Let $G$ denote a near-polygon distance-regular graph with diameter $d\geq 3$, valency $k$ and intersection numbers $a_1>0$, $c_2>1$. Let $θ_1$ denote the second largest eigenvalue for the adjacency matrix of $G$. We show $θ_1$ is at most $(k-a_1-c_2)/(c_2-1)$. We show the following are equivalent: (i) Equality is attained above; (ii) $G$ is $Q$-polynomial with respect to $θ_1$; (iii) $G$ is a dual polar graph or a Hamming graph.
The goal of this paper is to determine the number of perpendicularly inscribed polygons that intersect a given side of a regular polygon with an odd number of sides. This is done using circular permutations with repetition, and some special cases are calculated via circulant matrices and the Binomial Theorem. A method for finding such polygons, based on Banach's Fixed Point Theorem, is also developed.
We generalise the work of Segre (1965), Cameron - Goethals - Seidel (1978), and Vanhove (2011) by showing that nontrivial $m$-ovoids of the dual polar spaces $DQ(2d, q)$, $DW(2d-1,q)$ and $DH(2d-1,q^2)$ ($d\ge 3$) are hemisystems. We also provide a more general result that holds for regular near polygons.
This paper proves explicit formulas for the number of dissections of a convex regular polygon modulo the action of the cyclic and dihedral groups. The formulas are obtained by making use of the Cauchy-Frobenius Lemma as well as bijections between rotationally symmetric dissections and simpler classes of dissections. A number of special cases of these formulas are studied. Consequently, some known enumerations are recovered and several new ones are provided.
In this paper we study various versions of extension complexity for polygons through the study of factorization ranks of their slack matrices. In particular, we develop a new asymptotic lower bound for their nonnegative rank, shortening the gap between the current bounds, we introduce a new upper bound for their boolean rank, deriving from it some new numerical results, and we study their complex semidefinite rank, uncovering the possibility of non monotonicity of the ranks of regular $n$-gons.
The Hadwiger-Nelson problem asks for the minimum number of colors, so that each point of the plane can be assigned a single color with the property that no two points unit-distance apart are identically colored. It is now known that the answer is $5$, $6$, or $7$, Here we consider the problem in the context of Minkowski planes, where the unit circle is a regular polygon with $8$, $10$, or $12$ vertices. We prove that in each of these cases, one also needs at least five colors.
For a point $x$ on the Positive Grassmannian of two-dimensional subspaces in $\mathbb{R}^n$, define the loss function $E(x)$ as the ratio of its largest and smallest Plucker coordinates. We solve the extremal problem of minimizing the loss function $E(x)$ over the Grassmannian. This minimax problem was posed by Berman, et al. in their paper on error-correcting codes over the real numbers.
We consider outer billiard outside regular convex polygons. We deal with the case of regular polygons with $\{3,4,5,6,10\}$ sides, and we describe the symbolic dynamics of the map and compute the complexity of the language.
We describe presentations of braid groups of type $A$ arising from coloured quivers of mutation type $A$. We show that these can be interpreted geometrically as generalised triangulations of regular polygons.
A tiling is a decomposition of a polygon into finitely many non-overlapping triangles. We prove that if a regular n-gon, $n \geq 5$, $n \neq 28$, can be tiled with similar right triangles, then one of the angles of these triangles is in $\left\{\fracπ{n},\frac{2π}{n}, \fracπ {6}+\frac{2π}{3n}\right\}$. Some related results were previously obtained by M.Laczkovich and B. Szegedy.
The famous Hadwiger-Nelson problem asks for the minimum number of colors needed to color the points of the Euclidean plane so that no two points unit distance apart are assigned the same color. In this note we consider a variant of the problem in Minkowski metric planes, where the unit circle is a regular polygon of even and at most 22 vertices. We present a simple lattice-sublattice coloring scheme that uses 6 colors, proving that the chromatic number of the Minkowski planes above are at most 6. This result is new for regular polygons having more than 8 vertices.
