Symmetry groups of regular polyhedra over finite fields

1998 ◽  
pp. 43-53
Author(s):  
David Erschler
Keyword(s):  
Finite Fields ◽  
Symmetry Groups ◽  
Regular Polyhedra

scholarly journals Quantum correlations of colloidal particles as the root cause of topological features of the microcosm of oxyhydrate systems and their properties

2019 ◽  
Vol 57 (3) ◽  
pp. 1-32
Author(s):  
Yury I. Sukharev ◽  
Keyword(s):  
Colloidal Particles ◽  
Dimensional Space ◽  
Magic Numbers ◽  
Symmetry Groups ◽  
Weak Current ◽  
The Past ◽  
Regular Polyhedra ◽  
Topological Features ◽  
The Future ◽  
Crystallographic Structures

A fundamentally new method for calculating the crystallographic structures of oxyhydrate gels that transform over time is proposed. Simple features of caustics and wave fronts form two infinite series and and three exceptional features (Symmetry groups are groups of regular polyhedra in three – dimensional space, and exceptional features are symmetry groups of tetrahedron, octahedron, and icosahedron.) This follows from the Coxeter-Dynkin diagrams. The principles of the program "COXETER" for the implementation of the method of calculation of oxyhydrate crystallographic structures. Gel electric moments of higher even orders are the higher degrees of Laplace operators: sixth, eighth, tenth and so on orders. In this case, a General ratio is obtained, where new amplitudes and phases are obtained by adding oscillations with different phases, recorded formulas:, which are characterized by the values of currents depending on the spatial periodic structures, that is, an experimental connection between the current and the concentration of the nanoclusters of the system is obtained . From the analysis of the experimental data it follows that the main part of the time oscillations is determined by the light elements of the clusters, the quadrupoles of which give a relatively weak current surge with a small amplitude. Along with them there are elements of the third order, the fourth, fifth and possibly sixth. The nanoclusters are formed by the rule of "magic numbers" found experimentally in the hydroxide application in complex tin, yttrium, iron and other. The rules of "magic numbers", or the theory of magic numbers is widely used to describe nano-structures quite complicated. The role of Paterns is at the heart of everything, that is, at the heart of the structure of Reality. Reality is not collected in parts from the particles of matter in the course of evolution from the past to the future, and is all at once from the past to the future for a given pattern, that is, for specific PATTERNS, as defined by quantum theory. Irregularities and partial randomization of the structure of the core grids form gel defects, to which small mobile clusters are attracted by electrostatic or electromagnetic forces, which are then adsorbed and located on the "defects" in accordance with their dipole moments. This circumstance is determined by the values of"magic numbers". Cluster magic structures have a layered structure. The interior region such fulleroids be, for example, medium-structured clusters and their multipoles and octupole. If at the same time multipoles of the same name, it inevitably leads to fluctuations of cluster flows. The areas inside the fulleroid will not be filled completely; they always remain empty. This will also lead to permanent oscillations of the cluster “filling” and is related to the filling of the fluctuations of the cluster environment. Knowing the padding settings fulleroid (fluctuations) can be identified, what are its physico-chemical characteristics. Linear ordered REALITY, or in other terminology, historical oxyhydrate sequence – is the result of experimental wave interference (table 4) sums of historical (time) epochs-waves of equations Moreover, the Function satisfies the condition of the Wheeler-DeWitt equation, which allows to describe the birth of a 4-dimensional space-time gel oxyhydrate phase.

Ideal Uniform Polyhedra in and Covolumes of Higher Dimensional Modular Groups

2020 ◽  
pp. 1-28
Author(s):  
Ruth Kellerhals
Keyword(s):  
Modular Group ◽  
Independent Interest ◽  
Symmetry Groups ◽  
Reflection Groups ◽  
Regular Polyhedra ◽  
Higher Dimensional ◽  
Modular Groups ◽  
Riemann Zeta

Abstract Higher dimensional analogues of the modular group $\mathit{PSL}(2,\mathbb{Z})$ are closely related to hyperbolic reflection groups and Coxeter polyhedra with big symmetry groups. In this context, we develop a theory and dissection properties of ideal hyperbolic $k$ -rectified regular polyhedra, which is of independent interest. As an application, we can identify the covolumes of the quaternionic modular groups with certain explicit rational multiples of the Riemann zeta value $\unicode[STIX]{x1D701}(3)$ .

scholarly journals Aesthetic Patterns with symmetries of the Regular Polyhedron

2017 ◽  
Author(s):  
Peichang Ouyang ◽  
Liying Wang ◽  
Tao Yu ◽  
Xuan Huang
Keyword(s):  
Unit Sphere ◽  
Fast Algorithm ◽  
Dimensional Space ◽  
Regular Polyhedron ◽  
Fundamental Region ◽  
Symmetry Groups ◽  
Regular Polytopes ◽  
Order Restriction ◽  
Regular Polyhedra ◽  
Flexible Form

A fast algorithm is established to transform points of the unit sphere into fundamental region symmetrically. With the resulting algorithm, a flexible form of invariant mappings is achieved to generate aesthetic patterns with symmetries of the regular polyhedra. This method avoids the order restriction of symmetry groups, which can be similarly extended to treat regular polytopes in n-dimensional space for n>=4.

Finite Fields

1996 ◽  
Author(s):  
Rudolf Lidl ◽  
Harald Niederreiter
Keyword(s):  
Finite Fields

Maximum Distance Separable Hankel Rhotrices over Finite Fields

2018 ◽  
Vol 43 (1-4) ◽  
pp. 13-45
Author(s):  
Prof. P. L. Sharma ◽  
◽  
Mr. Arun Kumar ◽  
Mrs. Shalini Gupta ◽  
◽  
...  
Keyword(s):  
Finite Fields ◽  
Maximum Distance ◽  
Maximum Distance Separable

Notes on three-pass protocol

2020 ◽  
Vol 25 (4) ◽  
pp. 4-9
Author(s):  
Yerzhan R. Baissalov ◽  
Ulan Dauyl
Keyword(s):  
Finite Fields ◽  
Matrix Algebras ◽  
Associative Structures

The article discusses primitive, linear three-pass protocols, as well as three-pass protocols on associative structures. The linear three-pass protocols over finite fields and the three-pass protocols based on matrix algebras are shown to be cryptographically weak.

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