In the meantime, I will leave you with some things to think about. This blog post has gotten long enough so I think I will stop there! But I plan to write another followup post or three, because we have only just scratched the surface. This is another way to think about what we are doing when we modify matching edges of a square-we are really just cutting the torus along different lines. So a square tessellation is “what a torus looks like to an ant”, that is, what we get if we cut open a torus and glue infinitely many copies together so that each copy picks up exactly where the previous copy left off.īut there are lots of ways to cut a torus open so it lays flat! And all of them will produce some shape which tiles the plane just like a square. The ant cannot tell whether there are infinitely many copies of the letter “P”, or if there is only one letter “P” that it keeps coming back around to. Explore semi-regular tessellations using the Tessellation Interactivity below. Each vertex has the same pattern of polygons around it. Or perhaps it lives on a torus? (Or an infinitely long cylinder?) The point is that there is no way for the ant to tell the difference. Semi-regular tessellations (or Archimedean tessellations) have two properties: They are formed by two or more types of regular polygon, each with the same side length. After exploring for quite a while, the ant thinks its world looks something like this: Sometimes it finds places where the ground is black, and after making some maps the ant realizes that these places are shaped like a giant letter “P”. Sometimes it finds places where two lines cross at right angles. Occasionally there is a straight line drawn on the ground, extending off into the distance. The ant starts walking around, exploring its world. (You may know some tiny creatures in a similar situation who live on a sphere.) Unlike those tiny creatures on the sphere, however, the ant has nothing it can use to draw with, no objects to leave behind, etc., so it has no way to tell whether it has ever been to a particular location before. The ant is so small that it can’t tell that the surface it lives on is curved.

Now imagine a very tiny ant who lives by itself on the surface of the torus.

First, we glue the top and bottom edges together, resulting in a cylinder then bend the ends of the cylinder around to match up the left and right edges, resulting in a torus (a donut shape). Now, instead of matching up the edges of a bunch of copies of the same square, we can think about taking one square and gluing matching edges together.

Tessellation square maker