REMARKS ON THE ZIG-ZAG THEOREM

The classical Zig-zag Theorem [1] says that if an equilateral closed 2m-gon shuttles between two given circles of the Euclidean 3-space, then the vertices of the polygon can be moved smoothly along the circles without changing the lengths of the sides of the polygon. First we prove that the Zig-zag Theorem holds also in the hyperbolic, Euclidean and spherical n-spaces, and in fact the circles can be replaced by straight lines or any kind of cycles. In the second part of the paper we restrict our attention to planar zig-zag configurations. With the help of an alternative formulation of the Zig-zag Theorem, we establish two duality theorems for periodic zig-zags between two circles. This revised version was published online in August 2006 with corrections to the Cover Date.     

The lower bound of the number of non-overlapping triangles

András Bezdek proved that if a convexn-gon andn points are given, then the points and the sides of the polygon can be renumbered so that at least [n/3] triangles spanned by theith point and theith side (i=1,2,…n) are mutually non-overlapping. In this paper, we show that at least [n/2] mutually non-overlapping triangles can be constructed. This lower bound is best possible.     

Covering a Convex Polygon by Triangles

According to a theorem of A. V. Bogomolnaya, F. L. Nazarov and S. E. Rukshin, if n points are given inside a convex n-gon, then the points and the sides of the polygon can be numbered from 1 to n so that the triangles spanned by the ith point and the ith side(i=1....,n ) cover the polygon. In this paper, we prove that the same can be done without assuming that the given points are inside the convex n-gon. We also show that in the general case at least [(n/3)] mutually nonoverlapping triangles can be constructed in the same manner.     

Angle orders,regular n-gon orders and the crossing number

A finite poset P(X,<) on a set X={ x ₁,...,x _m} is an angle order (regular n-gon order) if the elements of P(X,<) can be mapped into a family of angular regions on the plane (a family of regular polygons with n sides and having parallel sides) such that x _ij if and only if the angular region (regular n-gon) for x _i is contained in the region (regular n-gon) for x _j. In this paper we prove that there are partial orders of dimension 6 with 64 elements which are not angle orders. The smallest partial order previously known not to be an angle order has 198 elements and has dimension 7. We also prove that partial orders of dimension 3 are representable using equilateral triangles with the same orientation. This results does not generalizes to higher dimensions. We will prove that there is a partial order of dimension 4 with 14 elements which is not a regular n-gon order regardless of the value of n. Finally, we prove that partial orders of dimension 3 are regular n-gon orders for n3.This research was supported by the Natural Sciences and Engineering Research Council of Canada, grant numbers A0977 and A2415.     

Regular polygons in taxicab geometry

A polygon of n sides will be called regular in taxicab geometry if it has n equal angles and n sides of equal taxicab length. This paper will show that there are no regular taxicab triangles and no regular taxicab pentagons. The sets of taxicab rectangles and taxicab squares will be shown to be the same, respectively, as the sets of Euclidean rectangles and Euclidean squares. A method of construction for a regular taxicab 2n-gon for any n will be demonstrated.     

On the perimeters of simple polygons contained in a disk

A simple n-gon is a polygon with n edges with each vertex belonging to exactly two edges and every other point belonging to at most one edge. Brass et?al. (Research Problems in Discrete Geometry, 2005) asked the following question: For n ???5 odd, what is the maximum perimeter of a simple n-gon contained in a Euclidean unit disk? In 2009, Audet et?al. (Discrete Comput Geom 41:208?C215) answered this question, and showed that the optimal configuration is an isosceles triangle with a multiple edge, inscribed in the disk. In this note we give a shorter and simpler proof of their result, which we generalize also for hyperbolic disks, and for spherical disks of sufficiently small radii.     

Lucas' theorem refined ∗

We prove a refined version of the classical Lucas' theorem: if p is a polynomial with zeros a ₁,…,a _n+1 all having modulus one and φis the Blaschke product whose zeros are those of the derivative p ¹, then the compression of the shift S(φ) has its numerical range circumscribed about by the (n+ l)-gon a ₁…a _n+1 with tangent points the midpoints of then+ l sides of the polygon. This is proved via a special matrix representation of S(φ) and is a generalization of the known case for n= 2.     

