A Note on the Geometry of Closed Loops

Shvalb, Nir; Frenkel, Mark; Shoval, Shraga; Bormashenko, Edward

doi:10.3390/math11081960

Cited by 2 publications

(1 citation statement)

References 21 publications

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“…Present work is generalizing the approach introduced in ref. 13, in which the coloring procedure was suggested for the points arranged on a closed contour and connected as a complete graph. The procedure exploited the sign of the slope of the straight line connecting the points placed on the Jordan curve, for coloring.…”

Section: Introductionmentioning

confidence: 99%

Ramsey Theory and Transformations of Coordinate Systems

Frenkel¹,

Shoval²,

Bormashenko³

2023

Preprint

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Application of the Ramsey Theory to the set of the points placed in the plane is discussed. Consider a set of N points located in the same plane. The straight lines connecting these points are yikx=αikx+βik i,k=1...N. Listed below values of slopes αik are possible: αik&gt;0; αik=0; αik is not defined; αik&lt;0. Following coloring procedure is introduced: we connect the pairs of points for which αik&gt;0, αik=0 or αik is not defined, with the red links, and the pairs of points for which αik&lt;0 place with green links. The suggested coloring procedure enables building of the complete bi-colored graph for any set containing N points located in the same plane. We apply the Ramsey theorem to the complete graph emerging from the suggested coloring. For the set containing N=6 points at least one monochromatic triangle will necessarily appear in the graph. The values of the slopes αik depend on the chosen coordinate system. The rotation of coordinate axes changes the coloring of the graph; however, at least one monochromatic triangle will be present for N=6. The introduced coloring procedure diminishes the order of the symmetry group of the regular hexagon, irrespectively to the orientation of the coordinate axes. We hypothesized that this will be true for arbitrary regular n-polygon, independently on the orientation of coordinate axes. The inverse bi-color Ramsey graphs arise, when we replace red links appearing in the source graph with red ones, and vice versa. The total number of triangles in the “direct” and “inverse” Ramsey graphs is the same. We considered the particular case of the set of points for which αik&gt;0 or αik&lt;0 is true in the given coordinates. In this particular case, the rotation of the Cartesian coordinate axes to the angle θ=π2 yields the inverse complete graph. Generalization of this coloring for an arbitrary number and arbitrary location of the source points is introduced.

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Section: Introductionmentioning

confidence: 99%

Ramsey Theory and Transformations of Coordinate Systems

Frenkel¹,

Shoval²,

Bormashenko³

2023

Preprint

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Riemannian Manifolds, Closed Geodesic Lines, Topology and Ramsey Theory

Bormashenko

2024

Mathematics

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We applied the Ramsey analysis to the sets of points belonging to Riemannian manifolds. The points are connected with two kinds of lines: geodesic and non-geodesic. This interconnection between the points is mapped into the bi-colored, complete Ramsey graph. The selected points correspond to the vertices of the graph, which are connected with the bi-colored links. The complete bi-colored graph containing six vertices inevitably contains at least one mono-colored triangle; hence, a mono-colored triangle, built of the green or red links, i.e., non-geodesic or geodesic lines, consequently appears in the graph. We also considered the bi-colored, complete Ramsey graphs emerging from the intersection of two Riemannian manifolds. Two Riemannian manifolds, namely (M1,g1) and (M2,g2), represented by the Riemann surfaces which intersect along the curve (M1,g1)∩(M2,g2)=ℒ were addressed. Curve ℒ does not contain geodesic lines in either of the manifolds (M1,g1) and (M2,g2). Consider six points located on the ℒ: {1,…6}⊂ℒ. The points {1,…6}⊂ℒ are connected with two distinguishable kinds of the geodesic lines, namely with the geodesic lines belonging to the Riemannian manifold (M1,g1)/red links, and, alternatively, with the geodesic lines belonging to the manifold (M2,g2)/green links. Points {1,…6}⊂ℒ form the vertices of the complete graph, connected with two kinds of links. The emerging graph contains at least one closed geodesic line. The extension of the theorem to the Riemann surfaces of various Euler characteristics is presented.

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A Note on the Geometry of Closed Loops

Cited by 2 publications

References 21 publications

Ramsey Theory and Transformations of Coordinate Systems

Ramsey Theory and Transformations of Coordinate Systems

Riemannian Manifolds, Closed Geodesic Lines, Topology and Ramsey Theory

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