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Автор: Grafiati

Опубліковано: 10 грудня 2022

Оновлено: 28 січня 2023

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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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Анотація:

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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Анотація:

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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Анотація:

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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11

Timofeenko, A. V. "The non-platonic and non-Archimedean noncomposite polyhedra." Journal of Mathematical Sciences 162, no. 5 (October 17, 2009): 710–29. http://dx.doi.org/10.1007/s10958-009-9655-0.

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12

Fujii, Shota, Rintaro Takahashi, Ji Ha Lee, and Kazuo Sakurai. "Correction: A tetramer micelle: the smallest aggregation number corresponding to the vertex number of regular polyhedra in platonic micelles." Soft Matter 14, no. 6 (2018): 1067. http://dx.doi.org/10.1039/c8sm90014k.

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Анотація:

Correction for ‘A tetramer micelle: the smallest aggregation number corresponding to the vertex number of regular polyhedra in platonic micelles’ by Shota Fujii et al., Soft Matter, 2018, DOI: 10.1039/c7sm02028g.

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13

Kovács, F. "Number and twistedness of strands in weavings on regular convex polyhedra." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 470, no. 2162 (February 8, 2014): 20130608. http://dx.doi.org/10.1098/rspa.2013.0608.

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This paper deals with two- and threefold weavings on Platonic polyhedral surfaces. Depending on the skewness of the weaving pattern with respect to the edges of the polyhedra, different numbers of closed strands are necessary in a complete weaving. The problem is present in basketry but can be addressed from the aspect of pure geometry (geodesics), graph theory (central circuits of 4-valent graphs) and even structural engineering (fastenings on a closed surface). Numbers of these strands are found to have a periodicity and symmetry, and, in some cases, this number can be predicted directly from the skewness of weaving. In this paper (i) a simple recursive method using symmetry operations is given to find the number of strands of cubic, octahedral and icosahedral weavings for cases where generic symmetry arguments fail; (ii) another simple method is presented to decide whether or not a single closed strand can run along the underlying Platonic without a turn (i.e. the linking number of the two edges of a strand is zero, and so the loop can be stretched to a circle without being twisted); and (iii) the linking number of individual strands in an alternate ‘check’ weaving pattern is determined.

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14

KRIVOSHAPKO, S. N. "POLYHEDRA AND QUASI- POLYHEDRA IN ARCHITECTURE OF CIVIL AND INDUSTRIAL ERECTIONS." Building and reconstruction 90, no. 4 (2020): 48–64. http://dx.doi.org/10.33979/2073-7416-2020-90-4-48-64.

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Анотація:

Innovative spatial forms appear and develop at the joint of science, art, and architecture. Geometry is the most important, fundamental components of architectural forming. Now, having passed the stages of passion for the large-span shells, the sky-scrapers, typical inexpensive buildings, architectural bionics and ergonomics; pneumatic, membrane, wire rope and shrouds erections, the architects and designers payed attention at analytically non-given forms of erections and at the polyhedron. It is noticeably especially at the last 10-15 years. In a paper, the problems of application of the polyhedron and their modifications in architecture, building, and technics are analyzed. They consider prisms, pyramids, prismatoids, Platonic and several Archimedean solids, quasi-polyhedrons, and some figures constituted on their base. Polyhedral domes, umbrella shells, and hipped plate constructions are presented too. Large quantity of the illustrations devoted to the architecture of buildings and erections, to the landscape architecture and to the sculptural compositions is presented for the confirmation of increasing interest to these structures. 31 titles of the used original sources are given.

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15

Siqueira, Paulo Henrique. "Visualization of Archimedian and Platonic polyhedra using a web environment in Augmented Reality and Virtual Reality." International Journal for Innovation Education and Research 9, no. 11 (November 1, 2021): 1–13. http://dx.doi.org/10.31686/ijier.vol9.iss11.2441.

