Intriguing Geometric Dissections

Any two polygons with equal area can be dissected into a finite number of pieces to form each other. One of the most interesting dissection or geometric equidecomposition puzzles is that discovered by Harry Lindgren in 1951 (see Fig. 2 further below). As you know, in this kind of puzzles, the geometric invariant is the area, since when a polygon is cut and its pieces are distributed differently, the overall area doesn’t change.

Lindgren was the first to discover how to cut a dodecagon into a minimal number of pieces that could pave a square, when rearranged differently. His solution is very elegant, he first built a regular Euclidean pavement by cutting a dodecagon as shown in fig. 1.a, then arranging the four pieces symmetrically on the plane (fig 1.b). The tessellation achieved with these pieces corresponds, by superposition, to a regular paving of squares. The example in fig. 2 shows how the puzzle appears once finished.

dodecagon fig. 1

The most remarkable aspect is that we can vary almost to infinity the basic cut, as illustrated in fig. 3 and 4. Obviously, it is impossible to cut the dodecagon into less than 6 pieces to make a square. But who knows? Maybe one of you will have an original idea?

Using geometric puzzles in your class is always a hit with the students. Whether you use them as an introduction for a new chapter , as a practice break, or to end a class. Please, feel free to download the templates (PDF file) to make the octagon dissection puzzles in your own class.