You’re probably familiar with the fact that you can draw a convex octagon with corners in an integer grid, fitting into a \(3\times 3\) square. It’s not regular, because its side lengths alternate between \(1\) and \(\sqrt 2\), but it has the same angles as a regular octagon and looks close enough to it for some purposes.

3x3 integer octagon and its 7-ply tiling of the plane

It also has another interesting property: if you place copies of it at every point of the integer grid, then each edge of each copy is also the edge of another copy. Therefore, if you let \(p\) be any point that avoids the edges of the octagons, count the number of octagons that cover any point of the plane, and then slide \(p\) around to another edge-avoiding point, the number of covering octagons always stays unchanged. Whenever \(p\) slides off from one octagon, it slides onto another one. Because the area of the integer octagon is seven units, and you’re placing one octagon for every unit square of the grid, the average covering depth is seven. And since the covering depth stays the same everywhere, it’s seven everywhere. You can think of this collection of octagons as forming a \(7\)-ply tiling of the plane, despite the fact that convex octagons cannot tile the plane in the usual \(1\)-ply sense of tiling.

More generally, define a \(k\)-ply tiling to be a covering of the plane by congruent copies of some prototile (allowing rotations, even though these are not necessary for the octagon) such that, except at a subset of the plane of measure zero (the boundaries of the prototiles), every point is covered by exactly \(k\) copies, and define the “ply” of a prototile to be the minimum \(k\) such that it has a \(k\)-ply tiling. The integer octagon has ply \(7\): once one octagon is placed anywhere in the plane, the rest of the tiling is forced to follow in the same way around it, in order to avoid creating seams where an octagon edge is not matched by another octagon and the ply changes. The same construction, using centrally symmetric octagons with integer vertices and longer sides, produces for any \(k\ge9\) a convex tile of ply \(k\).

This is the subject of my new short paper (or maybe extended abstract) “Multifold tiles of polyominoes and convex lattice polygons”, with many coauthors from the 2017 Bellairs Winter Workshop on Computational Geometry: Kota Chida, Erik Demaine, Martin Demaine, Adam Hesterberg, Takashi Horiyama, John Iacono, Hiro Ito, Stefan Langerman, Ryuhei Uehara, and Yushi Uno. You can find it in the book of abstracts of the 23rd Thailand–Japan Conference on Discrete and Computational Geometry, Graphs, and Games (TJCDCG3 2020+1), which was organized online by Chiang Mai University earlier this month.

As well as the family of octagon \(k\)-ply tilers described above, we found that cutting the bottom row of squares off a \(3\times 3\) octagon produces a \(5\)-ply hexagon tiler, this time requiring \(180^\circ\)-degree rotations for its tiling, and that stretching this hexagon can produce an \(8\)-ply convex tiler. We also found polyomino \(k\)-ply tilers for all \(k\ge 2\), and three heptominoes (the smallest possible polyominoes) that can each form \(k\)-ply tilings for all \(k\ge 2\) but not for \(k=1\). I imagine the details will become available in a more complete paper at some point but for now the abstract just announces these results and gives pictures of these heptominoes. We still don’t know whether there can exist convex polygons whose ply is one of \(\{2,3,4,6\}\).

The TJCDCG3 abstract book has many other intriguing results in discrete geometry, graph theory, and combinatorial game theory, so do check it out if you’re interested. Tiling-related highlights include a variation on Wang tiling adding connectivity constraints and inspired by a dungeon-making mini-game in The Legend of Zelda: Link’s Awakening (“Tiling the Plane Connectively with Wang Tiles”, Chao Yang), signal processing using high-dimensional substitution tilings (“Generating Frames via Discretized Substitution Tilings”, Luis S. Silvestre Jr. and Job A. Nable), a partial classification of edge-to-edge monohedral tilings of the sphere (“Tiling of the Sphere by Congruent Polygons”, Yohji Akama, Hoi Ping Luk, Erxiao Wang, and Min Yan), and tilings that can be used to make arrays of joined-up origami cranes (“Renzuru Tilings with Asymmetric Quadrilaterals”, Takashi Yoshino).

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