Equiangular Polygons Research Articles

Some characterizations of the circle related to circumscribed equiangular polygons

Some characterizations of the circle related to circumscribed equiangular polygons

Some Metric Properties of Semi-Regular Equilateral Nonagons

A simple polygon that either has equal all sides or all interior angles is called a semi-regular nonagon. In terms of this definition, we can distinguish between two types of semi-regular polygons: equilateral polygons (that have equal all sides and different interior angles) and equiangular polygons (that have equal interior angles and different sides). Unlike regular polygons, one characteristic element is not enough to analyze the metric properties of semi-regular polygons, and an additional one is needed. To select this additional characteristic element, note that the following regular triangles can be inscribed to a semi-regular equilateral nonagon by joining vertices: ∆A1 A4A7, △ A2 A5 A8, △A3 A6 A9. Now have a look at triangle △A1 A4A7. Let us use the mark φ=∡(a,b1) to mark the angle between side a of the semi-regular nonagon and side b1 of the inscribed regular triangle. In interpreting the metric properties of a semi-regular equilateral nonagon, in addition to its side, we also use the angle that such side creates with the side of one of the three regular triangles that can be inscribed to such semi-regular nonagon. We consider the way in which convexity, possibility of construction, surface area, and other properties depend on a side of the semi-regular nonagon and angle φ=∡(a,b1).

Some Metric Properties and a Constructive Task of a Semi-Regular 2n-Sides Polygon

A simple polygon that either has equal all sides or all interior angles is called a semi-regular polygon. In terms of this definition, we can distinguish between two types of semi-regular polygons: equilateral polygons (that have equal all sides and different interior angles) and equiangular polygons (that have equal interior angles and different sides). To analyze the metric properties of semi-regular polygons, knowing only one basic element, e.g. the length of a side, as in regular polygons, is not enough. Therefore, in addition to the side of a semi-regular polygon, we use another characteristic element of it to analyze the metric features, and that is the angle δ=∠(a,b) between the side of a semi-regular polygon PN and the side b of its inscribed regular polygon PN. Some metric properties of a semi-regular equilateral 2n-sides polygon are analyzed in this paper with respect to these two characteristic elements. Some of the problems discussed in the paper are: convexity, calculation of surface area, dependence on the length of sides a and δ, calculation of the radius of the inscribed circle depending on the sides a and angles δ, and calculation of the surface area in which the radius of the inscribed circle is known, as well as the relationship between them. It has been shown that the formula for calculating the surface area of regular polygons results from the formula for the surface area of 2n-side semi-regular, equilateral polygons. Further, by using these results, it has been shown that the cross-sections of regular polygons inscribed to semi-regular equilateral polygons, the vertices of equiangular semi-regular polygons, as well as the sides of the regular polygons inscribed to it, intersect in the same manner at the vertices of the equilateral semi-regular polygon. It has further been shown that the sides of the equiangular semi-regular polygon refer to each other as the sines of the angles created by the sides of the inscribed polygons and the side of the semi-regular polygon.

Coincidence of the barycentre and the geometric centre of weighted points

Recently, Gerhard J. Woeginger [1] gave a survey on the interesting history of results on equiangular n-vertex polygons with edge lengths in arithmetic progression. Such a polygon exists if, and only if, n has at least two distinct prime factors.

