Plus Equals #3

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.

On the left, a mosaic of square tiles of various shades. On the right, an intricate pattern of illustrated leaves.
Fig. 1–2: A colorful mosaic of square tiles and a seamless organic tile pattern

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.

On the left, a white square tile with a black circle in the middle of it. On the right, a 3×3 arrangement of nine of the same tile makes a simple pattern.
Fig. 3: A simple tile makes a simple pattern.

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).

On the left, a white square tile with black half-circles resting against each of its four edges. On the right, a 3×3 arrangement of the same tile makes a diagonal pattern of circles.
Fig. 4: This tile bisects two circles and moves the halves to their opposing tile edges to make a denser, diagonal version of the previous pattern.

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).

On the left, a white square tile with black quarter-circles in each of its four corners, with a black diamond in the middle. On the right, a 3×3 arrangement of the same tile makes an alternating pattern of circles and diamonds.
Fig. 5: This tile quarters the circle, moves the pieces to their opposing tile corners, and adds a diamond to the pattern.

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.

On the left, three white square tiles with halved shapes resting against each of the tiles’ four edges: circles, diamonds, and squares. On the right, a 3×3 arrangement of those same tiles makes a pattern of hybrid shapes.
Fig. 6: A set of three tiles designed using the same technique as Fig. 4, all of which can seamlessly border each other on any side
On the left, three white square tiles with halved shapes resting against each of the tiles’ four edges: circles, diamonds, and squares. On the right, another white square tile with a black circle in the middle of it. Dashed lines show that all four tiles are designed on a 3×3 grid.
Fig. 7: These tiles are all based on a 3×3 grid.

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.

A white square subdivided into a 3×3 with dashed lines. Black circles mark the points where the dashed lines meets the square’s edge. On the left and bottom of the square, x and y coordinates from 0 to 3 are marked.12301230
Fig. 8: A 3×3 grid has 12 points on its perimeter.

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).

A diagram demonstrating a process of elimination comparing the coordinates of each of the 12 perimeter points on a 3×3 to every other point’s coordinates. Coordinate pairs that make horizontal or vertical lines are crossed out.1,00,00,0—1,30,0—2,30,0—3,10,0—3,20,0—3,30,0—0,10,0—0,20,0—0,30,0—1,00,10,20,30,1—1,00,2—1,00,1—1,30,1—2,00,1—2,30,1—3,00,1—3,20,1—3,30,2—1,30,2—2,00,2—2,30,2—3,00,2—3,30,2—3,10,3—1,00,3—2,00,3—3,00,3—3,20,3—3,11,0—2,31,0—3,21,0—3,31,0—3,11,31,3—2,01,3—3,01,3—3,21,3—3,12,02,0—3,22,0—3,32,0—3,12,32,3—3,02,3—3,22,3—3,13,03,13,23,30,0—2,00,0—3,00,1—3,10,2—3,20,1—0,20,2—0,30,3—1,30,3—2,30,3—3,30,1—0,31,0—1,32,0—2,31,3—2,31,3—3,31,0—2,02,0—3,02,3—3,31,0—3,03,0—3,13,1—3,23,1—3,33,2—3,33,0—3,23,0—3,3
Fig. 9: The process for finding all possible unique pairs of a 3×3 grid’s perimeter points that make diagonal lines. For a pair of points to make a diagonal line, the x coordinates can’t match and the y coordinates can’t match.
A diagram showing 38 unique diagonal lines, each with its corresponding coordinates on a 3×3 grid0,0—1,30,0—2,30,0—3,10,0—3,20,0—3,30,1—1,00,1—1,30,1—2,00,1—2,30,1—3,00,1—3,20,1—3,30,2—1,00,2—1,30,2—2,00,2—2,30,2—3,00,2—3,10,2—3,30,3—1,00,3—2,00,3—3,00,3—3,10,3—3,21,0—2,31,0—3,11,0—3,21,0—3,31,3—2,01,3—3,01,3—3,11,3—3,22,0—3,12,0—3,22,0—3,32,3—3,02,3—3,12,3—3,2
Fig. 10: The 38 unique diagonal lines that can be made from the points on the perimeter of a 3×3 grid

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).

