Today we will talk about a single, very neglected pillow, which I will call 3/4. To make things look pretty, we will color it depending on its orientation, as follows.
Discriminated for centuries by the other, more curvy pillows, 3/4 only likes to hang out with other 3/4s and hide its straight edges as much as possible. Like so:
These two examples are portions of periodic tilings of the entire plane. This way, all straight edges are hidden, and we have periodically placed holes. This quickly becomes more interesting. Here, for instance, is the only tiling that has just circular holes (up to symmetries, of course):
The holes can be more complicated, like these double holes in subtly different periodic tilings.
Next month, we will analyze these patterns a bit. For today, we end with an example that indicates that the holes can become really big (as we will learn).
For today it’s enough to have learned that also the seemingly uninteresting can do pretty things.