Equilateral Polygons Research Articles

Packing an equilateral polygon in a thin strip

LetPm denote an equilateral polygon ofm sides with each side having length 1 and we allow the sides to cross and vertex repetitions. We consider the following question. What is the smallest widthtm of a horizontal strip in the Euclidean plane that contains aPm? This problem has its origins in Euclidean Ramsey theory. Whenm is even, it is easy to see thattm=0. For a polygon with an odd number of sides, we prove that $$\begin{gathered} t_{2n + 1} = \frac{{\sqrt {2n + 1} }}{{n + 1}} for 2n + 1 \equiv 3(\bmod 4) and \hfill \\ t_{2n + 1} = \sqrt {\frac{{2n + 1}}{{n^2 + 2n}}} for 2n + 1 \equiv 1(\bmod 4) \hfill \\ \end{gathered} $$ respectively. Applying the result to unit distance graphs, we prove the following: SupposeG is a unit-distance graph, i.e.G can be drawn on the planeR2 such that each edge ofG is a straight line segment of length 1. IfG has no odd circuits of length greater than or equal to 15, thenX(G) ≤ 6, whereX(G) is the chromatic number ofG.

Polygonal Harmonics of Silhouettes: Shape analysis

AbstractOne well documented method for finding the fractal dimension of a silhouette involves approximating the silhouette with polygons which have all but one side equal and the vertices of which fall upon the silhouette edge. These are constructed using a divider stepping method or a computer algorithm which mimics this process. If one does not complete the polygon with an unequal side, as is usually done, but continues the analysis by stepping around the silhouette several times, the dividers start to „walk in their own footsteps”︁ and describe an equilateral polygon in the silhouette. These phenomena, termed polygonal harmonics, are examined and discussed in detail. A rule is developed for harmonic stability and specific cases of the circle, triangle and smooth curves are examined to gain insight into the convergence process.

Translational drag coefficients of assemblies of spheres with higher-order hydrodynamic interactions

Calculations were made of the anisotropic translation drag coefficients of rigid assemblies of spheres in a variety of arrays, including lines, circular equilateral polygons, disks (polygons with a central sphere), and stacked squares. The objective was to compare results obtained using polynomial expansions for the two-body hydrodynamic interaction tensor truncated at the (a/r)1, (a/r)3, (a/r)5, and (a/r)7 levels. Calculations were also made with the three- and four-body hydrodynamic interaction tensors recently obtained by Mazur and van Saarlos, thereby examining the relative importance of two-, three-, and four-body interactions. There are substantial cancellations between terms of order (a/r)5 and (a/r)7, and between the three- and four-body hydrodynamic interactions. For the arrays studied here the Rotne–Prager [two-body, (a/r)3] hydrodynamic interaction tensor gives results which are at least as satisfactory as the results obtained from the full Mazur–van Saarlos [four-body, (a/r)7] interaction tensors.

A simple assembly for rapid measurements of Rf in paper chromatography

Network coherence is used to characterize the consensus dynamics with additive stochastic disturbances and can be described by Laplacian spectrum. In this paper, we mainly obtain the scalings of network coherence in the web graphs with a special feature that its fractal dimension is infinite. We then investigate the relationship between the scalings and fractal dimension. Based on the structures of web graphs, we obtain the relationships for Laplacian matrix and Laplacian eigenvalues between web graphs and their corresponding equilateral polygons. We also obtain analytical expressions for the sum of the reciprocals and square reciprocals of all nonzero Laplacian eigenvalues. Finally we calculate first and second order coherence and see that the scalings of network coherence with network size N are N and N3, which shows that the scalings are not related to the fractal dimension of web graphs. In addition, the scalings of network coherence in web graphs are larger than those performed on some fractal networks.

Mechanisms related to Poncelet's closure theorem

A closed equilateral polygon, whose vertices alternatively lie on two fixed circles, is moveable: The vertices can move along the circles without change of the common length of all sides. This phenomenon leads to a remarkable family of overconstrained linkages and is of interest for certain ball-bearings. In the case of coincident or parallel circle planes an algorithm is developed which allows the side length of such polygons to be calculated.

