Unitary Symmetry And Combinatorics.

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Unitary symmetry and combinatorics. Symmetry, Ornament and Modularity. Symmetry, Representations, and Invariants. Geometry and symmetry.

Symmetry and Perturbation Theory. Patterns in Permutations and Words. Equivalence, invariance, and symmetry. Groups and symmetry. Recommend Documents.

## Tessellation

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In particular, it is of interest in Euclidean, spherical, or Bolyai-Lobachevsky geometry to look at the geometric transformations that are isometries. An isometry is a transformation that preserves distance. Just as someone who makes a cubical box may not realize that the cube is a regular polyhedron one where all the vertices are alike and where all the faces are congruent regular polygons , a maker of a weaving may not know anything about isometries.

The isometries in the Euclidean plane are translations, rotations, reflections and glide reflections.

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Thus, a mathematician may state which isometries are used in the design of a rug, but that does not mean that the designer of the rug invoked any mathematical thinking. Social scientists have used symmetry ideas and the conservatism of the mechanisms of cultural transmission in their studies. Archeologists and anthropologists have attempted to use symmetry ideas to date artifacts pottery or fabrics and to study trade and patterns of commerce.

## Great software for Tilings, Patterns, Symmetry

A major tool in the analysis of symmetry has been the concept of a group. The group concept has a rich and complicated history with ties to the study of the theory of equations attempts to show that one could not find formulas to solve quintic polynomial equations. By late in the 19th century, group theory was being used as a tool to help crystallographers understand the symmetry of crystals and other naturally occurring structures. Out of this interest grew the work that made possible the classification of tilings and patterns using group theory considerations.

## A Computerized Method to Generate Complex Symmetric and Geometric Tiling Patterns | SpringerLink

One can also look at the symmetry of a single motif or ornament. Examples of such motifs illustrated from patterns used in batiks are shown below. A more complex pattern such as the one below can be built up from simple motifs. Such patterns have translational symmetry in one direction. Designs or patterns of this kind are known as strip , band , or frieze patterns. The motifs used to make such frieze patterns may be isolated from one another or coalesce into a "continuous" geometric design along the strip.

If a pattern has translations in two directions, then the pattern is often referred to as a wallpaper pattern. Whereas an artist may choose to create a pattern with absolute and strict adherence in all details to have symmetry in the pattern, this is not all that common for "tribal" artists or artisans.

Thus, if one looks carefully at a rug which at first view looks very symmetrical, it is common to see that at a more detailed level it is not quite totally symmetric either in the use of the design or of the colors used in different parts of the design. One can see the small liberties that are taken either because of the difficulty of making patterns exact by hand or because the artist wants consciously to make such small variations. In analyzing the symmetry of such a pattern it probably makes sense to idealize what the artist has done before applying some mathematical classification of the symmetry involved.

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In the patterns shown above no color appears. We have a black design on a white background. However, in discussing the symmetry of a pattern one can study the symmetry involved if color is disregarded or by taking color into account. If you look at the batik below from a symmetry point of view you must idealize model what is going on to use mathematics.