Hans Walser, [20260630]

Closing Figure with Polygons

1 Procedure

Starting at a vertex, we successively construct congruent regular polygons with n vertices until a polygon touches the starting polygon.

2 Examples

2.1 Equilateral Triangles

With equilateral triangles, we reach the closing figure (Fig. 1) after six steps. We denote the number of steps to the closing figure by m.

Fig. 1: Equilateral Triangles

During the closing process, we go around the center once. The number of turns is 1. The number of turns is best counted by connecting the centers of consecutive polygons. This creates a star with m points.

2.2 Squares

With squares, we reach the closing figure (Fig. 2) after only four steps. The number of turns is 1.

Fig. 2: Squares

2.3 Pentagons

Regular pentagons have overlaps. Therefore, the color transparent had to be chosen (Fig. 3). We need ten pentagons to reach the closing figure. The number of turns is 3.

Fig. 3: Pentagons

The two-dimensional figure (Fig. 3) can be interpreted as a top view of the rhombic icosahedron (Fig. 4).

Fig. 4: Rhombic icosahedron. Top view

Similarly, the two-dimensional figure (Fig. 3) can be interpreted as a top view of the rhombic triacontahedron (Fig. 5).

Fig. 5: Rhombbic triacontahedron. Top view

2.4 Hexagons

With regular hexagons, we even reach the closing figure (Fig. 6) in three steps. There are no overlaps. The number of turns is 1.

Fig. 6: Hexagons

2.5 Heptagons

With regular heptagons, overlaps occur again. We need m = 14 heptagons to reach the closing figure (Fig. 7). The number of turns is 5.

Fig. 7: Heptagons

2.6 Octagons

Regular octagons require eight steps (Fig. 8). The number of turns is 3.

Fig. 8: Octagons

Planimetrically, the closing figure consists of 24 rhombuses with equal edge lengths.

2.7 Nonagons

Regular nonagons require m = 18 steps (Fig. 9). The number of turns is 7.

Fig. 9: Nonagons

2.8 Decagons

Regular decagons require only five steps (Fig. 10). The number of turns is 2.

Fig. 10: Decagons

2.9 Eleven-gons

Regular eleven-gonss require m = 22 steps (Fig. 11). The number of turns is 9.

Fig. 11: Eleven-gons

2.10 Dodecagons

For regular dodecagons, we need 12 steps (Fig. 12). The number of turns is 5.

Fig. 12: Dodecagons

The closing figure, viewed planimetrically, consists of 60 rhombuses with equal edge lengths.

2.11 Thirteen-gons

For regular thirteen-gons, we need 26 steps (Fig. 13). The number of turns is 11.

Fig. 13: Thirteen-gons

2.12 Fourteen-gons

For regular foteen-gons, we only need 7 steps (Fig. 14). The number of turns is 3.

Fig. 14: Fourteen-gons

2.13 Fifteen-gons

For regular fifteen-gons, we need 30 steps (Fig. 15). The number of turns is 13.

Fig. 15: 15-gons

2.14 16-gons

For regular 16-gons, we need 16 steps (Fig. 16). The number of turns is 7.

Fig. 16: 16-gons

Planimetrically, the closing figure consists of 112 rhombuses with equal edge lengths.

2.15 17-gons

For regular 17-gons, we need 34 steps (Fig. 17). The number of turns is 15.

Fig. 17: 17-gons

2.16 18-gons

For regular 18-gons, we only need 9 steps (Fig. 18). The number of turns is 4.

Fig. 18: 18-gons

2.17 19-gons

For regular 19-gons, we need 38 steps (Fig. 19). The number of turns is 17.

Fig. 19 19-gons

2.18 20-gons

For regular 20-gons, we need 20 steps (Fig. 20). The number of turns is 9.

Fig. 20: 20-gons

The closing figure, viewed planimetrically, consists of 180 rhombuses with equal edge lengths.

3 Counts

The required number of steps m and the number of turns u vary considerably (Table 1)

#vertices n	#steps m	#turns u
3	6	1
4	4	1	Figure with four congruent squares
5	10	3
6	3	1
7	14	5
8	8	3	Figure with 24 rhombuses of equal edge length
9	18	7
10	5	2
11	22	9
12	12	5	Figure with 60 rhombuses of equal edge length
13	26	11
14	7	3
15	30	13
16	16	7	Figure with 112 rhombuses of equal edge length
17	34	15
18	9	4
19	38	17
20	20	9	Figure with 180 rhombuses of equal edge length

Table 1: Number of steps and number of turns

The following case distinction applies:

The number of vertices n is odd: We need 2n steps. The number of turns is n – 2.

The number of vertices n is even:

The number of vertices n is divisible by 4: We need n steps. The number of turns is n/2 – 1. The figure is a rhombic figure with n(n/2 – 1) rhombuses of equal edge length.

The number of vertices n is not divisible by 4: We only need n/2 steps. The number of turns is (n – 2)/4.