I have occasionally written about polyforms before. These are shapes obtained by putting simple shapes (like squares) together to form more complicated shapes. In the case of two squares, you ged dominoes, and more generally polyominoes. If you use other shapes, you get general polyforms.
If we, in the insatiable desire for more, allow the shapes to change size, we get even more general polyforms. The ones we study today I will call poly-worms. We start with an isosceles right triangle, halve it, and attach the smaller copy to the larger, edge-to-edge. Then we keep going, halving and attaching restlessly. Above you see the first four generations, giving us eight 4-worms, which come in mirror symmetric pairs.
Above is a template that allows you to print all eight 4-worms at once. Growing the polyworms further leads to problems with self-overlapping, but also to the tantalizing possibility of having polyworms with infinitely many sides. Maybe more about this in the finite future.
Let’s practice tiling with 4-worms. Below are combinations of 4-worms that can be used to tile the plane periodically by translating them. There are also two 4-worms that tile the plane by themselves. It should be amusing to study this for general polyworms.
Here are two puzzles, hopefully not too easy. The goal is to tile the left one with 8 and the right one with 16 4-worms. You will need the same number of each kind.
The Inner Frame 