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Relevant bibliographies by topics / Constructing regular polygons

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Academic literature on the topic 'Constructing regular polygons'

Author: Grafiati

Published: 4 June 2021

Last updated: 15 February 2022

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Journal articles on the topic "Constructing regular polygons"

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Helms, Charles N., and Robert E. Hart. "A Polygon-Based Line-Integral Method for Calculating Vorticity, Divergence, and Deformation from Nonuniform Observations." Journal of Applied Meteorology and Climatology 52, no. 6 (June 2013): 1511–21. http://dx.doi.org/10.1175/jamc-d-12-0248.1.

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AbstractTraditional observational analysis of derivative-based variables (e.g., vorticity) usually relies on interpolating observations and evaluating spatial derivatives either on a Cartesian grid or on a spherical grid. Great care must be taken in selecting the domain and the interpolation scheme to properly represent the features. There exist a number of alternative methods of calculating such variables by evaluating line integrals on triangular regions according to Green’s theorem. Since these methods rely on only three observations to perform calculations, they are overly sensitive to observations dominated by local phenomena as well as instrument noise. A few studies have attempted to minimize the impact of nonrepresentative or noisy observations by using higher-order polygons, but they have been limited to fitting regular polygons to near-regularly gridded data. The current study describes a new approach to calculating these fields by constructing higher-order polygons from a triangle tessellation and then applying Green’s theorem. Since the polygons are constructed using an automated triangle tessellation, the polygon construction process can proceed without the need for uniformly spaced data. The triangle tessellation employed here is unique for a given set of points, generating easily reproducible results. In addition, this method reduces the impact of noise associated with individual observations with only a minor loss in the length of the resolvable scale. An error analysis of the proposed method shows a large decrease in errors in comparison with purely triangle-based calculations. These improvements are present with a variety of data distributions (random and along research aircraft flight paths) and kinematic variables (vorticity, divergence, and deformation).

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Cho, Ju Yeon, and Sang Hun Song. "An Analysis of the Constructing Method Types of Regular Polygons in Elementary Gifted Class." School Mathematics 20, no. 1 (March 31, 2018): 171–84. http://dx.doi.org/10.29275/sm.2018.03.20.1.171.

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Verheyen, H. F. "A Single Die-Cut Element for Transformable Structures." International Journal of Space Structures 8, no. 1-2 (April 1993): 127–34. http://dx.doi.org/10.1177/0266351193008001-213.

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Any polyhedral structure composed of identical regular polygons can be turned into an expandable structure by replacing the polygons by pairs of polygons which can rotate about a common central hinge, and connecting a vertex of an upper polygon with the vertex of a lower polygon of an adjacent pair. Most of these structures will collapse in the fully expanded position by losing their rigidity near the final stage, and hence, become deployable. A triangular element which enables one to assemble and dismantle such structures is presented here, together with a series of examples of experimental shapes.

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Perkins, Martin. "89.52 Approximate constructions for regular polygons." Mathematical Gazette 89, no. 515 (July 2005): 290–92. http://dx.doi.org/10.1017/s0025557200177873.

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Detemple, Duane W. "Simple Constructions for the Regular Pentagon and Heptadecagon." Mathematics Teacher 82, no. 5 (May 1989): 361–65. http://dx.doi.org/10.5951/mt.82.5.0361.

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This article discusses two new Euclidean constructions to inscribe regular polygons of five and seventeen sides in a circle. The pentagon's construction shares many common elements with the familiar Ptolemaic method (see, e.g., Jacobs [1982, 254 55]), but an interesting variation occurs in the final steps. The explanation given for why the construction works is unusual in that it does not make use of the distance formula of the Pythagorean theorem.

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Munteanu, Marius, and Laura Munteanu. "Ruler and regular polygon constructions." Journal of Geometry 104, no. 3 (October 17, 2013): 515–37. http://dx.doi.org/10.1007/s00022-013-0184-0.

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Wang, C. Y. "In-Plane Vibration of a Deployable Segmented Ring." International Journal of Structural Stability and Dynamics 20, no. 07 (July 2020): 2071007. http://dx.doi.org/10.1142/s0219455420710078.

