Chaotic Attractors with Discrete Planar Symmetries

Carter, Nathan C.; Eagles, Richard L.; Grimes, Stephen; Hahn, Andrew C.; Reiter, Clifford A.

doi:10.1016/s0960-0779(97)00157-4

Cited by 49 publications

(18 citation statements)

References 12 publications

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“…So we obtain a different dual lattice. Proposition 2 [6]. Let be an arbitrary function, be a finite group realized by matrices acting on.…”

Section: Functions That Are Equivariant With Respect To Wallpaper Groupsmentioning

confidence: 99%

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Patterns with Symmetries of the Wallpaper Group on the Hyperbolic Space

Ouyang¹,

Li²,

Yu³

2014

TOCSJ

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Equivariant function with respect to symmetries of the wallpaper group is constructed by trigonometric functions. A proper transformation is established between Euclidean plane and hyperbolic spaces. With the resulting function and transformation, wallpaper patterns on the Poincaré and Klein models are generated by means of dynamic systems. This method can be utilized to produce infinity of beautiful pattern automatically.

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“…So we obtain a different dual lattice. Proposition 2 [6]. Let be an arbitrary function, be a finite group realized by matrices acting on.…”

Section: Functions That Are Equivariant With Respect To Wallpaper Groupsmentioning

confidence: 99%

“…For example, in [3,4], colorful images with symmetries of wallpaper groups were considered from the view point of dynamic systems. Many nice chaotic wallpaper attractors were produced by iterating equivariant functions [5][6][7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

Patterns with Symmetries of the Wallpaper Group on the Hyperbolic Space

Ouyang¹,

Li²,

Yu³

2014

TOCSJ

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“…We also employ this scheme to color attractors. For more details of the scheme, please refer to [2][3][4][5][11][12][13].…”

Section: Examples and Implementsmentioning

confidence: 99%

“…It is surprising that chaotic attractor is compatible with symmetry. Automatic generation of symmetrical patterns has been an active topic of recent study [2][3][4][5][6][7]. There are many methods dedicated to yield patterns with cyclic or dihedral symmetry.…”

Section: Introductionmentioning

confidence: 99%

Automatic Generation of Chaotic Attractors with Various Cyclic or Dihedral Symmetries

Wang¹,

Yan²,

Ouyang³

2014

TOCSJ

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Chaotic attractors are created by iterating functions that are equivariant with respect to the cyclic or dihedral groups. An improved color scheme based on the visit frequency of the pixels is proposed to render chaotic attractors. By normalization and scale transformation, aesthetic patterns which simultaneously have several kinds of cyclic or dihedral symmetries are generated. This method can be used to yield a great number of aesthetic patterns automatically.

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“…In recent years, automatic generation of symmetric tiling patterns by means of computers attracts many mathematicians' interest. Wallpaper repeating patterns [9] and hyperbolic repeating patterns [ 101 are generated by visualizing dynamical systems' chaotic attractors.…”

Section: Introductionmentioning

confidence: 99%

Chaotic attractors with symmetries of the triangle groups

International 2005 Computer Graphics

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The chaotic attractors with symmetries of the triangle p u p s are visualized in the hyperbolic plane. Using probabilistic iterated function systems that involve both &ne transformations of the plane and isometrjes of triangle groups, we can create intriguing and exotic attractor's images. The derived images are of self-similarity and of infinitely fine structure; they contain symmetries associated w i t h the triangle groups as well. The method provides a novel approach fur devising tiling patterns with symmetries.

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Chaotic Attractors with Discrete Planar Symmetries

Cited by 49 publications

References 12 publications

Patterns with Symmetries of the Wallpaper Group on the Hyperbolic Space

Patterns with Symmetries of the Wallpaper Group on the Hyperbolic Space

Automatic Generation of Chaotic Attractors with Various Cyclic or Dihedral Symmetries

Chaotic attractors with symmetries of the triangle groups

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