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Анотація:

This paper shows the development of a web environment for the construction of Archimedes and Plato polyhedra in Augmented Reality (AR) and Virtual Reality (VR). In this environment we used the geometric transformations of translation and rotation with the structure of hierarchies of HTML pages, without the use of the coordinates of each polyhedra vertex. The developed environment can be used in classroom to visualize the polyhedra in Augmented Reality, with the possibility of manipulations of the graphical representations by students in the environment created in Virtual Reality. Other studies that can be developed with the polyhedra modeled are areas, volumes and the relation of Euler. Another important content that can be developed is truncation, because seven Archimedes polyhedra are obtained by using truncation of Plato's polyhedrons. With this work, it becomes possible to develop didactic materials with a simple technology, free and with great contribution to improvement of the teaching of Geometry and other areas that use representation of 3D objects.

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16

Griffiths, Martin. "n-dimensional enrichment for Further Mathematicians." Mathematical Gazette 89, no. 516 (November 2004): 409–16. http://dx.doi.org/10.1017/s0025557200178258.

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Анотація:

There are infinitely many regular polygons, but we find, on extending the idea of polygons to three dimensions, that there are only five regular polyhedra, the Platonic solids. What happens then if we try to extend this idea beyond three dimensions? It turns out that, of the five Platonic solids, just the regular tetrahedron, cube and regular octahedron have analogues in all higher dimensions, the so-called regular polytopes. Brief descriptions of these mathematical objects are to be found in [1], for example.

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17

Montejano-Carrizales, Juan Martín, José Luis Rodríguez-López, Umapada Pal, Mario Miki-Yoshida, and Miguel José-Yacamán. "The Completion of the Platonic Atomic Polyhedra: The Dodecahedron." Small 2, no. 3 (March 2006): 351–55. http://dx.doi.org/10.1002/smll.200500362.

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18

Hopley, Ronald B. "Nested Platonic Solids: A Class Project in Solid Geometry." Mathematics Teacher 87, no. 5 (May 1994): 312–18. http://dx.doi.org/10.5951/mt.87.5.0312.

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Анотація:

Several years ago at a regional NCTM conference in Phoenix, the author was fascinated by a set of card board Platonic solids that were nested inside each other. The Platonic solids are polyhedra whose faces are congruent regular polygonal regions, such that the number of edges that meet at each vertex is the same for all vertices; only five are possible. since the set is no longer commercially available, the author decided to make a nested set for classroom demonstrations and instruction for students to make their own.

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19

Teich, Erin G., Greg van Anders, Daphne Klotsa, Julia Dshemuchadse, and Sharon C. Glotzer. "Clusters of polyhedra in spherical confinement." Proceedings of the National Academy of Sciences 113, no. 6 (January 25, 2016): E669—E678. http://dx.doi.org/10.1073/pnas.1524875113.

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Dense particle packing in a confining volume remains a rich, largely unexplored problem, despite applications in blood clotting, plasmonics, industrial packaging and transport, colloidal molecule design, and information storage. Here, we report densest found clusters of the Platonic solids in spherical confinement, for up to N=60 constituent polyhedral particles. We examine the interplay between anisotropic particle shape and isotropic 3D confinement. Densest clusters exhibit a wide variety of symmetry point groups and form in up to three layers at higher N. For many N values, icosahedra and dodecahedra form clusters that resemble sphere clusters. These common structures are layers of optimal spherical codes in most cases, a surprising fact given the significant faceting of the icosahedron and dodecahedron. We also investigate cluster density as a function of N for each particle shape. We find that, in contrast to what happens in bulk, polyhedra often pack less densely than spheres. We also find especially dense clusters at so-called magic numbers of constituent particles. Our results showcase the structural diversity and experimental utility of families of solutions to the packing in confinement problem.

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20

Coolsaet, Kris, and Stan Schein. "Some New Symmetric Equilateral Embeddings of Platonic and Archimedean Polyhedra." Symmetry 10, no. 9 (September 5, 2018): 382. http://dx.doi.org/10.3390/sym10090382.

