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

Geometric Figures Which Appear After VV Cutting in the Radial Cross Section of Generalized Möbius-Listing Bodies
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1. INTRODUCTION

Here we use all the traditional definitions and notation introduced in previous works [1,2,3]. In particular, VV cutting means cutting a polygon from vertex to vertex, with a chordal knife, which cuts the polygon in exactly two points. The following new parameters turn out to be decisive for these results.

  1. 1:

    VV1i  is the diagonal connecting the first and i-numbered vertices of the regular m-polygon. It is known from previous works that it suffices to consider i=2,...,m/2, where m/2 is an integer part of a fraction.

  2. 2:

    κi1 – number that is unique, for our needs, describing the given diagonal.

  3. 3:

    jgcdm,n – this is a very important and informative parameter characterizing this GMLmn body. If j=0 we have the GML body without twisting or classical toroidal body.

We have identified three cases in which different patterns of different planar geometric figures in the radial cross section of the GML bodies are obtained.

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. RESULTS

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 κ=k, in this case for arbitrary j=1,,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 k+1-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 k+1-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 j+2-gons appear after such VV cutting and the Möbius phenomenon always occurs.

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

2.2. Second Case

For arbitrary m when κj, then m/j similar κ+1-gons and one piece of mκ1mj-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 κ>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 κ=2,...,m/2 when j1, then two different groups, each of which consists of m/j pieces 3-gons and κ2 different groups, each of which consists of m/j  similar 4-gons and one m/j-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 κ=jβ<m/2, where βZ an integer and β>1, then one group consisting of m/j pieces 3-gons, β2 different groups, each of which consists of m/j pieces 4-gons, one group consisting of m/j pieces j+2-gons and one piece of m/j-gon appears after cutting! (Fig. 4)

Figure 4

Examples for the case 3.GA with different parameter values but always κ=jβ.

Case 3.GB. For arbitrary m and κ=jβ+l<m/2, β>1 and βZ where  l=1,2,...,β1, then one group consisting of m/j pieces 3-gons, β2 different groups each of which consists of m/j  pieces 4-gons, one group consisting of m/j pieces jl4-gons, one group consisting of m/j pieces l+2-gons and one piece of m/j-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<m/2, when κ=jβ+l and β=1, l=1,2,...,j1 then one group consisting of m/j pieces jl3-gons and one group consisting of m/j pieces l+2-gons and one piece of m/j-gon appears after cutting! (Fig. 6)

Figure 6

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

3. FINAL REMARK

It should be obligatorily noted that at present this regularity has been discovered and tested on many examples of parameters, but by this time there is no complete mathematical proof. Therefore, we call this regularity a “hypothetical regularity”. I also want to note that the situation is almost repeating itself, when in 2014 Johan Gielis and I found a general regularity about the number of GMLmn-cutting bodies in different ways, and only in 2019 we were able to fully prove this! [1]

ACKNOWLEDGMENTS

The author expresses his deep gratitude to Johan Gielis for the constant and very attentive discussion of these issues and the constant belief that I will find a regularity, as well as for valuable advice. The author is very grateful to Levan Roinishvili, who is his former student, for creating a convenient and easy-to-use computer program, with the help of which examples of figures appearing after such cuts were built.

REFERENCES

J. Gielis, I. Tavkhelidze. The General Case of Cutting of Generalized Möbius-Listing Surfaces and Bodies. 4Open, 2020, 3: 7. https://doi.org/10.1051/fopen/2020007
I. Tavkhelidze, J. Gielis. The Process of Cutting GMLmn Bodies With dm-Knives. Reports of Enlarged Sessions of the Seminar of I. Vekua Institute of Applied Mathematics, 2018, 32: 67–70.
I. Tavkhelidze, J. Gielis. Structure of the “dm-Knives” and Process of Cutting of GMLmn or GRTmn Bodies. Reports of Enlarged Sessions of the Seminar of I. Vekua Institute of Applied Mathematics, 2019, 33: 70–73.

Cite This Article

ris
TY  - CONF
AU  - Ilia Tavkhelidze
PY  - 2023
DA  - 2023/11/29
TI  - Geometric Figures Which Appear After VV Cutting in the Radial Cross Section of Generalized Möbius-Listing Bodies
BT  - Proceedings of the 1st International Symposium on Square Bamboos and the Geometree (ISSBG 2022)
PB  - Athena Publishing
SP  - 41
EP  - 45
SN  - 2949-9429
UR  - https://doi.org/10.55060/s.atmps.231115.004
DO  - https://doi.org/10.55060/s.atmps.231115.004
ID  - Tavkhelidze2023
ER  -
enw
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