A fractal tiling ( f -tiling) is a kind of rarely explored by similar polygonal tiles which possesses self-similarity and the boundary fractal. Based on isosceles right triangles, Dutch graphic artist M. C. Escher created an ingenious print Square Limit in fish are uniformly reduced size as they approach boundaries tiling. In this article, we present four families -tilings propose easy-to-implement method to achieve Escher-like drawings. By systematically investigating local star-shaped...

A fractal tiling (f-tiling) is a whose boundary fractal. This article presents two families of rare, infinitely many f-tilings. Each f-tiling constructed by reducing tiles fixed scaling factor, using single prototile, which segment regular polygon. The authors designed invariant mappings to automatically produce appealing seamless, colored patterns from such tilings.

A simple, fast method generates various visually appealing spiral patterns. The is based on the concept that patterns comprise a symmetry group of tilings. It employs invariant mappings and dynamical system to create seamless colored

This third installment of the Beautiful Math articles considers visualization aesthetic patterns with hyperbolic-triangle-group symmetries. A flexible form invariant mappings contributes to a simple, efficient way generate hyperbolic patterns. Combined conformal mappings, this method can yield an abundance exotic

The art of tiling originated very early in the history civilization. Almost every known human society has made use tilings some form or another. In particular, using only regular polygons have great visual appeal. Decorated with continuous and symmetrical patterns were widely used decoration field, such as mosaics, pavements, brick walls. science, these provide inspiration for synthetic organic chemistry. Building on previous CG&A “Beautiful Math” articles,...

Continuous wavelet transform (CWT) is a linear convolution of signal and function for fixed scale. This paper studies the algorithm CWT with Morlet as mother by using nonzero-padded convolution. The time domain filter, which non-causal sample function. By making generalized discrete Fourier (GDFT) inverse this we can get geometrically weighted periodic extension filter when evaluated outside its original support. From causal filter. In paper, GDFT-based CWT, has more concise form than that...

A fractal tiling is a which possesses self-similarity and the boundary of fractal. In this paper, we investigate dimension [Formula: see text]- text]-tilings. We first derive an explicit recursion formula for edges Then present analytical expression their dimensions using matrix methods. Results indicate that, as text] increases, boundaries text]-tilings will degenerate into general Euclidean curves. The method proposed in paper can be extended to compute other kinds tilings.

By constructing invariant mappings associated with wallpaper groups, this paper presents a simple and efficient method to generate colorful patterns. Although the constructed have form only two parameters, combined color scheme of orbit trap algorithm, such can create great variety aesthetic The resulting patterns are further projected by central projection onto sphere. This creates interesting spherical that possess infinite symmetries in finite space.

Abstract In this paper, using both hand-drawn and computer-drawn graphics, we establish a method to generate advanced Escher-like spiral tessellations. We first give way achieve simple tilings of cyclic symmetry. Then, introduce several conformal mappings three derived tilings. To obtain tessellations on the generated tilings, given pre-designed wallpaper motifs, specify tessellations’ implementation details. Finally, exhibit rich gallery According proposed method, one can produce great...

A fast algorithm is established to transform points of the unit sphere into fundamental region symmetrically. With resulting algorithm, a flexible form invariant mappings achieved generate aesthetic patterns with symmetries regular polyhedra.

Schwarz-Christoffel mapping is an elegant theory that can preserve the structure of a pattern perfectly. One most important elements conformal theory, has applications in many engineering fields, such as electromagnetism, aerodynamics, and thermal field theory. Building on their 2014 CG&A article titled "Beautiful Math, Part 3: Hyperbolic Aesthetic Patterns Based Conformal Mappings," authors apply professional numerical methods to achieve real polygon boundaries.

Abstract In this paper, we present a method for creating Escher‐like spherical patterns with regular polyhedron symmetries. Using the generators of symmetry groups associated polyhedra, first provide fast algorithms to construct tilings. Then, obtain patterns, specify texturing techniques decorate resulting Moreover, strategy create novel dynamic effect kaleidoscopes in which motifs have complete body. The has advantages simple implementation, calculation, good graphics, and artistic...

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

Dutch graphic artist M.C. Escher created many famous drawings with a deep mathematical background based on wallpaper symmetry, hyperbolic geometry, spirals, and regular polyhedra. However, he did not attempt any spiral in space. In this paper, we consider modified geometry by removing the condition that geodesic is orthogonal to unit circle Poincaré model. We show symmetry similarity property exist so creation of uncommon possible. To end, first establish theoretical foundation for proposed...

A fast algorithm is established to transform points of the unit sphere into fundamental region symmetrically. With resulting algorithm, a flexible form invariant mappings achieved generate aesthetic patterns with symmetries regular polyhedra. This method avoids order restriction symmetry groups, which can be similarly extended treat polytopes in n-dimensional space for n>=4.

Whirlpools , by the Dutch graphic artist M. C. Escher, is a woodcut print in which fish interlock as double spiral tessellation. Inspired this print, article we extend idea and present general method to create Escher-like interlocking drawings of N whirlpools. To end, first introduce an algorithm for constructing regular tiling T . Then, design suitable use copies compose K similar next specify realization details using wallpaper templates decorate enhance aesthetic appeal, propose several...

A fractal tiling or [Formula: see text]-tiling is a which possesses self-similarity and the boundary of fractal. By substitution rule tilings, this short paper presents very simple strategy to create great number text]-tilings. The Equithirds demonstrated show how achieve it in detail. method can be generalized every that constructed by rule.

Symmetry can be widely found in natural phenomenon. Regular polygons and polyhedra are the most basic important symmetrical structures 2D 3D Euclidean space. Four-dimensional regular polytopes (4-RPs) 4D analogs of three dimensions two dimensions. After introducing fundamental root systems 4-RPs, this article presents interesting methods to visualize 4-RPs using a region algorithm.

A fractal tiling or f ‐tiling is a which possesses self‐similarity and the boundary of fractal. ‐tilings have complicated structures strong visual appeal. However, so far, discovered are very limited since constructing such needs special talent. Based on idea hierarchically subdividing adjacent tiles, this paper presents general method to generate ‐tilings. Penrose tilings utilized as illustrators show how achieve it in detail. This can be extended treat large number that constructed by...