Ramified Rectilinear Polygons: Coordinatization by Dendrons

The Mathematical Intelligencer, 2017

2009

An algebraic foundation for the derivation of geometric construction schemes transforming arbitrary polygons with n vertices into k-regular n-gons is given. It is based on circulant polygon transformations and the asso ciated eigenpolygon decompositions leading to the definition of generalized Napole on vertices. Geometric construction schemes are derived exemplarily for different choices of n and k.

International Journal of Computational Geometry & Applications, 2010

In this paper we present purely combinatorial conditions that allow us to recognize the topological equivalence (or non-equivalence) of two given dissections. Using a computer program based on this result, we are able to generate a set which contains all topologically non-equivalent dissections of a p0-gon into convex pi-gons, i = 1…n, where n, p0,…,pn are integers such that n ≥ 2, pi ≥ 3. By analyzing generated structures, we are able to find all (up to similarity) dissections of a given type. Since the number of topologically non-equivalent dissections is huge even when the number of parts is small, it is necessary to find additional combinatorial conditions depending on the type of sought dissections, which will allow us to exclude the majority of generated structures. We present such conditions for some special dissections of a triangle into triangles. Finally we prove two new results concerning perfect dissections of a triangle into similar triangles.

2011

Canonical Polygons are plane polygons defined on a square lattice with limitations on length of sides. The smaller sets are enumerated, and metrical and non-metrical properties are defined and calculated.

2016

Starting with an arbitrary complex number z, we will introduce a construction of a polygon P z derived from a given polygon P. The inductively constructed sequence ( P z ) , associated to z and P, is studied, and its geometric properties are investigated. The complex numbers z for which the sequence ( T (k) z ) associated to a triangle T is “regular” are characterized, and the same is done for the sequence ( Q z ) associated to a quadrilateral Q. By suitable choices of z, also the well known Napoleon theorem and some of its generalisations can be detected from the above characterizations. M.S.C. 2010: 51M04, 51N20, 52A10, 97G80.

Theoretical Computer Science, 1995

Combinatorial structure of visibility is probably one of the most fascinating and interesting areas of engineering and computer science. The usefulness of visibility graphs in computational geometry and robotic navigation problems like motion planning, unknown-terrain learning, shortest-path planning, etc., cannot be overstressed. The visibility graph, apart from being an important data structure for storing and updating geometric information, is a valuable mathematical tool in probing and understanding the nature of shapes of polygonal and polyhedral objects. In this research we wish to initially focus our attention on a fundamental class of geometric objects. These geometric objects may be looked upon as building blocks for more complex geometric objects, and which offer an ideal balance between complexity and simplicity, namely simple polygons. A major theme of the proposed paper is the investigation of the combinatorial structure of the visibility graph. More importantly, the goals of this paper are: (i) To characterize the visibility graphs of simple polygons by obtaining necessary and sufficient conditions a graph must satisfy to qualify for the visibility graph of a simple polygon (ii) To obtain hierarchical relationships between visibility graphs of simple polygons of a given number of vertices by treating them as representing simple polygons that are deformations of one another. (iii) To exploit the potential of complete graphs to be natural coordinate systems for addressing the problem of reconstructing a simple polygon from visibility graph. We intend to achieve this by defining appropriate "betweenness" relationships on points with respect to the edges of the complete graphs.

1996

Recurrent relations and explicit formulae for the dichromate of Tutte and Negami's polynomial for chains of n-gons are presented. The graphs called chains of n-gons consist of n-gons connected with each other by edges. Two arbitrary n-gons either have only a common edge (i.e. they are adjacent), or have no common vertices. Each n-gon is adjacent to no more than two other n-gons and no three n-gons which share a common edge. Two terminal n-gons of a chain are adjacent to exactly one other n-gon.

Doklady Mathematics, 2008

J.UCS The Journal of Universal Computer Science, 1996

A new internal structure for simple polygons, the straight s k eleton, is introduced and discussed. It is composed of pieces of angular bisectores which partition the interior of a given n-gon P in a tree-like fashion into n monotone polygons. Its straight-line structure and its lower combinatorial complexity may make the straight skeleton preferable to the widely used medial axis of a polygon. As a seemingly unrelated application, the straight skeleton provides a canonical way of constructing a polygonal roof above a general layout of ground walls.

