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Автор: Grafiati
Опубліковано: 10 грудня 2022
Оновлено: 28 січня 2023

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Статті в журналах з теми "Platonic polyhedra"

1

Caglayan, Günhan. "Hanging Around with Platonic Solids." Mathematics Teacher 112, no. 5 (March 2019): 328–29. http://dx.doi.org/10.5951/mathteacher.112.5.0328.

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Анотація:
The Platonic solids, also known as the five regular polyhedra, are the five solids whose faces are congruent regular polygons of the same type. Polyhedra is plural for polyhedron, derived from the Greek poly + hedros, meaning “multi-faces.” The five Platonic solids include the tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron. Photographs 1a-d show several regular polyhedra
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2

Lee, J., J. Duffy, and J. Rooney. "An initial investigation into the geometrical meaning of the (pseudo-) inverses of the line matrices for the edges of platonic polyhedra." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 216, no. 1 (January 1, 2002): 25–30. http://dx.doi.org/10.1243/0954406021524882.

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It is well known that there are five regular (Platonic) polyhedra: the tetrahedron, the hexahedron (cube), the octahedron, the icosahedron and the dodecahedron. Each of these polyhedra has an associated dual polyhedron which is also Platonic. By considering the Platonic polyhedra to be constructed from lines, and then representing the lines in terms of both ray and axis coordinates, a further aspect of this duality is exposed. This is the duality of poles and polars associated with projective configurations of points, lines and planes. This paper shows that a line matrix may be constructed for any regular polyhedron, in such a way that its columns represent the normalized ray coordinates of the edges of the polyhedron. The (pseudo-) inverse of this line matrix may then be constructed, the rows of which represent the normalized axis coordinates of the corresponding dual polyhedron.
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3

Lengvarszky, Zsolt. "Compound Platonic Polyhedra in Origami." Mathematics Magazine 79, no. 3 (June 1, 2006): 190. http://dx.doi.org/10.2307/27642934.

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4

Lengvarszky, Zsolt. "Compound Platonic Polyhedra in Origami." Mathematics Magazine 79, no. 3 (June 2006): 190–98. http://dx.doi.org/10.1080/0025570x.2006.11953402.

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5

Gosselin, C. M., and D. Gagnon-Lachance. "Expandable Polyhedral Mechanisms Based on Polygonal One-Degree-of-Freedom Faces." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 220, no. 7 (July 1, 2006): 1011–18. http://dx.doi.org/10.1243/09544062jmes174.

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In this article, a new family of expandable mechanisms is presented. The proposed mechanisms are expandable polyhedra built using one-degree-of-freedom (one-DOF) planar linkages. The latter planar linkages have the shape of polygons and can be expanded while preserving their shape in any of their configurations. The planar mechanisms are used to form the faces of a polyhedron. They are assembled using spherical joints at the vertices of the polyhedron. The result is a one-DOF movable polyhedron which can be expanded while preserving its shape. The application of the principle on regular polyhedra is first presented. For the five Platonic solids, theoretical maximum expansion ratios are computed, simulation results are given, and two prototypes are shown. Then, two additional examples are provided to illustrate the application of the principle to irregular polyhedra.
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6

Lichtenberg, Donovan R. "Pyramids, Prisms, Antiprisms, and Deltahedra." Mathematics Teacher 81, no. 4 (April 1988): 261–65. http://dx.doi.org/10.5951/mt.81.4.0261.

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Each of the nine covers of the Mathematics Teacher for 1985 contained pictures of two polyhedra. The covers for January through May showed the five regular polyhedra, or Platonic solids, along with their truncated versions. The latter are semiregular polyhedra, or Archimedean solids. For the months of September through December the covers displayed the remaining eight Archimedean solids.
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7

Wohlhart, Karl. "Equally circumscribed cyclic polyhedra generalize Platonic solids." Mechanism and Machine Theory 133 (March 2019): 150–63. http://dx.doi.org/10.1016/j.mechmachtheory.2018.10.004.

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8

Tavakoli, Armin, and Nicolas Gisin. "The Platonic solids and fundamental tests of quantum mechanics." Quantum 4 (July 9, 2020): 293. http://dx.doi.org/10.22331/q-2020-07-09-293.

