1 What it's about

Dissection of regular dodecagons with side length n into equilateral triangles and squares with the same side length

2 Classical Dodecagon

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

Fig. 1: Regular dodecagon with side length 1

3 Rings

Figure 2 shows a ring whose inner contour corresponds to the outline of the regular dodecagon with side length 1 (Fig. 1). However, the outer contour is not a regular dodecagon. The sides have alternating lengths of 1 and 2.

The second ring in Figure 2, conversely, has a dodecagon as its inner contour with alternating side lengths of 1 and 2. The outer contour is a regular dodecagon with side length 2.

Fig. 2: Rings

The two rings can be nested inside each other to form a double ring, whose inner and outer contours are each a regular dodecagon.

Similarly, we can construct such a double ring for each side length (Figs. 3 to 5).

Fig. 3: Outer side length 3

Fig. 4: Outer side length 4

Fig. 5: Outer side length 5

These double rings are all constructed according to the same pattern. We can construct such a double ring for each side length n.

4 Combinations

We can now connect such double rings together, thus obtaining a dissection of a regular dodecagon with side length n.

However, two double rings can be connected in two different ways (Fig. 6).

Fig. 6: Two ways of connecting

In the second example, the outer double ring has been rotated by 30°.

For a dodecagon with side length n, including the innermost dodecagon, there are a total of n double rings involved. We therefore have n – 1 (“fence post problem”) possibilities of a relative rotation. According to our method, there are thus 2ⁿ^–1 possible solutions. Figures 7 and 8 show two extreme solutions.

Fig. 7: Solution without rotations

Fig. 8: Solution with four relative rotations

Weblinks

Hans Walser: Dodecagon Dissection

https://walser-h-m.ch/hans/Miniaturen/Z/Zwoelfeck-Zerlegung/Dodecagon_Dissection.html