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Анотація:

The icosahedron and the dodecahedron have the same graph structures as their algebraic conjugates, the great dodecahedron and the great stellated dodecahedron. All four polyhedra are equilateral and have planar faces—thus “EP”—and display icosahedral symmetry. However, the latter two (star polyhedra) are non-convex and “pathological” because of intersecting faces. Approaching the problem analytically, we sought alternate EP-embeddings for Platonic and Archimedean solids. We prove that the number of equations—E edge length equations (enforcing equilaterality) and 2 E − 3 F face (torsion) equations (enforcing planarity)—and of variables ( 3 V − 6 ) are equal. Therefore, solutions of the equations up to equivalence generally leave no degrees of freedom. As a result, in general there is a finite (but very large) number of solutions. Unfortunately, even with state-of-the-art computer algebra, the resulting systems of equations are generally too complicated to completely solve within reasonable time. We therefore added an additional constraint, symmetry, specifically requiring solutions to display (at least) tetrahedral symmetry. We found 77 non-classical embeddings, seven without intersecting faces—two, four and one, respectively, for the (graphs of the) dodecahedron, the icosidodecahedron and the rhombicosidodecahedron.

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21

Brinkmann, Gunnar, Pieter Goetschalckx, and Stan Schein. "Comparing the constructions of Goldberg, Fuller, Caspar, Klug and Coxeter, and a general approach to local symmetry-preserving operations." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2206 (October 2017): 20170267. http://dx.doi.org/10.1098/rspa.2017.0267.

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The use of operations on polyhedra possibly dates back to the ancient Greeks, who were the first to describe the Archimedean solids, which can be constructed from the Platonic solids by local symmetry-preserving operations (e.g. truncation) on the solid. By contrast, the results of decorations of polyhedra, e.g. by Islamic artists and by Escher, have been interpreted as decorated polyhedra—and not as new and different polyhedra. Only by interpreting decorations as combinatorial operations does it become clear how closely these two approaches are connected. In this article, we first sketch and compare the operations of Goldberg, Fuller, Caspar & Klug and Coxeter to construct polyhedra with icosahedral symmetry, where all faces are pentagons or hexagons and all vertices have three neighbours. We point out and correct an error in Goldberg’s construction. In addition, we transform the term symmetry-preserving into an exact requirement. This goal, symmetry-preserving, could also be obtained by taking global properties into account, e.g. the symmetry group itself, so we make precise the terms local and operation . As a result, we can generalize Goldberg’s approach to a systematic one that uses chamber operations to encompass all local symmetry-preserving operations on polyhedra.

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22

Kramer, Peter. "Platonic polyhedra tune the 3-sphere: harmonic analysis on simplices." Physica Scripta 81, no. 1 (January 2010): 019801. http://dx.doi.org/10.1088/1402-4896/81/1/019801.

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23

Kramer, Peter. "Platonic polyhedra tune the 3-sphere: harmonic analysis on simplices." Physica Scripta 79, no. 4 (March 31, 2009): 045008. http://dx.doi.org/10.1088/0031-8949/79/04/045008.

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24

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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Анотація:

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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25

Domokos, Gábor, Douglas J. Jerolmack, Ferenc Kun, and János Török. "Plato’s cube and the natural geometry of fragmentation." Proceedings of the National Academy of Sciences 117, no. 31 (July 17, 2020): 18178–85. http://dx.doi.org/10.1073/pnas.2001037117.

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Plato envisioned Earth’s building blocks as cubes, a shape rarely found in nature. The solar system is littered, however, with distorted polyhedra—shards of rock and ice produced by ubiquitous fragmentation. We apply the theory of convex mosaics to show that the average geometry of natural two-dimensional (2D) fragments, from mud cracks to Earth’s tectonic plates, has two attractors: “Platonic” quadrangles and “Voronoi” hexagons. In three dimensions (3D), the Platonic attractor is dominant: Remarkably, the average shape of natural rock fragments is cuboid. When viewed through the lens of convex mosaics, natural fragments are indeed geometric shadows of Plato’s forms. Simulations show that generic binary breakup drives all mosaics toward the Platonic attractor, explaining the ubiquity of cuboid averages. Deviations from binary fracture produce more exotic patterns that are genetically linked to the formative stress field. We compute the universal pattern generator establishing this link, for 2D and 3D fragmentation.

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26

Stefanides, Panagiotis. "Panagiotis Chr.Stefanides Invited to 4th TECHNIUM International Conference - Recognition of career." Technium: Romanian Journal of Applied Sciences and Technology 2, no. 3 (April 28, 2020): 1–16. http://dx.doi.org/10.47577/technium.v2i3.454.

