Escher-like Tilings

The artwork of M. C. Escher often incorporated mathematical motifs. Today we'll look at a fun way to make exciting tilings from conventional ones.

Setup

For this activity, you'll want paper squares that you can cut along with scissors and tape. The size of the squares isn't terribly important, but you'll want them small compared to a standard \(8.5" \times 11"\) sheet of paper but large enough to be able to cut and work with. I'll be using \(5" \times \, 3"\) index cards cut into \(3" \times\, 3"\) squares.

Challenges

The introductory challenge is a standard piece of popular math fare. You start by choosing one edge and cut into it from one corner to another, creating an interesting shape that can be removed. Then tape this shape to the opposite edge. Do the same with the remaining pair of edges, cutting from one and taping to the other. The result is a tile that has some jigsaw puzzle piece-like contours. You can then repeatedly trace your shape on paper with a marker, each time fitting it into the nooks and crannies of the pieces you've already traced.

Here are the steps I used to make a tile and my tracing:

The shapes you can make by cutting from one edge and taping to the opposite are plentiful. Kids enjoy trying to figure out what their shape might be: a dinosaur, cat, rocket, etc. Here's another:

However, there are other ways to go about this. For example, here is one way to go about it that involves cutting from one side and taping to an adjacent side:

Here we require four different orientations of the shape to tile whereas with our other two, one orientation suffices. What other ways are there to create these tiles from a square? What interesting sorts of symmetries can you create?

Notes

• Not every way of pairing edges works out, since you can create tiles that force corners that no tile can fill. In order to investigate possible tilings without cutting, it's useful to create a note-keeping tile of some sort. Here's one I've designed, with its mirror image in a darker gray:

  

  Using this piece you can think of cutting from a red edge and pasting to a yellow edge as requiring the red and yellow edges of neighboring note-keeping tiles border in a tessellation. (There is no real distinction between being cut from or taped to, as far as the symmetry of the tiling is concerned.) So far we haven't flipped a cut piece before taping it, but that is where having a mirror of the note-taking tile comes in handy.

• If we always cut from one edge and past to a distinct edge without flipping, there are only two ways to go about it, up to symmetry:

  The one on the left corresponds to our first couple examples (cut and tape opposite edges) while the one on the right is our last example (cut and tape adjacent edges).

• If we allow flipping of a piece before taping, this gives us access to three more symmetries:

• If we allow cutting from an edge and pasting to itself, the sensible way to do this is cutting from one half of the edge and pasting to the other, which would correspond to two of the same color arrows bordering but pointing in opposite directions. Allowing this same edge cut and tape but no flips gives two more symmetries:

  Allowing for flips gives another two more:

• The sensible way to interpret same-color arrows bordering and pointing in the same direction is that the corresponding edge was uncut, leaving a border along straight edges. This is potentially interesting but we'll skip it here. If you wish to explore it, you might consider taping two tiles together at these straight edges just to keep the shapes looking interesting.

Extra Challenges

• Because the equilateral triangle and regular hexagon are the only other regular polygons that tile the plane, they were great shapes to explore using this cut-and-tape technique. While the edges of the hexagon pair off nicely, the triangle must have at least one of its edges paired with itself.

• For a more challenging monotile, pentagon tilings can be an interesting place to look. Some nice pentagonal tiling patterns are:

  • Prismatic

  • Cairo

  • Floret

• If you want to create sets of two or more tiles that work nicely together, Archimedean tessellations are a nice place to start.

This post was sponsored by the Julia Robinson Mathematics Festival