The five regular convex polyhedra and their duals

Dual Models ◽  
1983 ◽  
pp. 7-13
Keyword(s):  
Convex Polyhedra ◽  
Regular Convex

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.

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.

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.

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 Data-based polyhedron model for optimization of engineering structures involving uncertainties

2021 ◽  
Vol 2 ◽  
Author(s):  
Zhiping Qiu ◽  
Han Wu ◽  
Isaac Elishakoff ◽  
Dongliang Liu
Keyword(s):  
Linear Programming ◽  
Convex Polyhedron ◽  
Linear Optimization ◽  
Optimal Solution ◽  
Composite Plates ◽  
Convex Polyhedra ◽  
Chebyshev Inequality ◽  
Engineering Structures ◽  
Load Bearing ◽  
Polyhedron Model

Abstract This paper studies the data-based polyhedron model and its application in uncertain linear optimization of engineering structures, especially in the absence of information either on probabilistic properties or about membership functions in the fussy sets-based approach, in which situation it is more appropriate to quantify the uncertainties by convex polyhedra. Firstly, we introduce the uncertainty quantification method of the convex polyhedron approach and the model modification method by Chebyshev inequality. Secondly, the characteristics of the optimal solution of convex polyhedron linear programming are investigated. Then the vertex solution of convex polyhedron linear programming is presented and proven. Next, the application of convex polyhedron linear programming in the static load-bearing capacity problem is introduced. Finally, the effectiveness of the vertex solution is verified by an example of the plane truss bearing problem, and the efficiency is verified by a load-bearing problem of stiffened composite plates.

Convex polyhedra in hyperbolic space and interpolation of functions

2011 ◽  
Vol 84 (3) ◽  
pp. 850-853
Author(s):  
O. P. Gladunova ◽  
E. D. Rodionov ◽  
V. V. Slavskii
Keyword(s):  
Hyperbolic Space ◽  
Convex Polyhedra ◽  
Interpolation Of Functions
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