Proceedings of the 1st International Symposium on Square Bamboos and the Geometree (ISSBG 2022)

Remarks:

In what follows, a polygon with three vertices is called a 3-gon, a polygon with 16 vertices is called a 16-gon etc. Furthermore, it should be noted that the methodology uses planar sections, whereas Generalized Möbius-Listing bodies are three dimensional. This means that the resulting ν-gons with the same color and shape form a single body after cutting.

2.1. First Case

The diagonal contains the center of symmetry of the polygon (Fig. 1).

Figure 1

Examples for the first case with different parameter values.

Then $m=2k$ and $\kappa =k$, in this case for arbitrary $j=1,\dots ,m$ and the center of symmetry of the polygon is located on this diagonal.

Case 1A. If $j=m=2k$ or $j=0$, then two different, but identical plane $\left(k+1\right)$-gons appear after such VV cutting, but the Möbius phenomenon never occurs.

Case 1B. If $j=k$ and $m/j=2$, then two identical plane $\left(k+1\right)$-gons appear after such VV cutting and the Möbius phenomenon always occurs.

Case 1C. If $j<k$, and $m/j$ is even number, then $m/j$ pieces identical $\left(j+2\right)$-gons appear after such VV cutting and the Möbius phenomenon always occurs.

Case 1D. If $j=2\beta$ is an even number and $\left[m/j\right]$ is an odd number, then two different groups appear after VV cutting, each of which consists of $\left[m/j\right]$ pieces of $\left[\beta +2\right]$-gons!

2.2. Second Case

For arbitrary $m$ when $\kappa \le j$, then $\left[m/j\right]$ similar $\left(\kappa +1\right)$-gons and one piece of $\left[m-\left(\kappa -1\right)\frac{m}{j}\right]$-gon appears after VV cutting! (Fig. 2)

Figure 2

Examples for the second case with different parameter values.

2.3. Third Case

For arbitrary $m$ when $\kappa >j$. This turned out to be the most difficult case to study, which has many branches and shows a strong connection with the structure of numbers and geometric shapes.

Case 3.I. This subcase is considered separately, since for any values of m and n (even when these numbers are coprime) it is realized! For arbitrary m and $\kappa =2,\mathrm{...},\left[m/2\right]$ when $j\equiv 1$, then two different groups, each of which consists of $\left[m/j\right]$ pieces 3-gons and $\left(\kappa -2\right)$different groups, each of which consists of $\left[m/j\right]$ similar 4-gons and one $\left[m/j\right]$-gon appears after such VV cutting! (Fig. 3)

Figure 3

Examples for the third case with different parameter values, but j = 1.

Case 3.GA. For arbitrary m and $\kappa =j\beta <\left[m/2\right]$, where $\beta \in Z$ an integer and $\beta >1$, then one group consisting of $\left[m/j\right]$ pieces 3-gons, $\left(\beta -2\right)$different groups, each of which consists of $\left[m/j\right]$ pieces 4-gons, one group consisting of $\left[m/j\right]$ pieces $\left(j+2\right)$-gons and one piece of $\left[m/j\right]$-gon appears after cutting! (Fig. 4)

Figure 4

Examples for the case 3.GA with different parameter values but always $\kappa =j\beta$.

Case 3.GB. For arbitrary m and $\kappa =j\beta +l<\left[m/2\right], \beta >1$ and $\beta \in Z$ where $l=1,2,\mathrm{...},\left(\beta -1\right),$ then one group consisting of $\left[m/j\right]$ pieces 3-gons, $\left(\beta -2\right)$different groups each of which consists of $\left[m/j\right]$ pieces 4-gons, one group consisting of $\left[m/j\right]$ pieces $\left[j-\left(l-4\right)\right]$-gons, one group consisting of $\left[m/j\right]$ pieces $\left(l+2\right)$-gons and one piece of $\left[m/j\right]$-gon appears after cutting! (Fig. 5)

Figure 5

Examples for the case 3.GB with different parameter values.

Case 3.GC. For arbitrary m and $j<\left[m/2\right]$, when $\kappa =j\beta +l \text{and} \beta =1,l=1,2,\mathrm{...},\left(j-1\right)$ then one group consisting of $\left[m/j\right]$pieces $\left[j-\left(l-3\right)\right]$-gons and one group consisting of $\left[m/j\right]$pieces $\left(l+2\right)$-gons and one piece of $\left[m/j\right]$-gon appears after cutting! (Fig. 6)

Figure 6

Examples for the case 3.GC with different parameter values.