2016

Thesis Submitted in Partial Fulfilment for the Degree of Master of Science In Pure Mathematics of Jomo Kenyatta University of Agriculture and Technology 2015

Journal of Physics: Conference Series, 2011

In two series of papers we construct quasi regular polyhedra and their duals which are similar to the Catalan solids. The group elements as well as the vertices of the polyhedra are represented in terms of quaternions. In the present paper we discuss the quasi regular polygons (isogonal and isotoxal polygons) using 2D Coxeter diagrams. In particular, we discuss the isogonal hexagons, octagons and decagons derived from 2D Coxeter diagrams and obtain aperiodic tilings of the plane with the isogonal polygons along with the regular polygons. We point out that one type of aperiodic tiling of the plane with regular and isogonal hexagons may represent a state of graphene where one carbon atom is bound to three neighboring carbons with two single bonds and one double bond. We also show how the plane can be tiled with two tiles; one of them is the isotoxal polygon, dual of the isogonal polygon. A general method is employed for the constructions of the quasi regular prisms and their duals in 3D dimensions with the use of 3D Coxeter diagrams.

Annales de l’institut Fourier, 2012

We study the space of 2-frieze patterns generalizing that of the classical Coxeter-Conway frieze patterns. The geometric realization of this space is the space of n-gons (in the projective plane and in 3-dimensional vector space) which is a close relative of the moduli space of genus 0 curves with n marked points. We show that the space of 2-frieze patterns is a cluster manifold and study its algebraic and arithmetic properties.

2016

We present two constructions in this paper : (a) A 10-vertex triangulation CP 2 10 of the complex projective plane CP 2 as a subcomplex of the join of the standard sphere (S 2 4) and the standard real projective plane (RP 2 6 , the decahedron), its automorphism group is A 4 ; (b) a 12-vertex triangulation (S 2 ×S 2) 12 of S 2 ×S 2 with automorphism group 2S 5 , the Schur double cover of the symmetric group S 5. It is obtained by generalized bistellar moves from a simplicial subdivision of the standard cell structure of S 2 ×S 2. Both constructions have surprising and intimate relationships with the icosahedron. It is well known that CP 2 has S 2 × S 2 as a twofold branched cover; we construct the triangulation CP 2 10 of CP 2 by presenting a simplicial realization of this covering map S 2 × S 2 → CP 2. The domain of this simplicial map is a simplicial subdivision of the standard cell structure of S 2 × S 2 , different from the triangulation alluded to in (b). This gives a new proof that Kühnel's CP 2 9 triangulates CP 2. It is also shown that CP 2 10 and (S 2 × S 2) 12 induce the standard piecewise linear structure on CP 2 and S 2 × S 2 respectively.

Acta Mathematica Hungarica, 1992

We easily conclude that the incidence structure of 2 and its flag structure determine each other. Fig. 1 shows two 1-adjacent flags l//k(vj, er, flk) and 2//k(Vj, es, f/k), eT es, symbolically and with picture as well (d = 3). Note that the polyhedron 2 can be given to a computer as the set of its flags, i.e. ordered d-tuples of natural numbers. For instance a 3dimensional tetrahedron is represented by 24 flags, a cube by 48 flags. The one-to-one correspondence between an n-face x ~ and the flag subset :~x-, defined above, makes possible the convenient translation of geometric facts into flag language, i.e. computer one, and vice versa. The main purpose of this section is to describe an algorithm for determining the automorphism group of 2, denoted by Aut 2. This automatically describes the automorphism group Aut ~ of the flag structure 2~. Ant 2 is a permutation group. Each permutation in Aut 2 consists of d component permutations. The n-th component permutes the n-faces of 2, 0 _< n _< d-1. Every elements of Aut 2 preserves the incidence structure of 2, i.e. if an n-face x ~ and an m-face y'~ are incident then for each permutation p from Aut2 the n-th and m-th components of p maps x n onto (x'~)P~ ym onto (ym)Pso that the images are also incident. Aut~ consists of all self-bijections of ~ which preserve n-adjacency relations for every n (0 _< n _< d-1). These bijections form a group by the composition as group operation. LEMMA 1.1. An automorphism a from AutO, resp. Aut2, is uniquely determined (if it exists) by a flag ~ and its a-image [a.