Symmetry Groups Associated with Tilings of a Flat Torus

2014 ◽  
Vol 70 (a1) ◽  
pp. C1428-C1428
Author(s):  
Mark Loyola ◽  
Ma. Louise Antonette De Las Peñas ◽  
Grace Estrada ◽  
Eko Santoso
Keyword(s):  
Symmetry Group ◽  
Full Rank ◽  
Flat Torus ◽  
Space Forms ◽  
Color Symmetry ◽  
Abstract Polytopes ◽  
Symmetry Properties ◽  
Transformation Geometry ◽  
Groups Of Isometries ◽  
Injective Maps

A flat torus E^2/Λ is the quotient of the Euclidean plane E^2 with a full rank lattice Λ generated by two linearly independent vectors v_1 and v_2. A motif-transitive tiling T of the plane whose symmetry group G contains translations with vectors v_1 and v_2 induces a tiling T^* of the flat torus. Using a sequence of injective maps, we realize T^* as a tiling T-of a round torus (the surface of a doughnut) in the Euclidean space E^3. This realization is obtained by embedding T^* into the Clifford torus S^1 × S^1 ⊆ E^4 and then stereographically projecting its image to E^3. We then associate two groups of isometries with the tiling T^* – the symmetry group G^* of T^* itself and the symmetry group G-of its Euclidean realization T-. This work provides a method to compute for G^* and G-using results from the theory of space forms, abstract polytopes, and transformation geometry. Furthermore, we present results on the color symmetry properties of the toroidal tiling T^* in relation with the color symmetry properties of the planar tiling T. As an application, we construct toroidal polyhedra from T-and use these geometric structures to model carbon nanotori and their structural analogs.

Symmetry groups associated with tilings on a flat torus

2015 ◽  
Vol 71 (1) ◽  
pp. 99-110 ◽  
Author(s):  
Mark L. Loyola ◽  
Ma. Louise Antonette N. De Las Peñas ◽  
Grace M. Estrada ◽  
Eko Budi Santoso
Keyword(s):  
Symmetry Group ◽  
Geometric Modeling ◽  
Three Dimensional ◽  
Symmetry Groups ◽  
Flat Torus ◽  
Color Symmetry ◽  
Symmetry Properties

This work investigates symmetry and color symmetry properties of Kepler, Heesch and Laves tilings embedded on a flat torus and their geometric realizations as tilings on a round torus in Euclidean 3-space. The symmetry group of the tiling on the round torus is determined by analyzing relevant symmetries of the planar tiling that are transformed to axial symmetries of the three-dimensional tiling. The focus on studying tilings on a round torus is motivated by applications in the geometric modeling of nanotori and the determination of their symmetry groups.

scholarly journals Designing Color Symmetry in Stigmergic Art

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1882
Author(s):  
Hendrik Richter
Keyword(s):  
Symmetry Group ◽  
Design Procedure ◽  
Design Principles ◽  
Generative Art ◽  
Color Symmetry ◽  
Symmetry Properties ◽  
Visual Results

Color symmetry is an extension of the symmetry imposed by isometric transformations and indicates that the colors of geometrical objects are assigned according to the symmetry properties of these objects. A color symmetry permutes the coloring of the objects consistently with their symmetry group. We apply this concept to bio-inspired generative art. Therefore, the geometrical objects are interpreted as motifs that may repeat themselves with a symmetry-consistent coloring. The motifs are obtained by design principles from stigmergy. We discuss the design procedure and present visual results.

Truss and Beam Finite Elements Revisited: A Derivation Based on Displacement-Field Decomposition

1996 ◽  
Vol 11 (4) ◽  
pp. 371-380 ◽  
Author(s):  
Alphose Zingoni
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Symmetry Group ◽  
Displacement Field ◽  
Three Dimensional ◽  
Beam Elements ◽  
Symmetry Properties ◽  
Mass Matrices ◽  
General Displacement

Where a finite element possesses symmetry properties, derivation of fundamental element matrices can be achieved more efficiently by decomposing the general displacement field into subspaces of the symmetry group describing the configuration of the element. In this paper, the procedure is illustrated by reference to the simple truss and beam elements, whose well-known consistent-mass matrices are obtained via the proposed method. However, the procedure is applicable to all one-, two- and three-dimensional finite elements, as long as the shape and node configuration of the element can be described by a specific symmetry group.

Symmetry properties of complex contact space form

2019 ◽  
pp. 2050157 ◽  
Author(s):  
Mohamed Belkhelfa ◽  
Fatima Zohra Kadi
Keyword(s):  
Space Form ◽  
Einstein Manifold ◽  
Contact Manifold ◽  
Sasakian Space Form ◽  
Space Forms ◽  
Sasakian Space Forms ◽  
Symmetry Properties ◽  
Symmetric Complex ◽  
Complex Contact Manifold ◽  
Contact Space

It is well known that a Sasakian space form is pseudo-symmetric [M. Belkhelfa, R. Deszcz and L. Verstraelen, Symmetry properties of Sasakian space-forms, Soochow J. Math. 31(4) (2005) 611–616], therefore it is Ricci-pseudo-symmetric. In this paper, we proved that a normal complex contact manifold is Ricci-semi-symmetric if and only if it is an Einstein manifold; moreover, we showed that a complex contact space form [Formula: see text] with constant [Formula: see text]-sectional curvature [Formula: see text] is properly Ricci-pseudo-symmetric [Formula: see text] if and only if [Formula: see text]; in this case [Formula: see text]. We gave an example of properly Ricci-pseudo-symmetric complex contact space form. On the other hand, we proved the non-existence of proper pseudo-symmetric ([Formula: see text]) complex contact space form [Formula: see text]