Going in Circles: Variations on the Money-Coutts Theorem

Given a polygon A ₁,...,A _n, consider the chain of circles: S ₁ inscribed in the angle A ₁, S ₂ inscribed in the angle A ₂ and tangent to S ₁, S ₃ inscribed in the angle A ₃ and tangent to S ₂, etc. We describe a class of n-gons for which this process is 2n-periodic. We extend the result to the case when the sides of a polygon are arcs of circles. The case of triangles is known as the Money-Coutts theorem.     

Minimal Convex k-Gons Containing a Given Convex Polygon

There is a k-gon of minimal area containing a given convex n-gon (k<n) such that k-1 sides of the n-gon lie on the sides of the k-gon. All midpoints of the sides of the k-gon belong to the n-gon. Bibliography: 3 titles.     

Minimal surfaces over stars

A JS surface is a minimal graph over a polygonal domain that becomes infinite in magnitude at the domain boundary. Jenkins and Serrin characterized the existence of these minimal graphs in terms of the signs of the boundary values and the side-lengths of the polygon. For a convex polygon, there can be essentially only one JS surface, but a non-convex domain may admit several distinct JS surfaces. We consider two families of JS surfaces corresponding to different boundary values, namely JS₀ and JS₁, over domains in the form of regular stars. We give parameterizations for these surfaces as lifts of harmonic maps, and observe that all previously constructed JS surfaces have been of type JS₀. We give an example of a JS₁ surface that is a new complete embedded minimal surface generalizing Scherk's doubly periodic surface, and show also that the JS₀ surface over a regular convex 2n-gon is the limit of JS₁ surfaces over non-convex stars. Finally we consider the construction of other JS surfaces over stars that belong neither to JS₀ nor to JS₁.     

Isometric embed-dings between classical Banach spaces,cubature formulas,and spherical designs

If an isometric embeddingl _p ^m →l _q ⁿ with finitep, q>1 exists, thenp=2 andq is an even integer. Under these conditions such an embedding exists if and only ifn?N(m, q) where $$\left( {\begin{array}{*{20}c} {m + q/2 - 1} \\ {m - 1} \\ \end{array} } \right) \leqslant N(m,q) \leqslant \left( {\begin{array}{*{20}c} {m + q - 1} \\ {m - 1} \\ \end{array} } \right).$$ To construct some concrete embeddings, one can use orbits of orthogonal representations of finite groups. This yields:N(2,q)=q/2+1 (by regular (q+2)-gon),N(3, 4)=6 (by icosahedron),N(3, 6)?11 (by octahedron), etc. Another approach is based on relations between embeddings, Euclidean or spherical designs and cubature formulas. This allows us to sharpen the above lower bound forN(m, q) and obtain a series of concrete values, e.g.N(3, 8)=16 andN(7, 4)=28. In the cases (m, n, q)=(3, 6, 10) and (3, 8, 15) some ε-embeddings with ε ～ 0.03 are constructed by the orbit method. The rigidness of spherical designs in Bannai's sense and a similar property for the embeddings are considered, and a conjecture of [7] is proved for any fixed (m, n, q).     

On a problem of Croft on optimally nested regular polygons

We present a solution for the largest regular m-gon contained in a regular n-gon. We find that the answer depends critically on the coprimality of m and n. We show that the optimal polygons are concentric if and only if gcd(m, n) > 1. Our principal result is a complete solution for the case where m and n share a common divisor. For the case of coprime m and n, we present partial results and a conjecture for the general solution. Our findings subsume some special cases which have previously been published on this problem.     

