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.

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.

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Related papers:

MultiTiling_TJM (Multifold tiles of polyominoes and convex lattice polygons)