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

Color groups arising from index-nsubgroups of symmetry groups

2015 ◽  
Vol 71 (2) ◽  
pp. 216-224 ◽  
Author(s):  
René P. Felix ◽  
Allan O. Junio
Keyword(s):  
Symmetry Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Symmetry Groups ◽  
Color Group ◽  
Color Symmetry ◽  
Intermediate Subgroup ◽  
Symmetrical Object ◽  
Necessary And Sufficient

One of the main goals in the study of color symmetry is to classify colorings of symmetrical objects through their color groups. The term color group is taken to mean the subgroup of the symmetry group of the uncolored symmetrical object which induces a permutation of colors in the coloring. This work looks for methods of determining the color group of a colored symmetric object. It begins with an indexnsubgroupHof the symmetry groupGof the uncolored object. It then considersH-invariant colorings of the object, so that the color groupH*will be a subgroup ofGcontainingH. In other words,H≤H*≤G. It proceeds to give necessary and sufficient conditions for the equality ofH*andG. IfH*≠Gandnis prime, thenH*=H. On the other hand, ifH*≠Gandnis not prime, methods are discussed to determine whetherH*isG,Hor some intermediate subgroup betweenHandG.

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

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.

scholarly journals Fundamental Domains for Rhombic Lattices with Dihedral Symmetry of Order 8

10.37236/7802 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Joseph Ray Clarence G. Damasco ◽  
Dirk Frettlöh ◽  
Manuel Joseph C. Loquias
Keyword(s):  
Symmetry Group ◽  
Fundamental Domain ◽  
Point Group ◽  
Symmetry Groups ◽  
Fundamental Domains ◽  
Open Case ◽  
Point Groups ◽  
Index 2 ◽  
Rhombic Lattice ◽  
Planar Lattices

We show by construction that every rhombic lattice $\Gamma$ in $\mathbb{R}^{2}$ has a fundamental domain whose symmetry group contains the point group of $\Gamma$ as a subgroup of index $2$. This solves the last open case of a question raised in a preprint by the authors on fundamental domains for planar lattices whose symmetry groups properly contain the point groups of the lattices.  

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.

Orientations – perfectly colored

2016 ◽  
Vol 49 (5) ◽  
pp. 1786-1802 ◽  
Author(s):  
G. Nolze ◽  
R. Hielscher
Keyword(s):  
Symmetry Group ◽  
Pole Figure ◽  
Crystal Orientation ◽  
Inverse Pole Figure ◽  
Major Goal ◽  
Matlab Toolbox ◽  
Orientation Data ◽  
Crystal Direction ◽  
Higher Symmetries ◽  
Point Groups

The inverse pole figure (IPF) coloring for a suitable evaluation of crystal orientation data is discussed. The major goal is a high correlation between encoding color and crystal orientation. Revised color distributions of the fundamental sectors are introduced which have the advantages of (1) being applicable for all point groups, (2) not causing color discontinuities within grains, (3) featuring carefully balanced regions for red, cyan, blue, magenta, green and yellow, and (4) an enlarged gray center in opposition to a tiny white center. A new set of IPF color keys is proposed which is the result of a thorough analysis of the colorization problem. The discussion considers several topics: (a) the majority of presently applied IPF color keys generate color discontinuities for specifically oriented grains; (b) if a unique correlation between crystal direction and color is requested, discontinuity-preventing keys are possible for all point groups, except for {\overline 4}, {\overline 3} and {\overline 1}; (c) for a specific symmetry group several IPF color keys are available, visualizing different features of a microstructure; and (d) for higher symmetries a simultaneous IPF mapping of two or three standard reference directions is insufficient for an unequivocal orientation assignment. All color keys are available in MTEX, a freely available MATLAB toolbox.

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.

scholarly journals Full Symmetry Groups and Exact Solutions to BKP and GKP Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Bo Ren ◽  
Jian-Yong Wang
Keyword(s):  
Exact Solutions ◽  
Symmetry Group ◽  
Symmetry Groups ◽  
Point Groups ◽  
Full Symmetry ◽  
The Relationship

We investigate the (2+1)-dimensional nonlinear BKP and GKP equations with the modified direct CK’s method. Then, we get its Lie point groups and the full symmetry group, and a relationship is constructed between the new solutions and the old one. Based on the relationship, the new solutions can be obtained by using a given solution of the equations.

scholarly journals Non-Perfect Colorings of Symmetrical Tilings with Color Groups of Index 4

2014 ◽  
Vol 70 (a1) ◽  
pp. C1426-C1426
Author(s):  
Beaunonie Gozo, Jr. ◽  
Ma. Louise Antonette De Las Peñas ◽  
Rene Felix
Keyword(s):  
Carbon Nanotubes ◽  
Electronic Properties ◽  
Symmetry Group ◽  
Proper Subgroup ◽  
Color Group ◽  
Systematic Construction ◽  
Chemical Structures ◽  
Structural Analogues ◽  
Transitive Set

In this work we present a method that will allow for the construction and enumeration of non-perfect colorings of symmetrical tilings. If G is the symmetry group of an uncolored symmetrical tiling, then a coloring of the symmetrical tiling is non-perfect if its associated color group is a proper subgroup of G. The process will facilitate a systematic construction of non-perfect colorings of a wider class of symmetrical tilings where the stabilizer of a tile in the symmetry group G of the uncolored symmetrical tiling is non-trivial and the set of tiles may not form a transitive set under the action of G. This poster discusses results on how to identify and characterize non-perfect colorings arising from the method with associated color groups of index 4. The approach obtained here provides an avenue to model and characterize various chemical structures with atoms of different proportions, and their symmetries. This is relevant particularly for understanding new and emerging structures, such as structural analogues of carbon nanotubes, where a lot of its physical and electronic properties depend on their symmetry.

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