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The in-plane vibrations of regular polygonal rings composed of rigid segments joined by torsional springs are studied for the first time. The nonlinear dynamical difference equations are formulated and solved by perturbation about the equilibrium state. As the number of segments increase, the frequencies, if aptly normalized, converge to the classical vibration frequencies of a continuous elastic ring. The vibration mode shapes are illustrated. The tiling of many identical polygons is discussed. Possible applications include the vibrations of space structures and graphene sheets.

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Kovács, Zoltán. "Automated Detection of Interesting Properties in Regular Polygons." Mathematics in Computer Science 14, no. 4 (June 25, 2020): 727–55. http://dx.doi.org/10.1007/s11786-020-00491-z.

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Abstract We demonstrate a systematic, automated way of discovery of a large number of new geometry theorems on regular polygons. The applied theory includes a formula by Watkins and Zeitlin on minimal polynomials of $$\cos \frac{2\pi }{n}$$ cos 2 π n , and a method by Recio and Vélez to discover a property in a plane geometry construction. This method exploits Wu’s idea on algebraizing the geometric setup and utilizes the theory of Gröbner bases. Also a bijective function is given that maps the investigated cases to the first natural numbers. Finally, several examples are shown that are all previously unknown results in planar Euclidean geometry.

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Huybers, P. "Form Generation Of Polyhedric Building Shapes." International Journal of Space Structures 11, no. 1-2 (April 1996): 173–81. http://dx.doi.org/10.1177/026635119601-223.

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The Platonic - or regular - and the Archimedean - or semi-regular - polyhedra can be considered as portions of space that are completely surrounded by one or more kinds of regular polygons. The numbers and positions in space of these polygons are strictly ruled by universal criteria. It is therefore possible to form these polyhedra by placing polygons around the centre of the coordinate system in distinct numbers, at certain distances and under certain angles in accordance with these rules. This is called here ‘rotation’ and the forelying paper describes a method where this is done for the regular and semi-regular polyhedra and for related figures that are found by derivation from these polyhedra. The figures that are rotated have not necessarily to be regular polygons, nor do have to be strictly planar. This method thus allows the rotation of arbitrary figures – also spatial ones – and the rotation procedure can even be used repeatedly, so that very complex configurations can be described.

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Nadyrshine, Neil, Lilia Nadyrshine, Rafik Khafizov, Nailia Ibragimova, and Karine Mkhitarian. "Parametric methods for constructing the Islamic ornament." E3S Web of Conferences 274 (2021): 09009. http://dx.doi.org/10.1051/e3sconf/202127409009.

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The article discusses the method of algorithmic construction of Islamic ornaments, which can be used as a decorative element in architectural design. Two necessary stages are highlighted in the generation of an ornamental motif by means of algorithmic design – the first includes operations on a base mesh consisting of a combination of regular or semiregular cells, for example, drawing rays from the centers of the edges of the original lattice or rotating figures relative to its nodes; the second stage involves the creation of a basic pattern of the ornamental design that fits into a regular or semi-regular polygon, and the decorative motif in the figure is made up of a primitive that is symmetrically reflected relative to any lines, usually rays, emanating from opposite corners of the polygon. The paper analyses the traditional patterns of Islamic ornaments, on the basis of which new designs were built, using visual programming tools (Rhino and Grasshopper).

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Dissertations / Theses on the topic "Constructing regular polygons"

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DeSouza, Chelsea E. "The Greek Method of Exhaustion: Leading the Way to Modern Integration." The Ohio State University, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=osu1338326658.

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Books on the topic "Constructing regular polygons"

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Chen, Fen. Regular Polygons: Applied New Theory of Trisection to Construct a Regular Heptagon for Centuries in the History of Mathematics. International School Math & Sciences Institut, 2001.

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Book chapters on the topic "Constructing regular polygons"

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Bewersdorff, Jörg. "The construction of regular polygons." In Galois Theory for Beginners, 63–80. Providence, Rhode Island: American Mathematical Society, 2006. http://dx.doi.org/10.1090/stml/035/07.

