Simuorb: a new method for generating and describing the intersection points of clique-arrangements

TL;DR

Simuorb addresses the distribution and counting of intersection points in clique-arrangements of diagonals from a regular $n$-gon by introducing a triplet parameterization $(p,q,r)$ with $s=n-(p+q+r)$ and a radius function $J_n(p,q,r)$ that determines origin-centered orbits. The method characterizes inscribed quadrilaterals as simple or complex and uses orbit equivalence to group points on the same radius, enabling efficient counting without graph construction. Central to the approach is a triple-loop algorithm that enumerates admissible triplets, computes orbit sizes and multiplicities, and uses arc-length criteria to distinguish cocircular point sets. The combination of explicit radius formulas, equivalence relations, and the Simuorb algorithm yields a robust, scalable framework for analyzing large clique-arrangements and their intricate multiplicity structures, with concrete numerical validation presented.

Abstract

This work, which may be seen as a companion paper to \cite{RS2}, handles the way the intersection points made by the diagonals of a regular polygon are distributed. It was stated recently by the authors that these points lie exclusively on circles centered on the origin and also the way their respective radii depend on the four indices of the vertices of the initial regular $n$-gon which characterize the two straight lines underlying the intersection points. Because these four vertices are located at preset positions on the the regular $n$-gon inscribed in the unit circle whose path-length perimeter is constant, it allows the orbits to be characterized by 3 parameters instead of 4, describing roughly the lengths of the paths between the first three vertices, whether the quadrilateral described by these four vertices is simple or complex. This approach enables us to deal with the orbits generated by the clique-arrangement, and to handle their cardinalities as well as the multiplicities of the associated intersection points. A reliable counting-algorithm based on this triplet strategy is provided in order to enumerate the intersection points without generating the associated graph. The orbits being simulated, we call this method \textit{Simuorb}. The procedure is robust, fast and allows a comprehensive understanding of what is happening in a clique-arrangement, whether it contains a large number of points or not.

Figures (10)

• Figure 1: A partial view of the cyclotomic clique-arrangement for $n=20$ together with its circular orbits
• Figure 2: The three configurations generating $z_{i,j,k,\ell}=\mathcal{D}_{i,j}\cap\mathcal{D}_{k,\ell}$ either outside or inside the unit circle
• Figure 3: The 3 cyclic quadrilaterals generated from 4 vertices $z_A,z_B,z_C,z_D$ and their associated 3 intersection points.
• Figure 4: Values of $\sqrt{J_{20}}$ outside the unit circle
• Figure 5: Values of $\sqrt{J_{20}}$ inside the unit circle

Theorems & Definitions (12)

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