On Stability of Thomson’s Vortex <Emphasis Type="Italic">N</Emphasis>-gon in the Geostrophic Model of the Point Bessel Vortices

A stability analysis of the stationary rotation of a system of N identical point Bessel vortices lying uniformly on a circle of radius R is presented. The vortices have identical intensity Γ and length scale γ^?1 > 0. The stability of the stationary motion is interpreted as equilibrium stability of a reduced system. The quadratic part of the Hamiltonian and eigenvalues of the linearization matrix are studied. The cases for N = 2,..., 6 are studied sequentially. The case of odd N = 2?+1 ≥ 7 vortices and the case of even N = 2_n ≥ 8 vortices are considered separately. It is shown that the (2? + 1)-gon is exponentially unstable for 0 < γR<R_*(N). However, this (2? + 1)-gon is stable for γR ≥ R_*(N) in the case of the linearized problem (the eigenvalues of the linearization matrix lie on the imaginary axis). The even N = 2n ≥ 8 vortex 2n-gon is exponentially unstable for R > 0.     

Convex nonagons with five intervertex distances

The vertices of a convex planar nonagon determine exactly five distances if and only if they are nine vertices of a regular 10-gon or a regular 11-gon. This result has important ties to related concerns, including the maximum number of points in the plane that determine exactly five distances and, for each n7, the samllest t for which there exists a convex n-gon whose vertices determine t distances and are not all on one circle.     

n-bathycenters

Does there exist a polygon with the property that for a suitable point p in the plane every ray with endpoint p intersects the polygon in exactly n connected components? Does there exist a polygon with the property that there are two such points, or three, or a segment of such points? For polygon P call a point p with the property that every ray from p intersects P in exactly n connected components n-isobathic with respect to P. Define the n-bathycenter of a polygon P as the set of all points p that are n-isobathic with respect to P. Further define a set S to be an n-bathycenter if there exists a polygon P of which S is the n-bathycenter. This paper deals with the characterization of 2- and 3-bathycenters, together with some results on the general case.     

A bijection between 2-triangulations and pairs of non-crossing Dyck paths

A k-triangulation of a convex polygon is a maximal set of diagonals so that no k+1 of them mutually cross in their interiors. We present a bijection between 2-triangulations of a convex n-gon and pairs of non-crossing Dyck paths of length 2(n−4). This solves the problem of finding a bijective proof of a result of Jonsson for the case k=2. We obtain the bijection by constructing isomorphic generating trees for the sets of 2-triangulations and pairs of non-crossing Dyck paths.     

On the minimality of polygon triangulation

The problem of triangulating a polygon using the minimum number of triangles is treated. We show that the minimum number of triangles required to partition a simplen-gon is equal ton+2w –d – 2, wherew is the number of holes andd is the maximum number of independent degenerate triangles of then-gon. We also propose an algorithm for constructing the minimum triangulation of a simple hole-freen-gon. The algorithm takesO(nlog² n+DK ²) time, whereD is the maximum number of vertices lying on the same line in then-gon andK is the number of minimally degenerate triangles of then-gon.     

The Existence and Application of Strongly Idempotent Self-orthogonal Row Latin Magic Arrays

Let N = {0, 1, · · ·, n ? 1}. A strongly idempotent self-orthogonal row Latin magic array of order n (SISORLMA(n) for short) based on N is an n × n array M satisfying the following properties: (1) each row of M is a permutation of N, and at least one column is not a permutation of N; (2) the sums of the n numbers in every row and every column are the same; (3) M is orthogonal to its transpose; (4) the main diagonal and the back diagonal of M are 0, 1, · · ·, n ? 1 from left to right. In this paper, it is proved that an SISORLMA(n) exists if and only if n ? {2, 3}. As an application, it is proved that a nonelementary rational diagonally ordered magic square exists if and only if n ? {2, 3}, and a rational diagonally ordered magic square exists if and only if n ≠ 2.     

On the Order Dimension of Convex Geometries

We study the order dimension of the lattice of closed sets for a convex geometry. We show that the lattice of closed subsets of the planar point set of Erd?s and Szekeres from 1961, which is a set of 2 ^n???2 points and contains no vertex set of a convex n-gon, has order dimension n???1 and any larger set of points has order dimension at least n.