scholarly journals Spherical F-Tilings by Triangles and $r$-Sided Regular Polygons, $r \ge 5$

10.37236/746 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Catarina P. Avelino ◽  
Altino F. Santos
Keyword(s):  
Symmetry Group ◽  
Equilateral Triangle ◽  
Regular Polygon ◽  
Isosceles Triangle ◽  
Combinatorial Structure ◽  
Subsequent Paper ◽  
Regular Polygons ◽  
Spherical Triangles ◽  
Wide Family

The study of dihedral f-tilings of the sphere $S^2$ by spherical triangles and equiangular spherical quadrangles (which includes the case of 4-sided regular polygons) was presented by Breda and Santos [Beiträge zur Algebra und Geometrie, 45 (2004), 447–461]. Also, in a subsequent paper, the study of dihedral f-tilings of $S^2$ whose prototiles are an equilateral triangle (a 3-sided regular polygon) and an isosceles triangle was described (we believe that the analysis considering scalene triangles as the prototiles will lead to a wide family of f-tilings). In this paper we extend these results, presenting the study of dihedral f-tilings by spherical triangles and $r$-sided regular polygons, for any $r \ge 5$. The combinatorial structure, including the symmetry group of each tiling, is given.

scholarly journals Dihedral F-Tilings of the Sphere by Equilateral and Scalene Triangles - II

10.37236/815 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
A. M. d'Azevedo Breda ◽  
Patrícia S. Ribeiro ◽  
Altino F. Santos
Keyword(s):  
Equilateral Triangle ◽  
Isosceles Triangle ◽  
Subsequent Paper ◽  
Euclidean Sphere ◽  
Regular Polygons ◽  
Combinatorial Description ◽  
Equilateral Triangles

The study of dihedral f-tilings of the Euclidean sphere $S^2$ by triangles and $r$-sided regular polygons was initiated in 2004 where the case $r=4$ was considered [5]. In a subsequent paper [1], the study of all spherical f-tilings by triangles and $r$-sided regular polygons, for any $r\ge 5$, was described. Later on, in [3], the classification of all f-tilings of $S^2$ whose prototiles are an equilateral triangle and an isosceles triangle is obtained. The algebraic and combinatorial description of spherical f-tilings by equilateral triangles and scalene triangles of angles $\beta$, $\gamma$ and $\delta$ $(\beta>\gamma>\delta)$ whose edge adjacency is performed by the side opposite to $\beta$ was done in [4]. In this paper we extend these results considering the edge adjacency performed by the side opposite to $\delta$.

Generalized Second Moment of Areas of Regular Polygons for Ludwick Type Material and its Application to Cantilever Column Buckling

2019 ◽  
Vol 19 (02) ◽  
pp. 1950010
Author(s):  
Joon Kyu Lee ◽  
Byoung Koo Lee
Keyword(s):  
Cross Sections ◽  
Equilateral Triangle ◽  
Regular Polygon ◽  
Regular Hexagon ◽  
Type Material ◽  
Regular Polygons ◽  
Column Buckling ◽  
Second Moment ◽  
Regular Pentagon

This study deals with the generalized second moment of area (GSMA) of regular polygon cross-sections for the Ludwick type material and its application to cantilever column buckling. In the literature, the GSMA for the Ludwick type material has only been considered for rectangular, elliptical and superellipsoidal cross-sections. This study calculates the GSMAs of regular polygon cross-sections other than those mentioned above. The GSMAs calculated by varying the mechanical constant of the Ludwick type material for the equilateral triangle, square, regular pentagon, regular hexagon and circular cross-sections are reported in tables and figures. The GSMAs obtained from this study are applied to cantilever column buckling, with results shown in tables and figures.