Relationship between torsional angles and ring-puckering coordinates

A relationship between torsional angles and ring puckering coordinates, applicable to any puckered ring, is deduced assuming infinitesimal displacements from a planar reference conformation. For equilateral polygons there is only one maximum torsion angle θ m and one phase angle φ m for each value of m, whereas for non-equilateral polygons there is one maximum θ m j,j+1 and a correction, ϵ m j,j+1 , to be applied to the phase angle φ m for each torsion angle θ j,j+1 . The resulting expressions for equilateral pentagons and hexagons are equivalent to the empirical ones proposed previously.

The constructibifity of regular and equilateral polygons on a square pinboard

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The Constructibility of Regular and Equilateral Polygons on a Square Pinboard

The Constructibility of Regular and Equilateral Polygons on a Square Pinboard

Pointing error in passively stabilized spacecraft caused by thermal bending

An important component of passively stabilized earth orbiting spacecraft is a long thinwalled cylinder that points at all times towards the earth. When the cylinder is exposed to the sun, the side facing the sun heats up more than the far side causing it to bend away from the sun. In general, the differential thermal expansion produces stresses in the material of the boom; however, there is a particular distribution that does not produce stresses. Equations for determining the bending radius are derived for the no-stress and the general cases. Sample calculations are presented; although the results obtained for the two cases do not, in general, differ very much, there are instances for which the discrepancy is significant. Nomenclature a = absorptivity A = area of boom cross section, in.2 d = boom diameter, in. e = external emissivity #i = internal emissivity E = Young's modulus of elasticity, psi / = moment of inertia, in.4 k = thermal conductivity, Btu-ft/hr-ft2-°F K = integer m = number of sides of equilateral polygon M = couple, Ib-in. P — force, Ib R = radius of curvature, in. RL = radius of curvature according to the linearized theory, in. Ri = projection of the distance between center fiber and edge of the z'th element on the axis b-b (Fig. 4), in. <s = boom thickness, in. t = temperature, °F £nax = maximum temperature, °F tmin = minimum temperature, °F At = tmax tmin

A note on equilateral polygons

Theorem. If a circle cut all the sides (produced if necessary) of an equilateral polygon, the algebraic sum of the intercepts between the vertices and the circle is zero; i.e., if any side AB of the polygon be cut by the circle in P and Q, then Σ(AP + BQ) = 0, the intercepts being signed by fixing a positive direction round the contour of the polygon.

1988. A real matrix representing an equilateral polygon in general Euclidean Space

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狭い場所に馴らされたイワシの遊泳状態に關して

A shoal of sardines, when they are fully habituated to a confined space such as a bait-net or an aquarium having a horizontal section of a circle or of an equilateral polygon, is usually found in a steady motion, each individual swimming in a horizontal circle about the central axis of the confined space. The author observed cinematographically the motion of sardines (Sardinia melanosticta T. & S.) habituated to a cylindrical tank 3.2m in diameter for more than two months, and learned that the motion is of constant angular velocity θ irrespective of the radial distance of each individual from the center, so that the relative position of each individual is undisturbed. The author obtained the values θ=8-10 degrees/sec. θ is more or less greater than these values, if the fish are not fully habituated to the space or are uneasy from some causes. It was also observed in a bait-net of the greatest size in practice, having the horizontal section of a regular octagon 3.6m long on each side, that the sardines in it swim with constant angular velocity independent of the radial distance. That the same is the case with a confined space of far wider area is of course doubtful, because, if it were, the fish at the margin of the space would be forced to travel with a formidable linear velocity, which is quite unnatural. Next, experiments were carried out concerning the layer of swimming of the sardines (Engraulis japonicus T. & S.) in a bait-net. The bait-net of the depth of 3.6m was prepared on one side with five horizontal curtains of net work at the depths of every 0.6m. When the curtains were shut instantaneously, the whole space of the bait-net, with the fish in it, was separated into six compartments of successive depths. The number of fish and the frequency distribution of their body-length in mm-classes were studied for each compartment with the following conclusions: 1) The fish of the small size-group are distributed denser in the upper layers than in the lower, when they are mixed with the group of larger size. 2) The difference in the layer of swimming of each group seems to be due to the difference in age of the fish. When the fish of the same age-group solely are kept in the net, the frequency distribution of the body-length does not show any difference among the different layers. 3) When a headlight lamp of motor car is lighted above the net at night, the fish in the net change the layer of swimming, many of them, especially the small ones, swimming upward at first and then, after a while, sinking down. In the present experiment with the bait-net of the depth of 3.6m, the fish swarmed at last denser in the lower layers than they are in the daytime.