Regular and Equiangular Polygons of a Hyperbolic Plane of Positive Curvature

Regular and Equiangular Polygons of a Hyperbolic Plane of Positive Curvature

Iterating Evolutes and Involutes

This paper concerns iterations of two classical geometric constructions, the evolutes and involutes of plane curves, and their discretizations: evolutes and involutes of plane polygons. In the continuous case, our main result is that the iterated involutes of closed locally convex curves with rotation number one (possibly, with cusps) converge to their curvature centers (Steiner points), and their limit shapes are hypocycloids, generically, astroids. As a consequence, among such curves only the hypocycloids are homothetic to their evolutes. The bulk of the paper concerns two kinds of discretizations of these constructions: the curves are replaced by polygons, and the evolutes are formed by the circumcenters of the triples of consecutive vertices (\(\mathcal {P}\)-evolutes), or by the incenters of the triples of consecutive sides (\(\mathcal {A}\)-evolutes). For equiangular polygons, the theory is parallel to the continuous case: we define discrete hypocycloids (equiangular polygons whose sides are tangent to hypocycloids) and a discrete Steiner point. The space of polygons is a vector bundle over the space of the side directions; our main result here is that both kinds of evolutes define vector bundle morphisms. In the case of \(\mathcal {P}\)-evolutes, the induced map of the base is 4-periodic, and the dynamics reduces to the linear maps on the fibers. We prove that the spectra of these linear maps are symmetric with respect to the origin. The asymptotic dynamics of linear maps is determined by their eigenvalues with the maximum modulus, and we show that all types of behavior can occur: in particular, hyperbolic, when this eigenvalue is real, and elliptic, when it is complex. We also study \(\mathcal {P}\)- and \(\mathcal {A}\)-involutes and prove that the side directions of iterated \(\mathcal {A}\)-involutes of polygons with odd number of sides behave ergodically; this generalizes well-known results concerning iterations of the construction of the pedal triangle. In addition to the theoretical study, we performed numerous computer experiments; some of the observations remain unexplained.

Side Lengths of Equiangular Polygons (as seen by a coding theorist)

A tuple a = (a1,…,an) of positive real numbers is said to be equiangular if there is an equiangular polygon with consecutive side lengths a1,…,an. It is well known that a is equiangular if and only if the polynomial a(x) = a1 + a2x + … + an-1xn-2 + anxn-1 vanishes at . Here we dispense with complex numbers and borrow an idea from the theory of cyclic codes to prove that a is equiangular if and only if a(x) is divisible by .

Nothing New about Equiangular Polygons

We survey results on equiangular n-vertex polygons with edge lengths in arithmetic progression. Such a polygon exists if and only if n has at least two distinct prime factors.

Arithmetic Polygons

We consider the question of the existence of equiangular polygons with edge lengths in arithmetic progression, and show that they do not exist when the number of sides is a power of two and do exist if it is any other even number. A few results for small odd numbers are given.

A New Metric to Quantify Resiliency in Networking

In network protocol engineering resiliency is still a relatively new and somewhat ill-defined concept. Insofar only few studies attempt to define some metric to measure and thus quantify resiliency. In this paper we propose to quantify network resiliency along multiple parameters and further we introduce a two dimensional graphical representation with multiple axes forming an equiangular polygon surface. This method allows to aggregate meaningfully several parameters and makes it easier to visually discern various tradeoffs thus greatly simplifying the process of protocol comparison. Finally, this method is flexible and can be applied to various networking contexts.

Improved data visualization techniques for analyzing macromolecule structural changes

The empirical phase diagram (EPD) is a colored representation of overall structural integrity and conformational stability of macromolecules in response to various environmental perturbations. Numerous proteins and macromolecular complexes have been analyzed by EPDs to summarize results from large data sets from multiple biophysical techniques. The current EPD method suffers from a number of deficiencies including lack of a meaningful relationship between color and actual molecular features, difficulties in identifying contributions from individual techniques, and a limited ability to be interpreted by color-blind individuals. In this work, three improved data visualization approaches are proposed as techniques complementary to the EPD. The secondary, tertiary, and quaternary structural changes of multiple proteins as a function of environmental stress were first measured using circular dichroism, intrinsic fluorescence spectroscopy, and static light scattering, respectively. Data sets were then visualized as (1) RGB colors using three-index EPDs, (2) equiangular polygons using radar charts, and (3) human facial features using Chernoff face diagrams. Data as a function of temperature and pH for bovine serum albumin, aldolase, and chymotrypsin as well as candidate protein vaccine antigens including a serine threonine kinase protein (SP1732) and surface antigen A (SP1650) from S. pneumoniae and hemagglutinin from an H1N1 influenza virus are used to illustrate the advantages and disadvantages of each type of data visualization technique.

Real-time traffic sign recognition from video by class-specific discriminative features

In this paper we address the problem of traffic sign recognition. Novel image representation and discriminative feature selection algorithms are utilised in a traditional three-stage framework involving detection, tracking and recognition. The detector captures instances of equiangular polygons in the scene which is first appropriately filtered to extract the relevant colour information and establish the regions of interest. The tracker predicts the position and the scale of the detected sign candidate over time to reduce computation. The classifier compares a discrete-colour image of the observed sign with the model images with respect to the class-specific sets of discriminative local regions. They are learned off-line from the idealised template sign images, in accordance with the principle of one-vs-all dissimilarity maximisation. This dissimilarity is defined based on the so-called Colour Distance Transform which enables robust discrete-colour image comparisons. It is shown that compared to the well-established feature selection techniques, such as Principal Component Analysis or AdaBoost, our approach offers a more adequate description of signs and involves effortless training. Upon this description we have managed to build an efficient road sign recognition system which, based on a conventional nearest neighbour classifier and a simple temporal integration scheme, demonstrates a competitive performance in the experiments involving real traffic video.

