file formats.

[Google Scholar search]

Related papers:

MultiTiling_TJM (Multifold tiles of polyominoes and convex lattice polygons)