Dissections and Area

Whenever need to explain what Mathematics is about, one of my favorite examples is the concept of area. The existence of an elementary notion of area hinges on the fact that any two Euclidean polygons have the same area if and only if they are scissor congruent, meaning that they can be cut into congruent pieces using straight cuts. To see this, it suffices to show that any polygon can be dissected into a square. Rect2square

The example above shows how to dissect a well-proportioned rectangle into a square. Here, well-proportioned means that the rectangle is not more than twice as tall than wide. If a rectangle is not well-proportioned, a few cuts parallel to the edges will make it so. Thus any two rectangles of the same area can be dissected int each other. We will use this later.


Next we show that any polygon can be dissected into triangles. By induction, it suffices to find a secant inside the polygon. To find this secant, pretend that the polygon is actually the floor plan of a room, and we are standing at one vertex V . The two adjacent walls lead to two vertices A and B which we can see. If we can see yet another vertex W from our position, we have found our secant VW. If we can’t see another vertex, nothing obstructs our view in the triangular region formed by A, B and V , and thus A and B can be joined by a secant.

As a further simplification, we cut all triangles into two pieces along one of their heights so that all triangles become right triangles.

Now we have a collection of right triangles, which will need to be dissected into a single square.

Triangle2squareTo do so, we dissect each right triangle into a rectangle. This can be done as shown above by dissecting the triangle into two pieces along a segment parallel to one of the legs and dividing the other leg into equal parts.

This leaves us with a collection of rectangles that most likely have different dimensions. So we dissect them into new rectangles that all have all height 1, using the example at the beginning. 

Then, the new rectangles can be lined up edge to edge along their sides of length 1 to form one very wide rectangle that finally can be dissected into a square.


As this was nice and easy, here a challenge: In our dissections, we were allowed to translate and rotate the pieces arbitrarily. What about if we forbid rotations? Can you dissect an equilateral triangle into finitely many pieces and translate them so that the result is the same triangle upside down? Or, can you cut a square, translate the pieces, and thereby achieve a 45 degree rotation of the square? 


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:

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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.

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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