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The Platonic solids is the name traditionally given to the five regular convex polyhedra, namely the tetrahedron, the octahedron, the cube, the icosahedron and the dodecahedron. Perhaps strongly boosted by the towering historical influence of their namesake, these beautiful solids have, in well over two millennia, transcended traditional boundaries and entered the stage in a range of disciplines. Examples include natural philosophy and mathematics from classical antiquity, scientific modeling during the days of the European scientific revolution and visual arts ranging from the renaissance to modernity. Motivated by mathematical beauty and a rich history, we consider the Platonic solids in the context of modern quantum mechanics. Specifically, we construct Bell inequalities whose maximal violations are achieved with measurements pointing to the vertices of the Platonic solids. These Platonic Bell inequalities are constructed only by inspecting the visible symmetries of the Platonic solids. We also construct Bell inequalities for more general polyhedra and find a Bell inequality that is more robust to noise than the celebrated Clauser-Horne-Shimony-Holt Bell inequality. Finally, we elaborate on the tension between mathematical beauty, which was our initial motivation, and experimental friendliness, which is necessary in all empirical sciences.
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9

Pál, Károly F., and Tamás Vértesi. "Platonic Bell inequalities for all dimensions." Quantum 6 (July 7, 2022): 756. http://dx.doi.org/10.22331/q-2022-07-07-756.

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In this paper we study the Platonic Bell inequalities for all possible dimensions. There are five Platonic solids in three dimensions, but there are also solids with Platonic properties (also known as regular polyhedra) in four and higher dimensions. The concept of Platonic Bell inequalities in the three-dimensional Euclidean space was introduced by Tavakoli and Gisin [Quantum 4, 293 (2020)]. For any three-dimensional Platonic solid, an arrangement of projective measurements is associated where the measurement directions point toward the vertices of the solids. For the higher dimensional regular polyhedra, we use the correspondence of the vertices to the measurements in the abstract Tsirelson space. We give a remarkably simple formula for the quantum violation of all the Platonic Bell inequalities, which we prove to attain the maximum possible quantum violation of the Bell inequalities, i.e. the Tsirelson bound. To construct Bell inequalities with a large number of settings, it is crucial to compute the local bound efficiently. In general, the computation time required to compute the local bound grows exponentially with the number of measurement settings. We find a method to compute the local bound exactly for any bipartite two-outcome Bell inequality, where the dependence becomes polynomial whose degree is the rank of the Bell matrix. To show that this algorithm can be used in practice, we compute the local bound of a 300-setting Platonic Bell inequality based on the halved dodecaplex. In addition, we use a diagonal modification of the original Platonic Bell matrix to increase the ratio of quantum to local bound. In this way, we obtain a four-dimensional 60-setting Platonic Bell inequality based on the halved tetraplex for which the quantum violation exceeds the 2 ratio.
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10

Naylor, Michael. "The Amazing Octacube." Mathematics Teacher 92, no. 2 (February 1999): 102–4. http://dx.doi.org/10.5951/mt.92.2.0102.

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Дисертації з теми "Platonic polyhedra"

1

Taylor, Brand R. "Topology and the Platonic Solids." Ohio University Honors Tutorial College / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=ouhonors1338993148.

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2

Chiu, Po-Jung, and 邱柏榮. "Polyhedra Expressed as the Intersection of Platonic Solids." Thesis, 2011. http://ndltd.ncl.edu.tw/handle/89247818405553855517.

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3

Castle, Toen. "Entangled graphs on surfaces in space." Phd thesis, 2013. http://hdl.handle.net/1885/11978.