Analytical light-ray tracing in two-dimensional objects for light-extraction problems in light-emitting diodes

Light extraction from two-dimensional objects is discussed. Analytical calculations in terms of three different parameters have been applied to equiangular polygons to trace light rays during multiple reflections in a polygon. Based on the result that there are a finite number of incident angles in a polygon for a light ray, it was found that the triangle has the least chance to trap light rays among the polygons. The discussion has been extended to parallelograms, which have an advantage in light extraction to rectangles. Placement of a possible light source in polygons is discussed.

The tiling conjecture for equiangular polygons

In [1], Derek Ball made three conjectures about equiangular polygons in which the length of each side is an integer. He called these integer equiangular polygons. The first two conjectures were proved in an earlier note, and the third is proved here. It can be stated as follows.The tiling conjecture: For each integer n ⩾ 3, there is a finite set Tn of tiles such that every integer equiangular n-gon can be tiled by sufficiently many congruent copies of tiles in Tn.

Binary Forms, Equiangular Polygons and Harmonic Measure

Binary Forms, Equiangular Polygons and Harmonic Measure

Binary forms, hypergeometric functions and the Schwarz-Christoffel mapping formula

In a previous paper, it was shown that if F F is a binary form with complex coefficients having degree n ⩾ 3 n \geqslant 3 and discriminant D F ≠ 0 {D_F} \ne 0 , and if A F {A_F} is the area of the region | F ( x , y ) | ⩽ 1 \left | {F(x,y)} \right | \leqslant 1 in the real affine plane, then | D F | 1 / n ( n − 1 ) A F ⩽ 3 B ( 1 3 , 1 3 ) {\left | {{D_F}} \right |^{1/n(n - 1)}}{A_F} \leqslant 3B(\frac {1} {3},\frac {1} {3}) , where B ( 1 3 , 1 3 ) B(\frac {1} {3},\frac {1} {3}) denotes the Beta function with arguments of 1/3. This inequality was derived by demonstrating that the sequence { M n } \{ {M_n}\} defined by M n = max | D F | 1 / n ( n − 1 ) A F {M_n} = \max |{D_F}{|^{1/n(n - 1)}}{A_F} , where the maximum is taken over all forms of degree n n with D F ≠ 0 {D_F} \ne 0 , is decreasing, and then by showing that M 3 = 3 B ( 1 3 , 1 3 ) {M_3} = 3B(\frac {1} {3},\frac {1} {3}) . The resulting estimate, A F ⩽ 3 B ( 1 3 , 1 3 ) {A_F} \leqslant 3B(\frac {1} {3},\frac {1} {3}) for such forms with integer coefficients, has had significant consequences for the enumeration of solutions of Thue inequalities. This paper examines the related problem of determining precise values for the sequence { M n } \{ {M_n}\} . By appealing to the theory of hypergeometric functions, it is shown that M 4 = 2 7 / 6 B ( 1 4 , 1 2 ) {M_4} = {2^{7/6}}B(\frac {1} {4},\frac {1} {2}) and that M 4 {M_4} is attained for the form X Y ( X 2 − Y 2 ) XY({X^2} - {Y^2}) . It is also shown that there is a correspondence, arising from the Schwarz-Christoifel mapping formula, between a particular collection of binary forms and the set of equiangular polygons, with the property that A F {A_F} is the perimeter of the polygon corresponding to F F . Based on this correspondence and a representation theorem for | D F | 1 / n ( n − 1 ) A F |{D_F}{|^{1/n(n - 1)}}{A_F} , it is conjectured that M n = D F n ∗ 1 / n ( n − 1 ) A F n ∗ {M_n} = D_{F_n^ * }^{1/n(n - 1)}{A_{F_n^*}} , where F n ∗ ( X , Y ) = ∏ k = 1 n ( X sin ⁡ ( k π n ) − Y cos ⁡ ( k π n ) ) F_n^*(X,Y) = \prod _{k = 1}^n \left (X \sin \left (\frac {k\pi }{n}\right ) - Y \cos \left (\frac {k\pi }{n}\right )\right ) , and that the limiting value of the sequence { M n } \{ {M_n}\} is 2 π 2\pi .

