Stared at some ceiling tiles today, which was a bad call for my productivity because I immediately wondered how the contractors cut and installed the cross pieces, and whether or not their was a standardizable piece that allowed optimal tiling.
So I started with some exploratory thought experiments. Pieces that have one intersection and one edge can perfectly cover one square (in pink below) so let’s go from there. Unfortunately as we add to it (green pieces), we create spaces that standard pieces don’t fit into. A failure.
What about a longer pieces that spans an intersection and two edges? This creates a very familiar weave-like pattern that definitely tessellates (below).
However, we aren’t close to a generalize-able rule. Indeed, one could imagine many other strange angled or partial length shapes that I have not tried yet, and I wondered if there were at least some property that influenced their tessellate-ability.
I began to think that it has to do with the number of intersections and edges that a piece crossed since the relationship between those helps to define how much space the cross pieces need to cover.
An inductive analysis can be used: a single tile has 4 corners and 4 edges, then four tiles have 9 corners and 12 edges, and then nine tiles have 16 corners and 24 edges (image below). This is solvable by looking for the pattern (n^2 corners to 2n^2 edges), but the answer is easier to come to if we simply look at the general situation of each corner and each edge instead.
In a fully-tiled situation, a corner sits at the intersection of 4 tiles while an edge sits at the intersection of 2 tiles. There we have it then! For an infinitely tiled surface, you will need a corner for every 4 tiles, but an edge every 2 – there will then be a ratio of edges to intersections/corners of 2:1. Thus, this must be a design constraint of the cross-pieces (which agrees with why my second thought experiment worked unlike the first).
While this does not give us a way to define all possible cross piece shapes (one could imagine a complicated cross-piece shape with 74 edges and 37 intersections but does not tessellate), it does allow us to check if a cross piece shape could potentially tessellate.