Truchet Tiles Math Art Project - A-maze-ing Art

Art Piece: Cable-Knit Truchet Tiles

Original Artists: Lisa Marks and Owen Rowm 

Group Members: Emily Scott, Eleiah Hengeveld, Adam Barlev

Lisa Marks and Owen Rowm collaborated to produce cable-knit truchet tiles. Since the original piece was fully knitted from yarn, we decided to incorporate fibers into the recreation. The fabric and yarn were purchased from a community fabric store called 'Our Social Fabric' which saves ends of rolls from being dumped in the landfill. 

  

Polyurethane spray adhesive was used to attach the white fabric, reminiscent of the cable-knit pattern, to 25mm wooden squares. Then attached magnets to the back. These magnets allow the tiles to be easily repositioned and rotated to express variations of the tiling pattern. 

  

This construction required us to create 100 identical tiles, a daunting task. What we discovered when making art of this nature is that the first few are very slow and awkward to make, but as the done pile increases, efficient ways to work with the material become apparent, and a form of mastery and meditative state engage. Hours passed and the sun moved across the sky as more and more squares came together. When we finally looked down at our finished handiwork, we recognized that the look was unique. At that moment, we knew all that effort was worth it.

  

To expand this piece and make it our own we started with the idea of making a new 10x10 grid but with a different tile. The tile used in the original piece had two rotations, so we challenged ourselves to create a tile for our piece that had 4 rotations. We also wanted to use colour more than it had been used in the original piece (the original piece used colour more simply, it had a white background with yellow lines). How we practically went about achieving these goals was we started by trying to find a tile that, when put into a grid, had lines that connected between tiles. To make sure this happened we made the points where our pattern touched the outside of the tile the same on all 4 sides of the square. Then we sketched a design that connected the lines and that had 4 distinct rotations. Using a digital drawing software, we were able to shrink, duplicate, and rotate the tile to see how the pattern we had chosen would look in the final piece. Next, we chose how to incorporate colour. We experimented with a few different ways of doing this, we wanted to insure that however we chose to use colour it didn’t interrupt the continuous flow we were trying to create between our tiles. Once we had chosen how we wanted to incorporate colour we printed 100 copies of our tile. We then glued them to cardboard backing, cut out each tile, arranged them in a 10x10 grid, and then placed Velcro dots on each tile to connect it to our board. 

 

Some considerations that came up for us were what was practical and achievable and how we could ensure the pattern made by our tiles was atheistically interesting. We knew early on that we wanted both of our pieces to have tiles that were able to rotate so we could demonstrate how different patterns could be made using the different rotations of the tiles. However, we considered multiple different ways of doing this including wooden tiles on pegs, magnets or Velcro on wooden tiles and cardboard tiles with Velcro which we ended up deciding on. A big part of the reason we ended up deciding to build it how we did was because, given that we had to crate 100 identical tiles, we felt that it would be wise to be pragmatic and realistic about which method would be the most achievable. Another thing that came up was how to choose a pattern that would be astatically interesting. What we focused on to achieve this was ensuring we had a pattern that felt like it flowed between the tiles. This meant ensuring our lines connected across tiles. We made sure to carry this through our art not only creating the pattern in a way that ensured connection but also choosing our method for the use of colour in a way we knew wouldn’t interrupt the flow of our piece.

 

We weren’t entirely sure how to relate our art project to math at first. It was just a beautifully interesting image that looked like something we could understand and recreate with our own touches. 

 

When we first stared at the image it reminded us of a maze. Could we determine the probability of any given pattern having a maze that’s solvable, what would be the conditions that dictate if the maze could be solvable? We were looking for a test for mazeability, like the vertical line test to see if a graph represents a function. While we were running through ideas on probability we were struck with the realization that none of the lines cross, and therefore the grid could very well represent the non-crossing partition problem in combinatorics. 

 

The grid very much represented the non-crossing partition problem, the lines had to follow the same rules for Catalan number problems. Each line on the edge has a starting and ending point. The lines don’t cross and therefore if the start of a line is +1, and the end of a line is -1, the sum must always be greater or equal to zero, i.e. the number of starting points is never less than the number of closing points. I started with an example of every line that could be finished with the first point. The results were initially promising, so I continued with a full example.

  

With a 2x2 there are 8 points for lines to start and end, thus 4 lines. The Catalan number for 4 is 14, but we have two rotations per square and four squares, thus 2^4=16. I was wondering which answers were repeated, the answer was 5 of them, giving only 12 unique answers. Meaning 2 were missing. I went about drawing all the answers to both problems for a 2x2, and I realized which answer our grid couldn’t produce, and it was limited by our geometry. It’s at this point I decided to switch to a lesson about geometry and orientations/permutations. 

We will begin by giving the class a single tile and asking them to determine how many unique orientations it can have through rotation. We will then introduce tiles with different symmetries and ask whether rotating, reflecting, or inverting them produces a genuinely different image. From there, we will scale the problem up to a grid of tiles. If each tile has a certain number of possible orientations, how many possible arrangements can the entire grid produce? Finally, we will return to the lines themselves and ask what restrictions the geometry of the tiles places on the patterns that can be created, particularly when the lines cannot cross.