The Independence of Cell Shape and Overall Form in Multicellular Algae and Land Plants: Cells Do Not Act as Building Blocks for Constructing Plant Organs

We evaluate here whether the mechanisms regulating cellular morphogenesis in planar structures are also responsible for dictating their overall shapes. A comparative survey of 15 multicellular algae and land plants was performed in order to identify the common features associated with cellular morphogenesis in their immature planar structures. The superficial layers of all such structures are partitioned into polygonal cellular arrays that are characterized by specific mean numbers of walls per cell, narrow ranges of cell sizes, and regular cell shapes. The ability to generate these polygonal arrays is independent of overall shape, evolutionary lineage, cell layer number, cell division pattern, and cell wall composition. Instead, geometric principles dictate that the superficial layers of plant structures exhibit mean numbers of ca. 6.0 walls per cell, which result from the tendency of cell plates in most dividing plant cells to avoid existing three-way vertices. In contrast, the superficial layers of algal structures exhibit mean numbers between 5.5 and 5.6 walls per cell, which depend on the ability of cleavage furrows in dividing algal cells to arise at existing three-way vertices. Sequential observations of developing prothalli of the fern Onoclea sensibilis were used to characterize the regulation of cell size and cell shape in this representative planar structure. Cell biological mechanisms appear to dictate cell size via the control over the symmetry, timing, and orientation of cell division. Biomechanical mechanisms regulate cell shape in a remarkable manner. Cell walls in the region of isodiametrical expansion pivot around their vertices so that the observed angles approach the ideal angles calculated for equiangular polygons. Additional divisions alter the wall numbers of the three cells around each vertex, but the observed angles continue to adjust so that they parallel the predicted ideal angles. Angle rotation is accompanied by differential amounts of relative wall expansion, which are inversely dependent on wall length. It appears as if the vertices of dividing cells undergoing isodiametrical expansion are attempting to reach the condition of local mechanical equilibrium. But the vertices in older cells undergoing preferential elongation along one axis tend to revert to the original rightangle configuration in the attempt to accommodate to global growth stresses. Thus, biomechanical mechanisms regulating cell shape allow the prothallus to approach mechanical stability. Last, the polygonal cellular array in planar structures is viewed as the morphogenetic equivalent of a ground state; i.e., it is generated without any evident input of specific genetic instructions. We conclude that the local regulation of cellular morphogenesis is completely independent of the global regulation of the overall form of planar structures.

Angular homeostasis: IV. Polygonal orbits.

Some properties are discussed of regular polygons that may result from angular homeostatic processes in stable orbit. To characterize these "homeostatic polygons" we need to discuss the winding number, the sidedness (integer, fractional and irrational), multiplicity, envelopes, and density. A regular (i.e., equilateral, equiangular) polygon may be closed in one revolution about its unique center, in multiple revolutions, or not at all. A homeostatic polygon can be generated only if all vertices are included in a single polygon, which occurs if and only if the number of vertices and the number of revolutions required to complete the polygon are relatively prime. For the homeostatic polygon to have a finite number of sides (without repeating itself) the angle subtended by any two successive vertices at the center must be a rational multiple of 2 pi. Biological implications of these properties are illustrated.

Topological Origin of the Surface Singularities in Glasses

ABSTRACTA model of CRN in two dimensions is presented, composed of equilateral but not equiangular polygons with a constant coordination number = 3; in a simplified version only pentagons, hexagons and heptagons are present in the network. We introduce the mean potential energy per atom averaged over a typical cell containing three adjacent polygons; we assume that although the thermal equilibrium is not attained for single atoms, it is attained on the level of these cells, so that we can apply the virial theorem for the cells. Then we minimize the free energy which contains the configuration entropy contribution. In terms of two variables P and δ (hexagon frequency and mean bond angle deviation) we get the surfaces of constant energy. Under stress the energy configurations cease to be one-connected, and the 0-th homotopy group is no more trivial. This can give rise to surface singularities (cracks).

A Theorem on Equiangular Convex Polygons Circumscribing a Convex Curve

The object of this note is to prove the following theorem, connecting the perimeter of a convex curve, with the perimeter of convex equiangular polygons circumscribing it.