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In the chemical world, as well as the physical, strands get tangled. When those strands form loops, the mathematical discipline of ‘knot theory’ can be used to analyse and describe the resultant tangles. However less has been studied about the situation when the strands branch and form entangled loops in either finite structures or infinite periodic structures. The branches and loops within the structure form a ‘graph’, and can be described by mathematical ‘graph theory’, but when graph theory concerns itself with the way that a graph can fit in space, it typically focuses on the simplest ways of doing so. Graph theory thus provides few tools for understanding graphs that are entangled beyond their simplest spatial configurations. This thesis explores this gap between knot theory and graph theory. It is focussed on the introduction of small amounts of entanglement into finite graphs embedded in space. These graphs are located on surfaces in space, and the surface is chosen to allow a limited amount of complexity. As well as limiting the types of entanglement possible, the surface simplifies the analysis of the problem – reducing a three-dimensional problem to a two-dimensional one. Through much of this thesis, the embedding surface is a torus (the surface of a doughnut) and the graph embedded on the surface is the graph of a polyhedron. Polyhedral graphs can be embedded on a sphere, but the addition of the central hole of the torus allows a certain amount of freedom for the entanglement of the edges of the graph. Entanglements of the five Platonic polyhedra (tetrahedron, octahedron, cube, dodecahedron, icosahedron) are studied in depth through their embeddings on the torus. The structures that are produced in this way are analysed in terms of their component knots and links, as well as their symmetry and energy. It is then shown that all toroidally embedded tangled polyhedral graphs are necessarily chiral, which is an important property in biochemical and other systems. These finite tangled structures can also be used to make tangled infinite periodic nets; planar repeating subgraphs within the net can be systematically replaced with a tangled version, introducing a controlled level of entanglement into the net. Finally, the analysis of entangled structures simply in terms of knots and links is shown to be deficient, as a novel form of tangling can exist which involves neither knots nor links. This new form of entanglement is known as a ravel. Different types of ravels can be localised to the immediate vicinity of a vertex, or can be spread over an arbitrarily large scope within a finite graph or periodic net. These different forms of entanglement are relevant to chemical and biochemical self-assembly, including DNA nanotechnology and metal-ligand complex crystallisation.
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4

Dohnalová, Eva. "Platonská a Archimédovská tělesa a jejich vlastnosti ve výuce matematiky na středních školách." Master's thesis, 2016. http://www.nusl.cz/ntk/nusl-347979.

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Title: Platonic and Archimedean solids and their properties in teaching of mathematics at secondary schools Author: Eva Dohnalová Department: Department of Didactics of Mathematics Supervisor: doc. RNDr. Jarmila Robová, CSc. Abstract: This work is an extension of my bachelor work and it is intended for all people interested in regular and semiregular polyhedra geometry. It is a comprehensive text which summarizes brief history, description and classification of regular and semiregular polyhedra. The work contains proofs of Descartes' and Euler's theorems and proofs about number of regular and semiregular polyhedra. It can be also used as a didactic aid in the instruction of regular and semiregular solids at secondary schools. This text is supplemented by illustrative pictures made in GeoGebra and Cabri3D. Keywords: Regular polyhedra, platonic solids, Platon, semiregular polyhedra, Archimedean solids, Archimedes, dulaism, Descartes' theorem, Euler's theorem.
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Книги з теми "Platonic polyhedra"

1

Beginner's book of multimodular origami polyhedra: The platonic solids. Mineola, N.Y: Dover, 2008.

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2

MacLean, Kenneth James Michael. A geometric analysis of the platonic solids and other semi-regular polyhedra: With an introduction to the phi ratio : for teachers, researchers and the generally curious. Ann Arbor, MI: Loving Healing Press, 2007.

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3

Sutton, Daud. Platonic and Archimedean Solids. Wooden Books, 2005.

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4

Platonic & Archimedean Solids (Wooden Books). Walker & Company, 2002.

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5

Arnstein, Bennett, and Rona Gurkewitz. Beginner's Book of Modular Origami Polyhedra: The Platonic Solids. Dover Publications, Incorporated, 2012.

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6

Arnstein, Bennett, and Rona Gurkewitz. Beginner's Book of Modular Origami Polyhedra: The Platonic Solids. Dover Publications, Incorporated, 2012.

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7

The platonic solids activity book. Berkeley, CA: Key Curriculum Press, 1991.

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8

MacLean, Kenneth J. M. A Geometric Analysis of the Platonic Solids and Other Semi-Regular Polyhedra. Loving Healing Press, 2015.

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9

MacLean, Kenneth J. M. A Geometric Analysis of the Platonic Solids and Other Semi-Regular Polyhedra (Geometric Explorations Series). Loving Healing Press, 2007.

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10

MacLean, Kenneth J. M. A Geometric Analysis of the Platonic Solids and Other Semi-Regular Polyhedra: With an Introduction to the Phi Ratio, 2nd Edition. Marvelous Spirit Press, 2019.

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Частини книг з теми "Platonic polyhedra"

1

Itoh, Jin-ichi, and Chie Nara. "Continuous Flattening of Platonic Polyhedra." In Lecture Notes in Computer Science, 108–21. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-24983-9_11.

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2

Wills, Jörg M. "Polyhedra Analogues of the Platonic Solids." In Shaping Space, 223–29. New York, NY: Springer New York, 2012. http://dx.doi.org/10.1007/978-0-387-92714-5_17.

