It all started with a question about polysticks: I wanted to see how to tile parts of the square grid with 3-sticks so that there are always three 3-sticks touching. A 3-stick is nothing but a T with all segments the same length: it has 3 arms that end at the hands, and are joined together at a head. By tiling I mean that the arms align with the edges of the standard square grid and don’t overlap. And, as I said, I want always three of them to hold hands. Here is an example (that you should imagine continued periodically):
The condition about holding hands in threes means that each such tiling has a dual tiling where the new 3-sticks have their heads wherever three hands come together, the same arms, and the previous heads are replaced by three hands. This also implies that a tiling and its dual will tile the same portion of the square grid, as below to the left and right.
We can also combine a tiling and its dual into a single figure by centrally scaling each 3-stick by 50%, and taking the union: The gaps created by the scaling makes room for the 3-sticks from the other tiling. You can examine the result up above in the middle. The new skeleton will have all 3-sticks hold hands in pairs instead of in triples. Here is another, more complicated example. The original periodic 3-stick tiling:
The inflated version that combines the original with its dual:
In the inflated version we can replace each 3-stick by a square so that the sides touch when the 3-sticks hold hands. The unused edge of the square is pushed inwards, turning the square into the familiar 3/4 pillow we admired the last time (the next image shows only a quarter of the previous piece. It repeats itself using horizontal and vertical translations).
As a final simplification, we can fill in the holes as follows: We replace the 3/4 pillows that border a hole by the polyomino they cover (thereby filling in the hole). Below are the two simplest polyominoes that surround a hole:
These polyominoes will tile the plane as before, because each 3/4 pillow must belong to exactly one hole. Our 3/4-pillow tiling now becomes a very simple polyomino tiling:
Thus periodic 3-stick-tilings with triple hand holding, 3-stick tilings with hand holding in pairs, 3/4-pillow tilings with holes, and tilings by polyominoes that surround holes are all the same thing.
You can use this to design much more intricate patterns, with holes of any size, for instance.