The symmetry groups of the regular tessellations of S 2 and S 3

1982 ◽  
Vol 383 (1785) ◽  
pp. 379-398 ◽  
Keyword(s):  
Irreducible Representations ◽  
Symmetry Groups ◽  
Character Tables ◽  
Point Groups

The construction of the symmetry groups is described for the regular complexes that tessellate S 2 and S 3 . For S 3 these groups are four-dimensional point groups, and they are described in this paper both in terms of their presentations and as subgroups of products of the binary polyhedral groups. The second description is used to obtain the irreducible representations of the symmetry groups; the character tables are also given.

SINIF CEBİRİ YAKLAŞIMI ALTINDA 𝑫𝟐𝒅 ve 𝑪𝟑𝒊 NOKTA GRUPLARI

2021 ◽  
Vol 8 (17) ◽  
pp. 141-149
Author(s):  
Melike DEDE ◽  
Harun AKKUS
Keyword(s):  
Symmetry Groups ◽  
Character Tables ◽  
Secular Equations ◽  
Point Groups ◽  
Cayley Tables

In this study, the point groups 𝐷2𝑑 and 𝐶3𝑖 which belong to tetragonal and trigonal crystal systems, respectively, are handled under the class sum approach. Symmetry groups were formed with symmetry elements that left these point groups unchanged and Cayley tables of related groups were obtained. Using these tables, the conjugates of the elements and the classes of the group were formed. Secular equations are written for each class sum obtained by the sum of the elements that make up the class. By solving these secular equations, the character vectors are obtained. Thus, the character tables were reconstructed with the calculated characters for both point groups under the class sum approach.

Automated Symmetry Exploitation in Engineering Analysis

2004 ◽  
Author(s):  
Krishnan Suresh
Keyword(s):  
Irreducible Representations ◽  
Point Symmetry ◽  
Symmetry Groups ◽  
Engineering Analysis ◽  
Symmetry Detection ◽  
Important Concept ◽  
Improve Accuracy ◽  
Speed Up ◽  
Boundary Mapping ◽  
Cell Construction

It is well known that one can exploit symmetry to speed-up engineering analysis and improve accuracy, at the same time. Not surprisingly, most CAE systems have standard ‘provisions’ for exploiting symmetry. However, these provisions are inadequate in that they needlessly burden the design engineer with time consuming and error-prone tasks of symmetry detection, symmetry cell construction and reformulation. In this paper, we propose and discuss an automated methodology for symmetry exploitation. First, we briefly review the theory of point symmetry groups that symmetry exploitation rests on. We then address symmetry detection and ‘symmetry cell’ construction. We then address an important concept of boundary mapping of symmetry cells, and relate it to the irreducible representations of point symmetry groups. By formalizing these concepts, we show how automated symmetry exploitation can be achieved, and discuss an implementation of the proposed work within the FEMLAB CAE environment.

scholarly journals Emergence of colour symmetry in free-vibration acoustic resonance of a nonlinear hyperelastic material

2013 ◽  
Vol 469 (2159) ◽  
pp. 20130275 ◽  
Author(s):  
Ryuichi Tarumi
Keyword(s):  
Free Vibration ◽  
Ritz Method ◽  
Geometrical Nonlinearity ◽  
Acoustic Resonance ◽  
Finite Amplitude ◽  
Hyperelastic Material ◽  
Irreducible Representations ◽  
Hyperelastic Materials ◽  
Point Groups ◽  
Linearized System

We investigated free-vibration acoustic resonance (FVAR) of two-dimensional St Venant–Kirchhoff-type hyperelastic materials and revealed the existence and structure of colour symmetry embedded therein. The hyperelastic material is isotropic and frame indifferent and includes geometrical nonlinearity in its constitutive equation. The FVAR state is formulated using the principle of stationary action with a subsidiary condition. Numerical analysis based on the Ritz method revealed the existence of four types of nonlinear FVAR modes associated with the irreducible representations of a linearized system. Projection operation revealed that the FVAR modes can be classified on the basis of a single colour (black or white) and three types of bicolour (black and white) magnetic point groups: , , and . These results demonstrate that colour symmetry naturally arises in the finite amplitude nonlinear FVAR modes, and its vibrational symmetries are explained in terms of magnetic point groups rather than the irreducible representations that have been used for linearized systems. We also predicted a grey colour nonlinear FVAR mode which cannot be derived from a linearized system.

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