Ekaterina Y. Arutyunova
Pope High School, 3001 Hembree Road, Marietta, GA, 30062, USA
ekaterina.arutyunova2001@gmail.com
Abstract
The paradox of cutting the Möbius strip is that longitudinal cutting is ambiguous as a result of cutting, unlike the cylindrical surface. The configurations and geometrical dimensions of the obtained strips depend on the distance from the cut to the edge of the surface. To date, the parameters and characteristics of the Möbius strip have been reliably identified, which determine the ambiguity of the cutting result, unlike the cylindrical surface. The interest in the cutting paradox is of both scientific and practical interest for cardiac surgery. If we manage to resolve this paradox, that is, to find out what parameters and characteristics lead to the ambiguity of the cutting, this will solve one of the problems in cardiac surgery.

Thus, the problem (solution of which is described in this article) is the identification, experimentally and theoretically, of factors that determine the uniqueness of the configurations obtained by cutting the Mobius strip and studying them.
Keywords
Topology; Cylindrical surface; Möbius Strip; One-sided Surface; Surface cutting;
Introduction
Möbius strip, also called Möbius loop is a model that can be obtained by turning a long strip of paper a half turn and then connecting its ends together.1 The study of Möbius strip and many other interesting objects is topology. Topology is a branch of mathematics that studies the invariable properties of an object during its continuous deformation (stretching, compression, bending without breaking integrity).
The father of this surface is August Ferdinand Möbius, a student of Gauss. He wrote a lot of geometry works, but became famous for the discovery of a one-sided surface in 1858.

Cylindrical surface and Möbius strip
For a better understanding of the differences in the Möbius strip from the usual cylindrical surface, it is necessary to compare them.

From the comparison it can be seen that the cylindrical surface and the Möbius surface are completely different geometric figures, although they are made of the same long strip.
Cutting a cylindrical surface and Möbius strip
As already known, any longitudinal cutting of a cylindrical surface gives two cylinders of different heights. At the longitudinal cutting of the Möbius strip, the figures have completely different configurations depending on the distance from the scissors to the edge of the tape.2 This paradox is especially interesting for many topologists.3
This experiment was performed to compare the obtained figures in the longitudinal cutting of a cylindrical surface and Möbius strip. The researcher hypothesizes that after experiments and studying the obtained geometric figures, the paradox can be explained analytically using the condition of the proportionality between the area (S) of the Möbius strip and width of the Möbius strip (λ), radius of the circle forming the Möbius strip (R); and condition of the proportionality between the length of the edge (L) and width of the Möbius strip (λ), radius of the circle forming the Möbius strip (R) .
Methods
To begin with, initial samples of the cylindrical surface and Möbius strip were made from a strips of paper of equal length and width (Figure 3).4 For ease of cutting made models, the striped paper was used.
Experiment #1
After preparing the models, the longitudinal cutting of the two surfaces begins into one-sixth of the initial width. As can be seen from the experiment, two cylindrical surfaces of different widths were obtained by longitudinal cutting of a cylindrical surface. By splitting the Möbius surface longitudinally, two connected tapes were obtained: one twice as large and with 3 twists, and one initial Möbius strip (Figure 4).
Experiment #2
Now we increase the cut strip to one-fifth of the initial width of the strip. From the experiment, two cylindrical surfaces of width differences were obtained by longitudinal cutting of a cylindrical surface.
At longitudinal cutting of the Möbius surface, two connected tapes were obtained: one twice as large and with 2 twists, and one Möbius strip of the original size (Figure 5).
Experiment #3
Longitudinally we cut the cylindrical surface and the Möbius surface, retreating from the edge of the strip by one-fourth of the original strip width. From the experimental results it can be seen that cutting a cylindrical surface, two new cylindrical surfaces of different heights were obtained. When cutting the Möbius surface, two connected tapes were obtained again: one twice as large and with 2 twists, and one Möbius strip of the original size.
Experiment #4
Longitudinally we cut the cylindrical surface and the Möbius surface, retreating from the edge of the strip by one-third of the initial strip width. From the experimental results it can be seen that cutting a cylindrical surface, two new cylindrical surfaces of different heights were obtained. When cutting the Möbius surface, two connected tapes were obtained: one twice the size and with 3 twists, and one Möbius strip of the original size.
Experiment #5
In the last experiment, we cut the cylindrical surface and the Möbius surface in half lengthwise. From the results of the experiment it is seen that by cutting a cylindrical surface in half, two equal cylinders were obtained. When cutting the Möbius surface in half, one surface was obtained. This surface is twice the original Möbius surface and has 4 twists.
Results and Discussion
Following the results of five experiments with cutting a cylindrical surface and Möbius surface, the summarizing result is shown in Table 2. From the table, it is possible to predict what will happen with the longitudinal cutting of a cylindrical surface. It is impossible to guess what happens with different longitudinal cutting of the Möbius strip.
Explanation the paradox of cutting Möbius strip
Analyzing the obtained experimental results, there is the question of the nature of this geometric topology paradox in experiments with the longitudinal cutting of the Möbius strip. One of the explanations for this paradox is to apply condition of the proportionality between the area (S) of the Möbius strip and width of the Möbius strip (λ), radius of the circle forming the Möbius strip (R); and condition of the proportionality between the length of the edge of the Möbius strip (L) and width of the Möbius strip (λ), radius of the circle forming the Möbius strip (R). These dependencies are obtained experimentally (for the case λ << R).

