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.

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 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.

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.

scholarly journals The chord length distribution function for regular polygons

2009 ◽  
Vol 41 (2) ◽  
pp. 358-366 ◽  
Author(s):  
H. S. Harutyunyan ◽  
V. K. Ohanyan
Keyword(s):  
Distribution Function ◽  
Regular Polygon ◽  
Length Distribution ◽  
Chord Length ◽  
Regular Hexagon ◽  
Regular Polygons ◽  
Chord Length Distribution ◽  
Regular Pentagon ◽  
Regular Triangle ◽  
Elementary Expression

In this paper we obtain an elementary expression for the chord length distribution function of a regular polygon. The formula is derived using δ-formalism in Pleijel identity. In the particular cases of a regular triangle, a square, a regular pentagon, and a regular hexagon, our formula coincides with the results of Sulanke (1961), Gille (1988), Aharonyan and Ohanyan (2005), and Harutyunyan (2007), respectively.

Torsion of beams whose cross-section is a regular polygon of n sides

1934 ◽  
Vol 30 (2) ◽  
pp. 139-149 ◽  
Author(s):  
B. R. Seth
Keyword(s):  
Cross Section ◽  
Regular Polygon ◽  
Regular Polygons ◽  
The Real ◽  
Image Position ◽  
Christoffel Transformation

E. Trefftz has discussed the problem of the torsion of a beam whose cross-section is bounded by a polygon with the help of the Schwarz-Christoffel transformation given bywhere a1, a2, …, an are external angles of the polygon in the w-plane, and ξ1, ξ2, …, ξn are the points on the real ξ-axis in the t-plane that correspond to the angular points of the polygon in the w-plane. In the case of regular polygons a further transformation of the upper half of the t-plane into the interior of a circle in the z-plane with the help of the transformationgreatly simplifies the problem, and some definite results can be obtained.

scholarly journals The chord length distribution function for regular polygons

2009 ◽  
Vol 41 (02) ◽  
pp. 358-366
Author(s):  
H. S. Harutyunyan ◽  
V. K. Ohanyan
Keyword(s):  
Distribution Function ◽  
Regular Polygon ◽  
Length Distribution ◽  
Chord Length ◽  
Regular Hexagon ◽  
Regular Polygons ◽  
Chord Length Distribution ◽  
Regular Pentagon ◽  
Regular Triangle ◽  
Elementary Expression

In this paper we obtain an elementary expression for the chord length distribution function of a regular polygon. The formula is derived using δ-formalism in Pleijel identity. In the particular cases of a regular triangle, a square, a regular pentagon, and a regular hexagon, our formula coincides with the results of Sulanke (1961), Gille (1988), Aharonyan and Ohanyan (2005), and Harutyunyan (2007), respectively.

scholarly journals Irregular Tilings of Regular Polygons with Similar Triangles

2021 ◽  
Author(s):  
Miklos Laczkovich
Keyword(s):  
Regular Polygon ◽  
Regular Polygons ◽  
Similar Triangles

AbstractWe say that a triangle T tiles a polygon A, if A can be dissected into finitely many nonoverlapping triangles similar to T. We show that if $$N>42$$ N > 42 , then there are at most three nonsimilar triangles T such that the angles of T are rational multiples of $$\pi $$ π and T tiles the regular N-gon. A tiling into similar triangles is called regular, if the pieces have two angles, $$\alpha $$ α and $$\beta $$ β , such that at each vertex of the tiling the number of angles $$\alpha $$ α is the same as that of $$\beta $$ β . Otherwise the tiling is irregular. It is known that for every regular polygon A there are infinitely many triangles that tile A regularly. We show that if $$N>10$$ N > 10 , then a triangle T tiles the regular N-gon irregularly only if the angles of T are rational multiples of $$\pi $$ π . Therefore, the number of triangles tiling the regular N-gon irregularly is at most three for every $$N>42$$ N > 42 .

Regular Polygons Keep the Beat

2018 ◽  
Vol 111 (6) ◽  
pp. 412-415
Keyword(s):  
Regular Polygon ◽  
Regular Polygons ◽  
Equilateral Triangles ◽  
Special Number

Finding regular polygons, such as equilateral triangles, squares, hexagons, and even octagons, in art or architecture is not unusual, but seeing a nonagon, or nine-sided regular polygon, such as the Mongolian drum (see photograph 1) is less common. Perhaps 9 is a special number in Mongolia: Shamans have 99 tengri, or spirits, (55 benevolent ones and 44 “dark” ones) on their drums, although the drums are more often circular.

Geometric Design of Micro-Reducer With Multi-Output Shafts Distributed in Regular Polygon Form

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
Yang-zhi Chen ◽  
Xiao-yan Fu ◽  
Jiang Ding ◽  
Shun-ke Liang
Keyword(s):  
Analytical Model ◽  
Regular Polygon ◽  
Geometric Design ◽  
Space Curve ◽  
Transmission Mechanism ◽  
Side Length ◽  
Design Parameters ◽  
The Novel ◽  
Regular Polygons ◽  
Computational Simulations

Based on the theory of space curve meshing, a space curve meshing wheel (SCMW) transmission mechanism has been invented by present authors in recent years. To extend applications of the SCMW, design methods for a novel micro-reducer with multioutput shafts distributed in regular polygon form is proposed in the paper. It is featured with three regular polygons nested. The middle regular polygon, named as reference regular polygon (RRP), is composed of transmission shafts. Three aspects are proposed as below to design the reducer: first, primary design parameters are determined by research and experience, and formulas of center distances are derived; second, an approach to establish the analytical model of the RRP simply and effectively is presented, which shows that the geometric dimensions of the reducer mainly depend on the side length of the RRP; and third, the novel micro-reducer is determined after the side length formulas of the RRP derived from the model. The simplicity and effectiveness of the formulas presented are demonstrated by a series of computational simulations.

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