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Анотація:

As I anticipate, it concerns another genus of Polyhedron, a very Special one Ontologically, and this is very important Ι understand: “....Στερεὰ δὲ σώματα λέγεσθαι χρὴ …. πέντε, ……., τὸ δὲ ἄλλο γένος ἅπαν ἔχει μορφὴν μίαν·…..…ψυχῆς γένος" http://remacle.org/bloodwolf/philosophes/platon/cousin/epinomisgrec.htm [..there are .. five solid bodies…. the other genus which in total has one form …the genus of the soul…] Plato’s Epinomis 981b. In 981a, of this work, Plato states that the composition of, soul and body bares a single form. Similarly, Plato in Timaeus [53 E] refers to the solids having each its own genus and in his Republic makes reference to the Construction of the Universal Planets [XIV 616 E -617A]. Interpretation for γένος genus – form] Proposed By Panagiotis Stefanides is the “Generator Polyhedron”, ohis recent Abstract. Searching, for many years, Plato's Timaeus Work, geometry related to the creation of the world- soul of the world] and presenting it to conferences nationally and internationally, I searched in the Liddell and Scott reference for the word “γένος” found in Plato's "Epinomis" 981b Discovered [Invention [ 03 April 2017].https://www.linkedin.com/…/generator-polyhedron-platonic-e…/ . “Generator Polyhedron” refers to the geometric characteristics of this Solid found to be the root upon which other Solid Polyhedra are based i.e. the Platonic/Eucleidean Solids [Icosahedron Dodecahedron etc.] The Geometry of this paper is part of book: [ISBN 978 – 618 – 83169 – 0 - 4], National Library of Greece , 04/05/2017, by Panagiotis Ch. Stefanides.

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27

Meurant, Robert C. "A New Order in Space — Platonic and Archimedian Polyhedra and Tilings." International Journal of Space Structures 6, no. 1 (March 1991): 11–32. http://dx.doi.org/10.1177/026635119100600102.

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28

Pedersen, Martin Cramer, and Stephen T. Hyde. "Polyhedra and packings from hyperbolic honeycombs." Proceedings of the National Academy of Sciences 115, no. 27 (June 20, 2018): 6905–10. http://dx.doi.org/10.1073/pnas.1720307115.

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Анотація:

We derive more than 80 embeddings of 2D hyperbolic honeycombs in Euclidean 3 space, forming 3-periodic infinite polyhedra with cubic symmetry. All embeddings are “minimally frustrated,” formed by removing just enough isometries of the (regular, but unphysical) 2D hyperbolic honeycombs {3,7}, {3,8}, {3,9}, {3,10}, and {3,12} to allow embeddings in Euclidean 3 space. Nearly all of these triangulated “simplicial polyhedra” have symmetrically identical vertices, and most are chiral. The most symmetric examples include 10 infinite “deltahedra,” with equilateral triangular faces, 6 of which were previously unknown and some of which can be described as packings of Platonic deltahedra. We describe also related cubic crystalline packings of equal hyperbolic discs in 3 space that are frustrated analogues of optimally dense hyperbolic disc packings. The 10-coordinated packings are the least “loosened” Euclidean embeddings, although frustration swells all of the hyperbolic disc packings to give less dense arrays than the flat penny-packing even though their unfrustrated analogues in H2 are denser.

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29

Fujii, Shota, Rintaro Takahashi, Lee Ji Ha, and Kazuo Sakurai. "A tetramer micelle: the smallest aggregation number corresponding to the vertex number of regular polyhedra in platonic micelles." Soft Matter 14, no. 6 (2018): 875–78. http://dx.doi.org/10.1039/c7sm02028g.

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30

Nickalls, R. W. D. "The quartic equation: alignment with an equivalent tetrahedron." Mathematical Gazette 96, no. 535 (March 2012): 49–55. http://dx.doi.org/10.1017/s0025557200003958.