A diagram showing four 3×3 grids with combinations of diagonal lines on them. The six lines on the first grid use one of the perimeter points more than once and leave one unused. An X above the grid indicates that that combination is disqualified. The second grid, also disqualified, has seven lines and reuses two perimeter points. The third and fourth grid both have checkmarks above them to indicate that they’re acceptable. Both have six lines, using each perimeter point just once.
Fig. 11: Line combinations are disqualified if the number of lines is anything other than six, or if any point is used more than once or left unused.

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.

A diagram demonstrating a process of elimination comparing the point coordinates of sets of four diagonal lines beginning with the coordinates “0,0–1,3.” Combinations with repeated points are crossed out.0,0—1,30,0—1,30,1—1,00,0—1,30,1—1,00,2—1,00,0—1,30,1—1,00,2—1,30,0—1,30,1—1,00,2—2,00,0—1,30,1—1,00,2—2,30,0—1,30,1—1,00,2—2,00,3—1,00,0—1,30,1—1,00,2—2,00,3—2,00,0—1,30,1—1,00,2—2,00,3—3,00,0—1,30,1—1,00,2—2,30,3—1,00,0—1,30,1—1,00,2—2,30,3—2,00,0—1,30,1—1,00,2—2,00,3—3,10,0—1,30,1—1,00,2—2,00,3—3,2
Fig. 12: The process for finding all unique combinations of four lines, with each combination making use of all four points on the grid’s zero column (List A). Combinations with repeated points are disqualified.

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.

A diagram demonstrating a process of elimination comparing the point coordinates of pairs of diagonal lines beginning with “1,0–2,3” and “1,0–3,1.” Combinations with repeated points are crossed out.1,0—2,31,0—2,31,3—2,01,0—2,31,3—3,01,0—2,31,3—3,11,0—2,31,3—3,21,0—2,32,0—3,11,0—2,32,0—3,21,0—2,32,0—3,31,0—3,11,0—3,11,3—2,01,0—3,11,3—3,01,0—3,11,3—3,21,0—3,12,0—3,21,0—3,12,0—3,31,0—3,12,3—3,01,0—3,12,3—3,21,0—3,21,0—3,21,0—3,21,0—3,21,0—3,21,0—3,21,0—3,21,0—3,21,0—2,31,0—2,31,0—2,31,0—3,11,0—3,21,0—3,31,0—2,31,0—2,31,0—2,32,3—3,02,3—3,12,3—3,21,0—3,11,0—3,11,0—3,21,0—3,31,0—3,11,3—3,11,0—3,11,0—3,12,0—3,12,3—3,11,0—3,21,0—3,21,0—3,21,0—3,2
Fig. 13: The process for finding all unique pairs of lines that don’t use the grid’s zero column (List B). Combinations with repeated points are disqualified.

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!

A diagram demonstrating a process of elimination comparing the point coordinates of sets of six diagonal lines beginning with “0,0–1,3; 0,1–1,0; 0,2–2,0; 0,3–3,0.” Combinations with repeated points are crossed out.0,0—1,30,1—1,00,2—2,00,3—3,00,0—1,30,1—1,00,2—2,00,3—3,01,0—2,31,3—2,00,0—1,30,1—1,00,2—2,00,3—3,01,0—2,31,33,00,0—1,30,1—1,00,2—2,00,3—3,01,0—2,31,3—3,10,0—1,30,1—1,00,2—2,00,3—3,01,0—2,31,3—3,20,0—1,30,1—1,00,2—2,00,3—3,01,0—2,32,0—3,10,0—1,30,1—1,00,2—2,00,3—3,01,0—2,32,0—3,20,0—1,30,1—1,00,2—2,00,3—3,01,0—2,32,0—3,30,0—1,30,1—1,00,2—2,00,3—3,01,0—3,11,33,00,0—1,30,1—1,00,2—2,00,3—3,01,0—3,11,3—3,20,0—1,30,1—1,00,2—2,00,3—3,01,0—3,12,0—3,20,0—1,30,1—1,00,2—2,00,3—3,01,0—3,12,0—3,30,0—1,30,1—1,00,2—2,00,3—3,01,0—3,12,3—3,00,0—1,30,1—1,00,2—2,00,3—3,01,0—3,12,3—3,2
Fig. 14: The process for combining List A and List B to find all unique combinations of six lines, with each combination using all 12 of the grid’s perimeter points. The vast majority of combinations are disqualified for containing repeated points.

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.

Rob Weychert
rob@robweychert.com