We confirm two conjectures of Lassak on the area of reduced spherical polygons. The area of every reduced spherical non-regular $n$-gon is less than that of the regular spherical $n$-gon of the same thickness. Moreover, the area of every reduced spherical polygon is less than that of the regular spherical odd-gons of the same thickness and whose number of vertices tends to infinity.
A $\textit{regular polygon surface}$ $M$ is a surface graph $(Σ, Γ)$ together with a continuous map $ψ$ from $Σ$ into Euclidean 3-space which maps faces to regular Euclidean polygons. When $Σ$ is homeomorphic to the sphere and the degree of every face of $Γ$ is five, we prove that $M$ can be realized as the boundary of a union of dodecahedra glued together along common facets. Under the same assumptions but when the faces of $Γ$ have degree four or eight, we prove that $M$ can be realized as the boundary of a union of cubes and octagonal prisms glued together along common facets. We exhibit co…
We show that for any finite coloring of the group $\mathbb{Z}_2 *\mathbb{Z}_2 *\mathbb{Z}_2$ and for any positive integer $k$, there always exists a monochromatic regular $k$-gon in $\mathbb{Z}_2 *\mathbb{Z}_2 *\mathbb{Z}_2$ with respect to the word length metric induced by the standard generating set; the edge length of which is estimated.
Let $P=A_1\ldots A_n$ be a generic polygon in three-dimensional space and let $v_1,v_2,\ldots,v_n$ be vectors $\overline{A_1A_2},\overline{A_2A_3},\ldots,\overline{A_nA_1}$, respectively. $P$ will be called \emph{regular}, if there exist vectors $u_1,\ldots,u_n$ such that cross products $[u_1,u_2],[u_2,u_3],\ldots,[u_n,u_1]$ are equal to vectors $v_2,v_3,\ldots,v_1$, respectively. In this case the polygon $P'$, defined be vectors $u_2-u_1,u_3-u_2,\ldots,u_1-u_n$ will be called the \emph{derived polygon} or the \emph{derivative} of the polygon $P$. In this work we formulate conditions for regul…
Quadrance between two points A_1 = [x_1,y_1] and A_2 = [x_2,y_2] is the number Q (A_1, A_2) := (x_2 - x_1)^2 + (y_2 - y_1)^2. In this paper, we present some interesting results arise from this notation. In Section 1, we will study geometry over finite fields under quadrance notations. The main purpose of Section 1 is to answer the question, for which a_1,...,a_n, we have a polygon A_1...A_n such that Q(A_i,A_{i+1})=a_i for i = 1,...,n. In Section 2, using tools developed in Section 1, we define a family of association schemes over finite field space F_q x F_q where q is a prime power. These sc…
In \emph{$k$-bootstrap percolation}, we fix $p\in (0,1)$, an integer $k$, and a plane graph $G$. Initially, we infect each face of $G$ independently with probability $p$. Infected faces remain infected forever, and if a healthy (uninfected) face has at least $k$ infected neighbors, then it becomes infected. For fixed $G$ and $p$, the \emph{percolation threshold} is the largest $k$ such that eventually all faces become infected, with probability at least $1/2$. For a large class of infinite graphs, we show that this threshold is independent of $p$. We consider bootstrap percolation in tilings o…
A semi-regular tiling of the hyperbolic plane is a tessellation by regular geodesic polygons with the property that each vertex has the same vertex-type, which is a cyclic tuple of integers that determine the number of sides of the polygons surrounding the vertex. We determine combinatorial criteria for the existence, and uniqueness, of a semi-regular tiling with a given vertex-type, and pose some open questions.
Given a polytope P in $\mathbb{R}^n$, we say that P has a positive semidefinite lift (psd lift) of size d if one can express P as the linear projection of an affine slice of the positive semidefinite cone $\mathbf{S}^d_+$. If a polytope P has symmetry, we can consider equivariant psd lifts, i.e. those psd lifts that respect the symmetry of P. One of the simplest families of polytopes with interesting symmetries are regular polygons in the plane, which have played an important role in the study of linear programming lifts (or extended formulations). In this paper we study equivariant psd lifts …