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The lower polynomials are inextricably linked to the symmetries of polyhedra and Platonic solids [1, 2, 3], and the quartic is no exception; its alter ego is the regular tetrahedron [4]. In this article we present a solution to the problem of aligning the vertices of a tetrahedron with the roots of a particular quartic. After establishing the size of a quartic-equivalent tetrahedron, we derive a triple-angle expression for the alignment rotation, analogous to that for the cubic [5].

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31

Kramer, Peter. "Platonic polyhedra tune the three-sphere: II. Harmonic analysis on cubic spherical manifolds." Physica Scripta 82, no. 1 (July 7, 2010): 019802. http://dx.doi.org/10.1088/1402-4896/82/1/019802.

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32

Fusco, G., G. F. Gronchi, and P. Negrini. "Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem." Inventiones mathematicae 185, no. 2 (December 16, 2010): 283–332. http://dx.doi.org/10.1007/s00222-010-0306-3.

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33

Nestorović, Miodrag, and Vladimir Mišković. "Advanced development of space structures in domains of 3D transformation." SAJ - Serbian Architectural Journal 3, no. 3 (2011): 116–39. http://dx.doi.org/10.5937/saj1102116n.

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The emergence of the new architectural solutions and structural forms of Mengeringhausen, Tsuboi, Safdie, Foster, Calatrava and other creators of magnificent structures, may be taken as an initiation and explosion of inventiveness which has continued up to till present. Consequently, the topic of this paper is to show a part of broad range of structural systems which have not been sufficiently disclosed in Serbia and surroundings, in spite of their attractiveness in contemporary architecture, in terms of space transformations, materialization and technology. The basic properties of all analyzed space structures lies in their geometric shape (Archimedean and Platonic polyhedra, polyhedron structures, and bionic of structures as well), which applies regularity, symmetry, speed of mounting, as well as modularity of the original matrices. Solutions and analyses shown deal with multifunctional space matrices, which make its potential very important both in architectural design and in structural theory. The topic of this paper is to consider the development of matrix structure in context of architectural forms in future, emphasizing the importance of structural geometry and its possible applications.

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34

Kramer, Peter. "Platonic polyhedra tune the three-sphere: II. Harmonic analysis on cubic spherical three-manifolds." Physica Scripta 80, no. 2 (July 20, 2009): 025902. http://dx.doi.org/10.1088/0031-8949/80/02/025902.

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35

Kramer, Peter. "Platonic polyhedra tune the three-sphere: III. Harmonic analysis on octahedral spherical three-manifolds." Physica Scripta 81, no. 2 (January 29, 2010): 025005. http://dx.doi.org/10.1088/0031-8949/81/02/025005.

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36

Fenucci, M., and G. F. Gronchi. "On the stability of periodic N-body motions with the symmetry of Platonic polyhedra." Nonlinearity 31, no. 11 (October 5, 2018): 4935–54. http://dx.doi.org/10.1088/1361-6544/aad644.

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37

Vukičević, Damir, and Alexandru T. Balaban. "Note on ordering and complexity of Platonic and Archimedean polyhedra based on solid angles." Journal of Mathematical Chemistry 44, no. 3 (February 13, 2008): 725–30. http://dx.doi.org/10.1007/s10910-008-9361-z.

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38

Fenucci, Marco, and Àngel Jorba. "Braids with the symmetries of Platonic polyhedra in the Coulomb (N+1)-body problem." Communications in Nonlinear Science and Numerical Simulation 83 (April 2020): 105105. http://dx.doi.org/10.1016/j.cnsns.2019.105105.

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39

Neumann, W., and J. Komrska. "Crystal shape amplitudes of platonic polyhedra. II. The regular pentagonal dodecahedron and the icosahedron." Physica Status Solidi (a) 150, no. 1 (July 16, 1995): 113–26. http://dx.doi.org/10.1002/pssa.2211500110.

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40

Duan, Jinwei, Lin Cui, and Ying Wang. "Rational design of DNA platonic polyhedra with the minimal components number from topological perspective." Biochemical and Biophysical Research Communications 523, no. 3 (March 2020): 627–31. http://dx.doi.org/10.1016/j.bbrc.2019.12.113.

