Groups, Platonic solids and Bell inequalities

Quantum ◽

10.22331/q-2021-11-29-593 ◽

2021 ◽

Vol 5 ◽

pp. 593

Author(s):

Katarzyna Bolonek-Lasoń ◽

Piotr Kosiński

Keyword(s):

Finite Groups ◽

Bell Inequalities ◽

Platonic Solids ◽

Archimedean Solids

The construction of Bell inequalities based on Platonic and Archimedean solids (Quantum 4 (2020), 293) is generalized to the case of orbits generated by the action of some finite groups. A number of examples with considerable violation of Bell inequalities is presented.

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Pyramids, Prisms, Antiprisms, and Deltahedra

Mathematics Teacher ◽

10.5951/mt.81.4.0261 ◽

1988 ◽

Vol 81 (4) ◽

pp. 261-265

Author(s):

Donovan R. Lichtenberg

Keyword(s):

Mathematics Teacher ◽

Platonic Solids ◽

Regular Polyhedra ◽

Archimedean Solids

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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Golden Section and Golden Rectangles When Building Icosahedron, Dodecahedron and Archimedean Solids Based On Them

Geometry & Graphics ◽

10.12737/article_5d2c1ceb9f91b1.21353054 ◽

2019 ◽

Vol 7 (2) ◽

pp. 47-55 ◽

Cited By ~ 1

Author(s):

В. Васильева ◽

V. Vasil'eva

Keyword(s):

Computer Modeling ◽

Golden Section ◽

Platonic Solids ◽

Divine Proportion ◽

Section Ratio ◽

Archimedean Solids ◽

Regular Dodecahedron ◽

History Of ◽

Dual Connection ◽

Regular Icosahedron

A brief history of the development of the regular polyhedrons theory is given. The work introduces the reader to modelling of the two most complex regular polyhedrons – Platonic solids: icosahedron and dodecahedron, in AutoCAD package. It is suggested to apply the method of the icosahedron and dodecahedron building using rectangles with their sides’ ratio like in the golden section, having taken the icosahedron’s golden rectangles as a basis. This method is well-known-of and is used for icosahedron, but is extremely rarely applied to dodecahedron, as in the available literature it is suggested to build the latter one as a figure dual to icosahedron. The work provides information on the first mentioning of this building method by an Italian mathematician L. Pacioli in his Divine Proportion book. In 1937, a Soviet mathematician D.I. Perepelkin published a paper On One Building Case of the Regular Icosahedron and Regular Dodecahedron, where he noted that this “method is not very well known of” and provided a building based “solely on dividing an intercept in the golden section ratio”. Taking into account the simplicity and good visualization of the building based on golden rectangles, a computer modeling of icosahedron and dodecahedron inscribed in a regular hexahedron is performed in the article. Given that, if we think in terms of the golden section concepts, the bigger side of the rectangle equals a whole intercept – side of the regular hexahedron, and the smaller sides of the icosahedron and dodecahedron rectangles are calculated as parts of the golden section ratio (of the bigger part and the smaller one, respectively). It is demonstrated how, using the scheme of a wireframe image of the dual connection of these polyhedrons as a basis, to calculate the sides of the rectangles in the golden section ratio in order to build an “infinite” cascade of these dual figures, as well as to build the icosahedron and dodecahedron circumscribed about the regular hexahedron. The method based on using the golden-section rectangles is also applied to building semiregular polyhedrons – Archimedean solids: a truncated icosahedron, truncated dodecahedron, icosidodecahedron, rhombicosidodecahedron, and rhombitruncated icosidodecahedron, which are based on icosahedron and dodecahedron.

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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 ◽

10.1098/rspa.2017.0267 ◽

2017 ◽

Vol 473 (2206) ◽

pp. 20170267 ◽

Cited By ~ 3

Author(s):

Gunnar Brinkmann ◽

Pieter Goetschalckx ◽

Stan Schein

Keyword(s):

Symmetry Group ◽

Local Symmetry ◽

Icosahedral Symmetry ◽

Platonic Solids ◽

Global Properties ◽

Archimedean Solids

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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The Platonic solids and fundamental tests of quantum mechanics

Quantum ◽

10.22331/q-2020-07-09-293 ◽

2020 ◽

Vol 4 ◽

pp. 293 ◽

Cited By ~ 2

Author(s):

Armin Tavakoli ◽

Nicolas Gisin

Keyword(s):

Quantum Mechanics ◽

Natural Philosophy ◽

Bell Inequality ◽

Bell Inequalities ◽

Convex Polyhedra ◽

Classical Antiquity ◽

Platonic Solids ◽

Mathematical Beauty ◽

Regular Convex ◽

And Mathematics

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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Uniform polyhedra

Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences ◽

10.1098/rsta.1954.0003 ◽

1954 ◽

Vol 246 (916) ◽

pp. 401-450 ◽

Cited By ~ 67

Keyword(s):

Spherical Triangle ◽

Platonic Solids ◽

The Third ◽

Archimedean Solids ◽

The Future ◽

Uniform Polyhedron

Uniform polyhedra have regular faces meeting in the same manner at every vertex. Besides the five Platonic solids, the thirteen Archimedean solids, the four regular star-polyhedra of Kepler (1619) and Poinsot (1810), and the infinite families of prisms and antiprisms, there are at least fifty-three others, forty-one of which were discovered by Badoureau (1881) and Pitsch (1881). The remaining twelve were discovered by two of the present authors (H.S.M.C. and J.C.P.M.) between 1930 and 1932, but publication was postponed in the hope of obtaining a proof that there are no more. Independently, between 1942 and 1944, the third author (M.S.L.-H.) in collaboration with H. C. Longuet-Higgins, rediscovered eleven of the twelve. We now believe that further delay is pointless; we have temporarily abandoned our hope of obtaining a proof that our enumeration is complete, but we shall be much surprised if any new uniform polyhedron is found in the future. We have classified the known figures with the aid of a systematic notation and we publish drawings (by J.C.P.M.) and photographs of models (by M .S.L.-H.) which include all those not previously constructed. One remarkable new polyhedron is contained in the present list, having eight edges at a vertex. This is the only one which cannot be derived immediately from a spherical triangle by Wythoff's construction.

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