New Theorems for Moments of Inertia

1993 ◽  
Vol 21 (4) ◽  
pp. 355-366 ◽  
Author(s):  
David L. Wallach
Keyword(s):  
Regular Polygon ◽  
Moment Of Inertia ◽  
Regular Polygons ◽  
Centre Of Mass ◽  
Moments Of Inertia ◽  
The Moment

The moment of inertia of a plane lamina about any axis not in this plane can be easily calculated if the moments of inertia about two mutually perpendicular axes in the plane are known. Then one can conclude that the moments of inertia of regular polygons and polyhedra have symmetry about a line or point, respectively, about their centres of mass. Furthermore, the moment of inertia about the apex of a right pyramid with a regular polygon base is dependent only on the angle the axis makes with the altitude. From this last statement, the calculation of the centre of mass moments of inertia of polyhedra becomes very easy.

scholarly journals Doubly Semiequivelar Maps on Torus and Klein Bottle

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Anand K. Tiwari ◽  
Amit Tripathi ◽  
Yogendra Singh ◽  
Punam Gupta
Keyword(s):  
Symmetry Group ◽  
Euclidean Plane ◽  
Klein Bottle ◽  
Universal Cover ◽  
Regular Polygons

A tiling of the Euclidean plane, by regular polygons, is called 2-uniform tiling if it has two orbits of vertices under the action of its symmetry group. There are 20 distinct 2-uniform tilings of the plane. Plane being the universal cover of torus and Klein bottle, it is natural to ask about the exploration of maps on these two surfaces corresponding to the 2-uniform tilings. We call such maps as doubly semiequivelar maps. In the present study, we compute and classify (up to isomorphism) doubly semiequivelar maps on torus and Klein bottle. This classification of semiequivelar maps is useful in classifying a category of symmetrical maps which have two orbits of vertices, named as 2-uniform maps.

KNOTTING OF REGULAR POLYGONS IN 3-SPACE

1994 ◽  
Vol 03 (03) ◽  
pp. 263-278 ◽  
Author(s):  
KENNETH C. MILLETT
Keyword(s):  
Regular Polygon ◽  
Regular Polygons ◽  
Homfly Polynomial ◽  
Linear Maps ◽  
Compact Variety ◽  
Linear Embedding

The probability that a linear embedding of a regular polygon in R3 is knotted should increase as a function of the number of sides. This assertion is investigated by means of an exploration of the compact variety of based oriented linear maps of regular polygons into R3. Asymptotically, an estimation of the probability of knotting is made by means of the HOMFLY polynomial.

scholarly journals Computing symmetry groups of polyhedra

2014 ◽  
Vol 17 (1) ◽  
pp. 565-581 ◽  
Author(s):  
David Bremner ◽  
Mathieu Dutour Sikirić ◽  
Dmitrii V. Pasechnik ◽  
Thomas Rehn ◽  
Achill Schürmann
Keyword(s):  
Linear Programming ◽  
Symmetry Group ◽  
Integer Linear Programming ◽  
Combinatorial Structure ◽  
Symmetry Groups ◽  
Graph Automorphism ◽  
Practical Experiences

AbstractKnowing the symmetries of a polyhedron can be very useful for the analysis of its structure as well as for practical polyhedral computations. In this note, we study symmetry groups preserving the linear, projective and combinatorial structure of a polyhedron. In each case we give algorithmic methods to compute the corresponding group and discuss some practical experiences. For practical purposes the linear symmetry group is the most important, as its computation can be directly translated into a graph automorphism problem. We indicate how to compute integral subgroups of the linear symmetry group that are used, for instance, in integer linear programming.

scholarly journals THE THE APPLICATION OF GENERIC GREEN SKILLS IN TESSELLATION OF REGULAR POLYGONS FOR ECONOMIC AND SOCIAL SUSTAINABILITY

2019 ◽  
Vol 4 (3) ◽  
pp. 121-124
Author(s):  
ABDULLAH Musa Cledumas ◽  
YUSRI BIN KAMIN ◽  
RABIU HARUNA ◽  
SHUAIBU HALIRU
Keyword(s):  
Regular Polygon ◽  
Social Sustainability ◽  
Regular Polygons ◽  
Environmentally Sustainable ◽  
Clothing Design ◽  
Proposed Model ◽  
The Right ◽  
Modelling Approach