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41

Williams, W., A. R. Muxworthy, and M. E. Evans. "A micromagnetic investigation of magnetite grains in the form of Platonic polyhedra with surface roughness." Geochemistry, Geophysics, Geosystems 12, no. 10 (October 2011): n/a. http://dx.doi.org/10.1029/2011gc003560.

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42

Al-Siyabi, Abeer, Nazife Ozdes Koca, and Mehmet Koca. "Icosahedral Polyhedra from D6 Lattice and Danzer’s ABCK Tiling." Symmetry 12, no. 12 (November 30, 2020): 1983. http://dx.doi.org/10.3390/sym12121983.

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It is well known that the point group of the root lattice D6 admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group H3, its roots, and weights are determined in terms of those of D6. Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by projections of the sets of lattice vectors of D6 determined by a pair of integers (m1, m2) in most cases, either both even or both odd. Vertices of the Danzer’s ABCK tetrahedra are determined as the fundamental weights of H3, and it is shown that the inflation of the tiles can be obtained as projections of the lattice vectors characterized by the pair of integers, which are linear combinations of the integers (m1, m2) with coefficients from the Fibonacci sequence. Tiling procedure both for the ABCK tetrahedral and the <ABCK> octahedral tilings in 3D space with icosahedral symmetry H3, and those related transformations in 6D space with D6 symmetry are specified by determining the rotations and translations in 3D and the corresponding group elements in D6. The tetrahedron K constitutes the fundamental region of the icosahedral group and generates the rhombic triacontahedron upon the group action. Properties of “K-polyhedron”, “B-polyhedron”, and “C-polyhedron” generated by the icosahedral group have been discussed.

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43

Bruce King, R. "Platonic tessellations of Riemann surfaces as models in chemistry: non-zero genus analogues of regular polyhedra." Journal of Molecular Structure 656, no. 1-3 (August 2003): 119–33. http://dx.doi.org/10.1016/s0022-2860(03)00335-1.

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44

Fusco, Giorgio, and Giovanni F. Gronchi. "Platonic Polyhedra, Periodic Orbits and Chaotic Motions in the $$N$$ N -body Problem with Non-Newtonian Forces." Journal of Dynamics and Differential Equations 26, no. 4 (December 2014): 817–41. http://dx.doi.org/10.1007/s10884-014-9401-2.

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45

de La Vaissière, B., P. W. Fowler, and M. Deza. "Codes in Platonic, Archimedean, Catalan, and Related Polyhedra: A Model for Maximum Addition Patterns in Chemical Cages." Journal of Chemical Information and Computer Sciences 41, no. 2 (March 2001): 376–86. http://dx.doi.org/10.1021/ci000129s.

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46

Wester, Ture. "3-D Form And Force Language Proposal For A Structural Basis." International Journal of Space Structures 11, no. 1-2 (April 1996): 221–31. http://dx.doi.org/10.1177/026635119601-227.

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The usually declared archetype for structural behaviour is the lattice (bar-and-node) structure. It is true that structural manifestations in general - with great clarity - may be reduced to the two interconnected structural nuclei, the pin-jointed bar and the hinged node. The relation to the geometrical archetypes, the five regular Platonic polyhedra, is obviously unsatisfactory, as only the three triangulated polyhedra have particular structural qualities in being kinematically stable. The two remaining, the cube and the dodecahedron, are unclear and usually considered as structurally inferior or incomplete. This “Fullerian cosmology” – as it may be called – has dominated basic structural understanding for many years. The structural dual to the lattice type, the plate structure, which fits perfectly with the geometrical plane-to-point duality, not only at the level of topology and kinematic stability, but also at the level of metric geometry, magnitude of forces and elastic properties, is rarely taken into consideration. This structural duality, which was discovered around 20 years ago, has mostly been used to describe the particular qualities of the plate structure and not for its unifying qualities with the lattice structure, creating an entirety at the archetypical level of 3-D structures. This quality forms the basis for some simple unique correlations betwen geometry and structural mechanics, but it also implies that the usual basic hierarchy of 3-D geometry has to be altered accordingly. The paper will discuss some of the possibilities of creating a Form-and-Force Language on this basis.

