1 What it's about

Dissection of equiangular dodecagons into equilateral triangles and squares of equal side length

Fibonacci numbers

2 Classical dodecagon

Figure 1 shows a dissection of the regular dodecagon into twelve equilateral triangles and six squares of equal side length.

Fig. 1: Dissection of the regular dodecagon

The dissection can be divided into a kernel and a ring (Fig. 2).

Fig. 2: Kernel and ring

We extend the regular dodecagon by adding another ring (Fig. 3). The ring has a thickness of 1.

Fig. 3: Extension by a ring

The resulting dodecagon is no longer regular. The sides alternate between lengths of 1 and 2. However, the dodecagon is equiangular.

We now extend with a ring of thickness 2 (Fig. 4). The small squares in the extension ring combine to form a larger square.

Fig. 4: Extension with a ring of thickness 2

The next extension has a thickness of 3 (Fig. 5).

Fig. 5: Extension with a ring of thickness 3

The next extension has a thickness of 5 (Fig. 6).

Fig. 6: Extension with a ring of thickness 5

The next extension has a thickness of 8 (Fig. 7).

Fig. 7: Extension with a ring of thickness 8

In each extension step, the small squares combine in a checkerboard pattern to form a larger square. The thicknesses of the extension rings are:

1, 1, 2, 3, 5, 8

This is the beginning of the Fibonacci sequence.

Isoceles trapezoids lie between the checkerboard-like squares. Their side lengths are three consecutive numbers in the Fibonacci sequence (Fig. 8).

Explanation: The first two expansion steps have a thickness of 1.

From the isosceles trapezoids, we can read off the Fibonacci recursion (Fig. 8).

Fig. 8: Fibonacci recursion