Colorings of hyperbolic plane crystallographic patterns

2006 ◽  
Vol 221 (10) ◽  
Author(s):  
Ma. Louise Antonette N. De Las Peñas ◽  
Rene P. Felix ◽  
Glenn R. Laigo
Keyword(s):  
Symmetry Group ◽  
Hyperbolic Plane ◽  
Systematic Approach ◽  
Basic Problem ◽  
Color Symmetry ◽  
Regular Tiling

AbstractIn color symmetry the basic problem has always been to classify symmetrically colored symmetrical patterns [13]. An important step in the study of color symmetry in the hyperbolic plane is the determination of a systematic approach in arriving at colored symmetrical hyperbolic patterns. For a given uncolored semi-regular tiling with symmetry group

Local and global color symmetries of a symmetrical pattern

2019 ◽  
Vol 75 (5) ◽  
pp. 730-745
Author(s):  
Agatha Kristel Abila ◽  
Ma. Louise Antonette De Las Peñas ◽  
Eduard Taganap
Keyword(s):  
Symmetry Group ◽  
Local Symmetry ◽  
Global Symmetry ◽  
Color Symmetry ◽  
Usual Approach ◽  
Symmetrical Pattern ◽  
Symmetry Theory ◽  
Global Symmetry Group ◽  
Local Symmetries ◽  
Global And Local

This study addresses the problem of arriving at transitive perfect colorings of a symmetrical pattern {\cal P} consisting of disjoint congruent symmetric motifs. The pattern {\cal P} has local symmetries that are not necessarily contained in its global symmetry group G. The usual approach in color symmetry theory is to arrive at perfect colorings of {\cal P} ignoring local symmetries and considering only elements of G. A framework is presented to systematically arrive at what Roth [Geom. Dedicata (1984), 17, 99–108] defined as a coordinated coloring of {\cal P}, a coloring that is perfect and transitive under G, satisfying the condition that the coloring of a given motif is also perfect and transitive under its symmetry group. Moreover, in the coloring of {\cal P}, the symmetry of {\cal P} that is both a global and local symmetry, effects the same permutation of the colors used to color {\cal P} and the corresponding motif, respectively.

Point groups and color symmetry

1975 ◽  
Vol 142 (1-2) ◽  
pp. 1-23 ◽  
Author(s):  
Marjorie Senechal
Keyword(s):  
Symmetry Group ◽  
Color Group ◽  
Color Symmetry ◽  
Point Groups

AbstractA theory of polycolor symmetry is derived from elementary considerations. It is shown that each color group is characterized by a symmetry group

APPLYING SYMMETRY PROPERTIES OF FINITE SYSTEM TO ITS FIELD THEORY DESCRIPTION

2004 ◽  
Vol 18 (26) ◽  
pp. 3443-3450
Author(s):  
B. A. SEREDYUK ◽  
K. D. TOVSTYUK ◽  
N. K. TOVSTYUK
Keyword(s):  
Symmetry Group ◽  
Particle System ◽  
Finite System ◽  
Wave Functions ◽  
Point Symmetry ◽  
Relaxation Processes ◽  
Point Symmetry Group ◽  
Field Techniques ◽  
Self Energy ◽  
Symmetry Properties

The self-energy is received using the theory group and field techniques, based on the wave functions of the classes of point symmetry group and accounting for the two-particle system. It contains components responsible not only for relaxation processes but also for the oscillating ones, caused by a different degree of occupation of the class of point symmetry group by particles of the structure. Analyzing the character of energy oscillations depending on the degree of the class occupation, the mechanisms of diffusion, catalysis, chemical reactions and Le-Shatelje principle are described.

Universal Alternating Semiregular Polytopes

2020 ◽  
pp. 1-24
Author(s):  
B. Monson ◽  
Egon Schulte
Keyword(s):  
Automorphism Group ◽  
Symmetry Group ◽  
Local Structure ◽  
Convex Polytope ◽  
Main Concern ◽  
Abstract Polytopes ◽  
Classical Setting

Abstract In the classical setting, a convex polytope is said to be semiregular if its facets are regular and its symmetry group is transitive on vertices. This paper continues our study of alternating semiregular abstract polytopes, which have abstract regular facets, still with combinatorial automorphism group transitive on vertices and with two kinds of regular facets occurring in an alternating fashion. Our main concern here is the universal polytope ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$ , an alternating semiregular $(n+1)$ -polytope defined for any pair of regular $n$ -polytopes ${\mathcal{P}},{\mathcal{Q}}$ with isomorphic facets. After a careful look at the local structure of these objects, we develop the combinatorial machinery needed to explain how ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$ can be constructed by “freely assembling” unlimited copies of  ${\mathcal{P}}$ , ${\mathcal{Q}}$ along their facets in alternating fashion. We then examine the connection group of ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$ , and from that prove that ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$ covers any $(n+1)$ -polytope ${\mathcal{B}}$ whose facets alternate in any way between various quotients of ${\mathcal{P}}$ or  ${\mathcal{Q}}$ .

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