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47

Komrska, J., and W. Neumann. "Crystal shape amplitudes of platonic polyhedra. I. General aspects and the shape amplitudes of the tetrahedron, cube, and octahedron." Physica Status Solidi (a) 150, no. 1 (July 16, 1995): 89–111. http://dx.doi.org/10.1002/pssa.2211500109.

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48

de La Vaissiere, B., P. W. Fowler, and M. Deza. "ChemInform Abstract: Codes in Platonic, Archimedean, Catalan, and Related Polyhedra: A Model for Maximum Addition Patterns in Chemical Cages." ChemInform 32, no. 22 (May 26, 2010): no. http://dx.doi.org/10.1002/chin.200122185.

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49

Kriechbaum, Manfred. "Analytical calculation of the radius of gyration of regular shapes and polyhedra." Acta Crystallographica Section A Foundations and Advances 70, a1 (August 5, 2014): C415. http://dx.doi.org/10.1107/s2053273314095849.

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The radius of gyration (Rg) is one of the most common parameters to be extracted from small-angle X-ray/neutron scattering (SAXS, SANS) measurements of nanoparticles and combines information about size, shape, symmetry and homogeneity in one single value. The analytical expressions for Rg are well known for simple geometric shapes (spheres, ellipsoids, cylinders, cubes). In this work, the analytical equations for Rg for other homogeneous (constant electron or scattering length density) shapes like cones, pyramids, paraboloids, hemispheres or tori are derived and are compiled in this poster. In this approach, the Rg of different 3-dimensional objects can be composed of a 2-dimensional cross-sectional (Rc) and of a perpendicular (h) contribution. Thus, Rg2is the linear sum of both: Rg2= f1*Rc2+ f2*h2, with h being the height or diameter of the object in the perpendicular direction to the cross-section and f1 and f2 being multiplicative factors with values depending on the geometric shape. The cross-sectional area can be (semi-)circular, (semi-)elliptic, n-polygonal or rhombic, resulting in a conical, pyramidal, ellipsoidal or paraboloidal 3D-shape, depending on the perpendicular component. A mirror-symmetry in the cross-sectional plane may be present (e.g. ellipsoids, bi-cones or bi-pyramids) or absent (e.g. hemispheres or single cones or pyramids). General equations of Rc for regular (equilateral) n-polygons will be given, but also for non-equilateral polygonal (rectangular, triangular) and rhombic cross-sections. Furthermore, the analytical equations of Rg of nanoscaled particles of high symmetry, in particular of convex polyhedra like the 5 Platonic solids (tetra-, hexa-, octa-, dodeca- and icosa-hedron) or the 13 Archimedean solids and their duals (Catalan solids) are presented, for the solid, for the hollow (faces only) and as well as for the skeletal (edges only) and dot (vertices only) shape.

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50

Klenk, Simon, Wolfgang Frey, Martina Bubrin та Sabine Laschat. "Tetra-μ3-iodido-tetrakis[(tri-n-butylphosphane-κP)copper(I)]". Acta Crystallographica Section E Structure Reports Online 70, № 4 (5 березня 2014): m117—m118. http://dx.doi.org/10.1107/s1600536814003390.

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The title complex, [Cu4I4(C12H27P)4], crystallizes with six molecules in the unit cell and with three independent one-third molecule fragments, completed by application of the relevant symmetry operators, in the asymmetric unit. The tetranuclear copper core shows a tetrahedral geometry (site symmetry 3..). The I atoms also form a tetrahedron, with I...I distances of 4.471 (1) Å. Both tetrahedra show an orientation similar to that of a pair of self-dual platonic bodies. The edges of the I-tetrahedral structure are capped to the face centers of the Cu-tetrahedron andvice versa. The Cuface...I distances are 2.18 Å (averaged) and the Iface...Cu distances are 0.78 Å (averaged). As a geometric consequence of these properties there are eight distorted trigonal–bipyramidal polyhedra evident, wherein each trigonal face builds up the equatorial site and the opposite Cu...I positions form the axial site. As expected, then-butyl moieties are highly flexible, resulting in large elongations of their anisotropic displacement parameters. Some C atoms of then-butyl groups were needed to fix alternative discrete disordered positions.

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