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3

Rajwade, A. R. "Metrical Properties of the five Platonic Polyhedra." In Texts and Readings in Mathematics, 33–39. Gurgaon: Hindustan Book Agency, 2001. http://dx.doi.org/10.1007/978-93-86279-06-4_5.

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4

Gronchi, Giovanni Federico. "Periodic Orbits of the N-body Problem with the Symmetry of Platonic Polyhedra." In Mathematical Models and Methods for Planet Earth, 143–55. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-02657-2_12.

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5

Wei, Guowu, and Jian S. Dai. "Duality of the Platonic Polyhedrons and Isomorphism of the Regular Deployable Polyhedral Mechanisms (DPMs)." In Advances in Reconfigurable Mechanisms and Robots I, 759–71. London: Springer London, 2012. http://dx.doi.org/10.1007/978-1-4471-4141-9_68.

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6

"Polyhedra: Platonic Solids." In Connections, 255–94. WORLD SCIENTIFIC, 2001. http://dx.doi.org/10.1142/9789812811394_0007.

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"More Platonic Solids Design." In Origami Polyhedra Design, 135–45. A K Peters/CRC Press, 2009. http://dx.doi.org/10.1201/b11163-12.

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8

"Sunken Platonic Solids Design." In Origami Polyhedra Design, 147–70. A K Peters/CRC Press, 2009. http://dx.doi.org/10.1201/b11163-13.

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9

"Platonic and Related Polyhedra." In Origami Polyhedra Design, 61. A K Peters/CRC Press, 2009. http://dx.doi.org/10.1201/b11163-8.

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10

"Part II: Platonic and Related Polyhedra." In Origami Polyhedra Design, 75–184. A K Peters/CRC Press, 2009. http://dx.doi.org/10.1201/b11163-6.

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Тези доповідей конференцій з теми "Platonic polyhedra"

1

Wei, Guowu, and Jian S. Dai. "An Overconstrained Eight-Bar Linkage and its Associated Fulleroid-Like Deployable Platonic Mechanisms." 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/detc2014-34499.

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This paper for the first time presents an overconstrained spatial eight-bar linkage and its application to the synthesis of a group of Fulleroid-like deployable Platonic mechanisms. Structure of the proposed eight-bar linkage is introduced, and constrain and mobility of the linkage are revealed based on screw theory. Then by integrating the proposed eight-bar linkage into Platonic polyhedron bases, synthesis of a group of Fulleroid-like deployable Platonic mechanism is carried out and illustrated by the synthesis and construction of a Fulleroid-like deployable tetrahedral mechanism. Further, mobility of the Fulleroid-like deployable Platonic mechanisms is formulated via constraint matrices by following Kirchhoff’s circulation law for mechanical networks and kinematics of the mechanisms is presented with numerical simulations illustrating the intrinsic kinematic properties of the group of Fulleroid-like deployable Platonic mechanisms. Ultimately, a prototype of the Fulleroid-like deployable hexahedral mechanism is fabricated and tested verifying mobility and kinematic characteristics of the proposed deployable polyhedral mechanisms. This paper thus presents a novel overconstrained spatial eight-bar linkage and a new geometrically intuitive method for synthesizing Fulleroid-like regular deployable polyhedral mechanisms.
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2

Wei, Guowu, and Jian Dai. "Synthesis and Construction of a Family of One-DOF Highly Overconstrained Deployable Polyhedral Mechanisms (DPMs)." In ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/detc2012-70918.

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This paper presents a family of one-DOF highly overconstrained regular and semi-regular deployable polyhedral mechanisms (DPMs) that perform radially reciprocating motion. Based on two fundamental kinematic chains with radially reciprocating motion, i.e. the PRRP chain and a novel plane/semi-plane-symmetric spatial eight-bar linkage, two methods, i.e. the virtual-axis-based (VAB) method and the virtual-centre-based (VCB) method are proposed for the synthesis of the family of regular and semi-regular DPMs. Procedure and principle for synthesizing the mechanisms are presented and selected DPMs are constructed based on the five regular Platonic polyhedrons and the semi-regular Archimedean polyhedrons, Prism polyhedrons and Johnson polyhedrons. Mobility of the mechanisms is then analysed and verified using screw-loop equation method and degree of overconstraint of the mechanisms are investigated by combing the Euler’s formula for polyhedrons and the Grübler-Kutzbach formula for mobility analysis of linkages.
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