Abstract This paper proposes an improved modelling approach for tessellating regular polygons in such a way that it is environmentally sustainable. In this paper, tessellation of polygons that have been innovated through the formed motifs, is an innovation from the traditional tessellations of objects and animals. The main contribution of this work is the simplification and innovating new patterns from the existing regular polygons, in which only three polygons (triangle, square and hexagon) that can free be tessellated are used, compared to using irregular polygons or other objects.  This is achieved by reducing the size of each polygon to smallest value and tessellating each of the reduced figure to the right or to left to obtain a two different designs of one unit called motif. These motifs are then combined together to form a pattern. In this innovation it is found that the proposed model is superior than tessellating ordinary regular polygon, because more designs are obtained, more colours may be obtained or introduced to give meaningful tiles or patterns. In particular Tessellations can be found in many areas of life. Art, architecture, hobbies, clothing design, including traditional wears and many other areas hold examples of tessellations found in our everyday surroundings.

scholarly journals Harmonic interpolation based on Radon projections along the sides of regular polygons

2013 ◽  
Vol 11 (4) ◽  
Author(s):  
Irina Georgieva ◽  
Clemens Hofreither ◽  
Christoph Koutschan ◽  
Veronika Pillwein ◽  
Thotsaporn Thanatipanonda
Keyword(s):  
Harmonic Function ◽  
Finite Number ◽  
Interpolation Problem ◽  
Numerical Experiments ◽  
Regular Polygon ◽  
Harmonic Polynomial ◽  
Regular Polygons ◽  
Symbolic Summation ◽  
Radon Projections ◽  
Harmonic Interpolation

AbstractGiven information about a harmonic function in two variables, consisting of a finite number of values of its Radon projections, i.e., integrals along some chords of the unit circle, we study the problem of interpolating these data by a harmonic polynomial. With the help of symbolic summation techniques we show that this interpolation problem has a unique solution in the case when the chords form a regular polygon. Numerical experiments for this and more general cases are presented.

On covering a regular polygon with a triangle

1962 ◽  
Vol 58 (1) ◽  
pp. 8-11 ◽  
Author(s):  
H. G. Eggleston
Keyword(s):  
Equilateral Triangle ◽  
Regular Polygon ◽  
Side Length ◽  
Covering Problem ◽  
Image Position ◽  
Least Area

In this note we consider a particular type of covering problem. Let T be given a plane set and Tθ be the set obtained from T by a rotation about some point in the plane through an angle θ in the clockwise sense. If a set K is such that for every θ there is a translation which transforms Tθ into a subset of K then we say that K is a rotation cover of T. The problem considered here is to determine for fixed n, l the triangle of least area which is a rotation cover for an n-sided regular polygon of side length l. In each case the solution is an equilateral triangle the altitude of which is given byThis solution is unique.

scholarly journals XIII.—The Formation, Rupture, and Closure of Ovarian Follicles in Ferrets and Ferret-Polecat Hybrids, and some Associated Phenomena.

1919 ◽  
Vol 52 (2) ◽  
pp. 303-363 ◽  
Author(s):  
Arthur Robinson
Keyword(s):  
Equilateral Triangle ◽  
Isosceles Triangle ◽  
Ovarian Follicles ◽  
The Face ◽  
Yellowish White

A few words of explanation are necessary at the outset in connection with the terms used in association with the animals upon which these investigations were made.Dealers in ferrets supply two types of animals—one with yellowish-white fur and pink eyes, which they call “ferrets,” and a second type in which the yellowish fur is intermingled with a varying admixture of brownish or brown-black hairs; the second type they call “polecats.” The polecats of the ferret-dealers are not, however, true polecats, for Miss Frances Pitt, who has bred ferrets and polecats and ferret-polecat hybrids in a scientific manner, and to whom I am indebted for the greater and best part of my information on this subject, points out that the so-called polecat of the dealers is never so dark and handsome in colour as the true polecat. Moreover, as Miss Pitt states, in the true polecat the light marks in front of the ears do not usually join across the forehead, whereas in the polecat of the dealers they form a band across the face; further, when viewed from in front and above, the head of the true polecat has the form of a fairly equilateral triangle, whilst in ferrets and the so-called polecats of the dealers the head, on the whole, has the form of an isosceles triangle.

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