My partner and I bought a house last year. After decades of renting, we’re excited to finally have a space we can truly make our own, and our eyes are more open than ever before to the finer details of home furnishings and interior decorating. My knowledge of wood species, LED lightbulbs, and every single dish drain on the market has reached dizzying new heights. My dreams are routinely invaded by patio furniture, window treatments, and paint samples.
One aspect of home improvement that hasn’t come up for us yet but has nevertheless been on my mind (perhaps due to a recent visit to Fonthill Castle) is tiling. I’m a sucker for a good creative constraint, and I love the way tiling’s spatial framework is both rigidly composed and rife with opportunity. Whether they’re used to create colorful mosaic grids (Fig. 1) or seamless organic patterns (Fig. 2), the creative potential unique to tiles has long fascinated me.
In my design career, I’ve created my share of tile patterns, especially in the early days when the background of pretty much every website was a neo-Warholian eternity of stars or balloons or skulls or whatever. In patterns, as in any other aesthetic endeavor, beauty is not easily formulated or defined. Music theory and the golden ratio notwithstanding, the right brain guards its secrets well. But I’ve found that patterns that stimulate my left brain tend to obey a recognizable principle: the harder the individual tiles are to detect, the better. This often equates to complexity: greater complexity equals greater seamlessness equals greater wonder induced.
Achieving that kind of complexity in tile pattern design has typically evaded me, to say nothing of making it actually look good. But recently I started thinking about how complex, seamless tile patterns could be derived from relatively simple systems, and it wasn’t long before combinatorics entered the fray.
The simplest square tiling pattern is a single shape—say, a circle—that sits in the middle of each tile (Fig. 3). Nothing mysterious about that.
A denser diagonal version of that pattern can be made with two copies of the shape. Cut the first shape in half horizontally and move its top half to the bottom edge of the tile and its bottom half to the top edge of the tile. Cut the second copy in half vertically and move its left half to the right edge of the tile and its right half to the left edge of the tile (Fig. 4).
To make an even fancier pattern, cut a single shape in quarters, move each quarter to its opposing corner of the tile, and add a different shape to the middle for a little spice (Fig. 5).
What these processes reveal is that making a seamless pattern relies on the opposing edges of the tile having a relationship with each other. Whatever happens on the right edge is continued on the left edge, and whatever happens on the top edge is continued on the bottom edge. Understanding this enables us to take things a step further by creating sets of tile designs that are all compatible with each other (Fig. 6).
And once there’s a set of things, I want to know how big it can be.
Sharp-eyed observers may have noticed that most of the tiles I’ve made so far have been designed with the help of a 3×3 grid (Fig. 7). I’m going to use that grid as the basis for a system which can produce many more tiles, each one able to seamlessly border any other one on any side. To help obscure the underlying grid, each tile design will consist entirely of diagonal lines, with each line beginning at one of the points on the perimeter of the grid and ending on another. The 3×3 grid gives us 12 perimeter points to work with (Fig. 8), and my first task is to find every unique diagonal line I can make with them.
This is done by naming each perimeter point by its grid coordinates and then listing the points in ascending numeric order. For each point, compare it to every point that comes after it in the list. For each comparison, if the x coordinates don’t match and the y coordinates don’t match—congratulations!—the two points make a diagonal line (Fig. 9). This process reveals that 38 unique diagonal lines can be made from the 3×3 grid’s 12 perimeter points (Fig. 10).
The next part is where things get tricky. I want to add another combinatorial layer to the diagonal lines I just made by finding every possible set that makes use of all 12 perimeter points and uses each point only once. That means every qualifying set will have six lines, but not every set of six lines will qualify (Fig. 11).
The combinatorics of my two previous Plus Equals explorations were concerned with finding sequence variations, which isn’t really relevant to this task, and my initial ideas for finding all these sets of six lines were tying my brain in knots. So I reached out to someone with a substantially hardier brain: my friend Paul Lopata, who has a PhD in mathematical physics and an abiding affection for puzzles. After we batted the problem back and forth a bit, Paul got us on the right track. His solution involves generating two lists of line combinations, and then using them to arrive at a final list.
We’ll call the first one List A, and it contains combinations of four lines. Importantly, in each combination, all four points in the grid’s zero column are represented. To find every such combination, I’ll use a similar process to the one I used earlier to find the full set of diagonal lines. For each line originating at 0,0, compare it to each line originating at 0,1. For each comparison, if no points appear more than once, combine the lines and repeat the process by comparing the combination to every line originating at 0,2. Once again, for each comparison, if no points are repeated, add the 0,2 line to the combination and compare it to each line originating at 0,3. This process (Fig. 12) ultimately finds a total of 549 unique combinations of four lines, with each combination making use of all four points on the grid’s zero column.
List B contains every pair that can be made out of lines that don’t use the grid’s zero column. To find them, list the lines in ascending numeric order. For each line, compare it to every line that comes after it. For each comparison, if no points are repeated, combine the lines and add the pair to List B (Fig. 13). The complete list ultimately contains 53 unique pairs.
The process for building the final list of line combinations should feel pretty familiar by now: For each combination in List A, compare it to each combination in List B. For each comparison, if no points are repeated, merge the combinations (Fig. 14). Of the nearly 30,000 comparisons made between the two lists, less than two percent result in mergers, but it still makes a sizable amount of tiles. The final list contains 425 unique combinations of six diagonal lines, with each combination using all 12 of the grid’s perimeter points. I now have a complete set of tiles, and each one can seamlessly border any other one on any side!
Before I knew how many tiles this system would generate, I imagined adding one more combinatorial layer to see how many different compound patterns the tiles could make, but that number surely stretches closer to infinity than is practical to ponder. Still, this issue’s gallery is arranged to give a sense of the possibilities. In addition to displaying each individual tile in isolation, it demonstrates the pattern the tile makes and how it seamlessly integrates with adjacent tile patterns. The final image’s intricate tangle of lines is a mosaic of all 425 tiles.
I don’t expect to redecorate my bathroom with these tiles, but then again, something tells me they’d inspire some good shower thoughts.