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Ghourabi, Fadoua, Tetsuo Ida, and Kazuko Takahashi. "Interactive construction and automated proof in Eos system with application to knot fold of regular polygons." In Origami⁶, 55–65. Providence, Rhode Island: American Mathematical Society, 2015. http://dx.doi.org/10.1090/mbk/095.1/06.

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"The construction of regular polygons." In Galois Theory for Beginners, 85–103. Providence, Rhode Island: American Mathematical Society, 2021. http://dx.doi.org/10.1090/stml/095/07.

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Conference papers on the topic "Constructing regular polygons"

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Burton, Greg. "A Hybrid Approach to Polygon Offsetting Using Winding Numbers and Partial Computation of the Voronoi Diagram." In ASME 2014 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/detc2014-34303.

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In this paper we present a new, efficient algorithm for computing the “raw offset” curves of 2D polygons with holes. Prior approaches focus on (a) complete computation of the Voronoi Diagram, or (b) pair-wise techniques for generating a raw offset followed by removal of “invalid loops” using a sweepline algorithm. Both have drawbacks in practice. Robust implementation of Voronoi Diagram algorithms has proven complex. Sweeplines take O((n + k)log n) time and O(n + k) memory, where n is the number of vertices and k is the number of self-intersections of the raw offset curve. It has been shown that k can be O(n2) when the offset distance is greater than or equal to the local radius of curvature of the polygon, a regular occurrence in the creation of contour-parallel offset curves for NC pocket machining. Our O(n log n) recursive algorithm, derived from Voronoi diagram algorithms, computes the velocities of polygon vertices as a function of overall offset rate. By construction, our algorithm prunes a large proportion of locally invalid loops from the raw offset curve, eliminating all self-intersections in raw offsets of convex polygons and the “near-circular”, k proportional to O(n2) worst-case scenarios in non-convex polygons.

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Ida, Tetsuo, Fadoua Ghourabi, and Kazuko Takahashi. "Knot Fold of Regular Polygons: Computer-Assisted Construction and Verification." In 2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC). IEEE, 2013. http://dx.doi.org/10.1109/synasc.2013.9.

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Irick, Kevin W., and Nima Fathi. "Response Effects Due to Polygonal Representation of Pores in Porous Media Thermal Models." In ASME 2021 Verification and Validation Symposium. American Society of Mechanical Engineers, 2021. http://dx.doi.org/10.1115/vvs2021-65231.

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Abstract Physics models — such as thermal, structural, and fluid models — of engineering systems often incorporate a geometric aspect such that the model resembles the shape of the true system that it represents. However, the physical domain of the model is only a geometric representation of the true system, where geometric features are often simplified for convenience in model construction and to avoid added computational expense to running simulations. The process of simplifying or neglecting different aspects of the system geometry is sometimes referred to as “defeaturing.” Typically, modelers will choose to remove small features from the system model, such as fillets, holes, and fasteners. This simplification process can introduce inherent error into the computational model.Asimilar event can even take place when a computational mesh is generated, where smooth, curved features are represented by jagged, sharp geometries. The geometric representation and feature fidelity in a model can play a significant role in a corresponding simulation’s computational solution. In this paper, a porous material system — represented by a single porous unit cell — is considered. The system of interest is a two-dimensional square cell with a centered circular pore, ranging in porosity from 1% to 78%. However, the circular pore was represented geometrically by a series of regular polygons with number of sides ranging from 3 to 100. The system response quantity under investigation was the dimensionless effective thermal conductivity, k*, of the porous unit cell. The results show significant change in the resulting k* value depending on the number of polygon sides used to represent the circular pore. In order to mitigate the convolution of discretization error with this type of model form error, a series of five systematically refined meshes was used for each pore representation. Using the finite element method (FEM), the heat equation was solved numerically across the porous unit cell domain. Code verification was performed using the Method of Manufactured Solutions (MMS) to assess the order of accuracy of the implemented FEM. Likewise, solution verification was performed to estimate the numerical uncertainty due to discretization in the problem of interest. Specifically, a modern grid convergence index (GCI) approach was employed to estimate the numerical uncertainty on the systematically refined meshes. The results of the analyses presented in this paper illustrate the importance of understanding the effects of geometric representation in engineering models and can help to predict some model form error introduced by the model geometry.

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