WHERE

WHERE

(Lm – length of the edge of the Möbius strip, Lc – length of the edge of the cylindrical surface, Sm – area of the Möbius strip, Sc – area of the cylindrical surface, R – radius of the circle forming the Möbius strip, λ – half-width of the Möbius strip).
The ambiguity of the figures obtained when cutting a Mobius strip is due to the presence of a part of the expression. This part of the expression is associated with twisting. There is no square part () of the expression for the cylindrical surface, because the cylinder is curved, but not twisted.
The impact on the Cardiac Surgery
In cardiac surgery, there is problem of determining the size, shape, thickness of the material for the cardiac muscle patch, which is attached to the myocardium in the place of the scar from a myocardial infarction or aneurysm (Figure 9).
In accordance with the discovery of Dr. Francisco Torrent-Guasp and Dr. Gerald Buckberg myocardium is a thick curved-twisted strip with a Möbius strip topology (Figure 10).5-6

This cardiological discovery complicates the issue of determining the configuration of the patch in terms of its location on the myocardium (on the curved-twisted surface with the topology of Möbius strip. The factor influencing the solution of this question is the paradox of the cutting the Möbius strip.
Therefore, the results of the research described in this article can be applied in cardiac surgery to determine the configuration and size of the cardiac muscle patch.
Conclusion
This article describes the conduct of experimental and theoretical studies of the paradox of cutting Möbius strip; comparison with the cutting of the cylindrical surface; generalization of the results. The consequence of this, it is the identified characteristics and parameters of the Möbius strip, the values of which determine the configuration and the geometric dimensions of the resulting figures after longitudinal cutting. Significant parameter from experiments and analysis is the width of the cut strip during the longitudinal cutting of the Möbius strip.
The results of these studies have practical cardiac surgery significance.
Acknowledgements
I acknowledge my father, Yurii Arutyunov, for his help and support in research.
References Fuchs D. Möbius strip. Variations on the Old Theme. J. Quantum. 1979, 1, 2-9. Marushina T.D. Cutting tapes. Studies in the field of natural sciences. [Online] 2014, 7. http://science.snauka.ru/2014/07/7567 (accessed Feb 8, 2019). Sangalov M.E. Using the experiment in the study of topological properties of surfaces, International scientific.-practical. conference “Modern directions of theoretical and applied research ’2011», Odessa, Black Sea, UA, 2011. ‘’Modern directions of theoretical and applied research: Odessa, Black Sea, 2011; pp 73–77. Kordemsky B. Topological experiments with their own hands. J. Quantum. 1974, 3, 73-75. Buckberg G. Basic science review: the helix and the heart. J Thorac Cardiovasc Surg. 2002, 124(5), 863-83. Kocica M.; Corno A.; Carreras-Costa F.; Ballester-Rodes M.; Moghbel M.; Cueva C.; Lackovic V.; Kanjuh V.; Torrent-Guasp F. The helical ventricular myocardial band: global, three-dimensional, functional architecture of the ventricular myocardium. Eur. J. Cardio-thoracic surgery. 2006, 4, 21-40.
Authors
Ekaterina Arutyunova conducts researches in the field of Physics and Mathematics with her father. She plans to continue and develop her science projects in college at the respective major.