Dissect and Conquer

Many basic mathematical concepts are easy to convey to the layperson. For instance, most people are ok with numbers, distances, and right angles. An example of a concept that I found very hard to explain is that of a group action, and the related concept of a fundamental domain. Equivalence classes in general seem to be completely out of this world.

Lattice 01

Periodic tilings give many examples. The colored square tiling above for instance is periodic with respect to a group of (color respecting) translations, all of which can be written as a combination of the two orange arrows at the bottom left, or their reversed arrows. The collection of all these translations is called the lattice of the tiling.

More complicated looking tilings can have simpler lattices. For instance, the tiling by the differently sized yellow and blue squares below has the same lattice as the tiling by the outlined orange squares.

2squares 01

The not so simple consequence of this simple observation is the following dissection of a large square into two smaller squares:

2squaressol 01

The reason why this works is that both the large square and the union of the two smaller squares are a fundamental domain for the common lattice of the two tilings. You can think about the orange grille as a cookie cutter, and the yellow and blue squares as periodic dough. Cutting a blue and a yellow dough square with that cutter gives you five pieces that just fill one larger square of the cutter. There are many different ways to place the cutter over the dough, and all are allowed, as long as cutter and dough have the same lattice. This means that you can translate the cutter, but not rotate.

This method is well known among dissectionists. My favorite example is the dissection of a regular octagon into a square.

Octagonsol 01

To explain how to find it, we tile the plane with octagons and yellow squares. This tiling has the same lattice as a tiling by two unequal squares, where we choose the smaller purple squares to be exactly the same size as the yellow squares.

Octagon 01

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