scholarly journals Toric Geometry of the Regular Convex Polyhedra

2017 ◽  
Vol 2017 ◽  
pp. 1-15
Author(s):  
Fiammetta Battaglia ◽  
Elisa Prato
Keyword(s):  
Convex Polyhedra ◽  
Toric Geometry ◽  
Regular Tetrahedron ◽  
Regular Octahedron ◽  
Regular Dodecahedron ◽  
Regular Convex ◽  
Regular Icosahedron

We describe symplectic and complex toric spaces associated with the five regular convex polyhedra. The regular tetrahedron and the cube are rational and simple, the regular octahedron is not simple, the regular dodecahedron is not rational, and the regular icosahedron is neither simple nor rational. We remark that the last two cases cannot be treated via standard toric geometry.

scholarly journals Number and twistedness of strands in weavings on regular convex polyhedra

2014 ◽  
Vol 470 (2162) ◽  
pp. 20130608 ◽  
Author(s):  
F. Kovács
Keyword(s):  
Graph Theory ◽  
Structural Engineering ◽  
Closed Surface ◽  
Convex Polyhedra ◽  
Recursive Method ◽  
Simple Method ◽  
Linking Number ◽  
Regular Convex ◽  
Weaving Pattern ◽  
Symmetry Operations

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.

scholarly journals The correlation functions of the regular tetrahedron and octahedron

2014 ◽  
Vol 47 (4) ◽  
pp. 1445-1448 ◽  
Author(s):  
Salvino Ciccariello
Keyword(s):  
Correlation Functions ◽  
Rational Functions ◽  
Regular Tetrahedron ◽  
Numerical Determination ◽  
Algebraic Form ◽  
Autocorrelation Functions ◽  
Regular Octahedron ◽  
Size Dispersion ◽  
Arctangent Function

The expressions of the autocorrelation functions (CFs) of the regular tetrahedron and the regular octahedron are reported. They have an algebraic form that involves the arctangent function and rational functions of r and (a + br 2)1/2, a and b being appropriate integers and r a distance. The CF expressions make the numerical determination of the corresponding scattering intensities fast and accurate even in the presence of a size dispersion.

scholarly journals Symplectic Toric Geometry and the Regular Dodecahedron

2015 ◽  
Vol 2015 ◽  
pp. 1-5 ◽  
Author(s):  
Elisa Prato
Keyword(s):  
Toric Geometry ◽  
Toric Manifold ◽  
Platonic Solids ◽  
Simple Polytope ◽  
Singular Space ◽  
Regular Dodecahedron

The regular dodecahedron is the only simple polytope among the platonic solids which is not rational. Therefore, it corresponds neither to a symplectic toric manifold nor to a symplectic toric orbifold. In this paper, we associate to the regular dodecahedron a highly singular space called symplectic toric quasifold.

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

2019 ◽  
Vol 7 (2) ◽  
pp. 47-55 ◽  
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.

scholarly journals An Affine Regular Icosahedron Inscribed in an Affine Regular Octahedron in a GS-Quasigroup

KoG ◽  
2017 ◽  
pp. 3-5
Author(s):  
Zdenka Kolar-Begović
Keyword(s):  
Golden Section ◽  
Regular Octahedron ◽  
Geometric Concepts ◽  
Regular Icosahedron ◽  
The Given ◽  
Quasigroup Operation

A golden section quasigroup or shortly a GS-quasigroup is an idempotent quasigroup which satises the identities a\dot (ab \dot c) \dot c = b; a\dot (a \dot bc) \dot c = b. The concept of a GS-quasigroup was introduced by VOLENEC. A number of geometric concepts can be introduced in a general GS-quasigroup by means of the binary quasigroup operation. In this paper, it is proved that for any affine regular octahedron there is an affine regular icosahedron which is inscribed in the given affine regular octahedron. This is proved by means of the identities and relations which are valid in a general GS-quasigrup. The geometrical presentation in the GS-quasigroup C(\frac{1}{2} (1 +\sqrt{5})) suggests how a geometrical consequence may be derived from the statements proven in a purely algebraic manner.

Wright-Fisher Geometry

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Normal Form ◽  
Euclidean Space ◽  
Convex Polyhedron ◽  
Convex Polyhedra ◽  
Linearly Independent ◽  
Regular Convex ◽  
Normal Vectors ◽  
Manifolds With Corners ◽  
Natural Domains ◽  
Manifold With Corners

This chapter introduces the geometric preliminaries needed to analyze generalized Kimura diffusions, with particular emphasis on Wright–Fisher geometry. It begins with a discussion of the natural domains of definition for generalized Kimura diffusions: polyhedra in Euclidean space or, more generally, abstract manifolds with corners. Amongst the convex polyhedra, the chapter distinguishes the subclass of regular convex polyhedra P. P is a regular convex polyhedron if it is convex and if near any corner, P is the intersection of no more than N half-spaces with corresponding normal vectors that are linearly independent. These definitions establish that any regular convex polyhedron is a manifold with corners. The chapter concludes by defining the general class of elliptic Kimura operators on a manifold with corners P and shows that there is a local normal form for any operator L in this class.

Notes on the Regular Icosahedron and the Regular Dodecahedron

1942 ◽  
Vol 26 (270) ◽  
pp. 153
Author(s):  
W. Hope-Jones ◽  
Lester S. Hill
Keyword(s):  
Regular Dodecahedron ◽  
Regular Icosahedron

Dodecahedral slices and polyhedral pieces

2010 ◽  
Vol 94 (529) ◽  
pp. 5-17
Author(s):  
Doug French ◽  
David Jordan
Keyword(s):  
Icosahedral Symmetry ◽  
Surprising Result ◽  
Regular Pentagon ◽  
Regular Dodecahedron ◽  
Regular Icosahedron ◽  
Image Position

Figure 1 shows how a regular dodecahedron can be dissected into three slices by two planes through the two sets of vertices, each set defining a regular pentagon parallel to the top and bottom faces. A surprising result emerges if we calculate the ratio of the volumes of the three slices. We first prove this result directly and then show it by a dissection argument using simple polyhedral pieces of five types. These pieces can be used to build many polyhedra, including the regular dodecahedron, the regular icosahedron, the great dodecahedron, the small and great stellated dodecahedra and all the Archimedean polyhedra which have icosahedral symmetry.

scholarly journals The Platonic solids and fundamental tests of quantum mechanics

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 293 ◽  
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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