Beautiful Math, Part 5: Colorful Archimedean Tilings from Dynamical Systems

Ouyang, Peichang; Zhao, Wen Jun; Huang, Xuan

doi:10.1109/mcg.2015.135

Cited by 9 publications

(8 citation statements)

References 14 publications

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“…Sebagai contoh, teknologi seperti komputer dapat digunakan untuk menghasilkan gambargambar dengan pola simetri yang dijadikan sebagai motif pada dinding, tekstil, keramik, karpet, jendela kaca patri dan sebagainya. Pola-pola ini lebih lanjut memiliki dampak ekonomi dan estetika [5].…”

Section: Pendahuluanunclassified

Pembangkitan Pola Simetri Rotasi 90◦ Dari Simulasi Sistem Dinamik

Cania¹,

Syafwan²,

Bahri³

2021

Jurnal Matematika UNAND

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Dalam makalah ini dibahas pola simetri jenis p4 berwarna yang memiliki simetri rotasi 90◦ dan simetri translasi dengan periode T masing-masing sepanjang sumbu-x dan sumbu-y. Pola simetri p4 ini dibangkitkan dengan menggunakan aplikasi Matlab melalui simulasi sistem dinamik diskrit dengan terlebih dahulu melakukan analisis terhadap syarat dan pemilihan fungsi-fungsi dinamiknya. Dalam hal ini, setiap titik pada bidang dijadikan sebagai titik awal pada iterasi sistem dinamik, dan jumlah iterasi yang dihasilkan dari kriteria konvergensi dalam bentuk norm Euclidian menentukan warna yang diberikan pada titik tersebut. Dengan menggunakan beberapa kombinasi nilai-nilai parameter pada fungsi-fungsi dinamik, diperoleh pola-pola simetri p4 yang lebih menarik dan variatif.Kata Kunci: Isometri, Pola Simetri p4, Sistem Dinamik

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Section: Pendahuluanunclassified

Pembangkitan Pola Simetri Rotasi 90◦ Dari Simulasi Sistem Dinamik

Cania¹,

Syafwan²,

Bahri³

2021

Jurnal Matematika UNAND

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“…Graphical presentations of the phase portraits not only reveal the complex behaviours of such systems, but also quite often have high artistic appeal (Chung et al, 2001). Many different methods of creating phase portraits with aesthetic appeal have been proposed (Chung and Chan, 1993;Chung et al, 2001;Lu et al, 2007;Ouyang et al, 2015). One of these (which will be used in this paper) has been proposed by Chung and Chan (1993).…”

Section: Symmetrical Patterns From Dynamicsmentioning

confidence: 99%

“…In this way, we are able to obtain patterns that possess wallpaper symmetries (Chung and Chan, 1993;Lu et al, 2007), spherical symmetries (Chung and Chan, 1995), frieze symmetries (Lu et al, 2010), Archimedean tilings (Ouyang et al, 2015), etc. In recent years, another interesting approach has been proposed, namely, the use of iteration processes from fixed point theory.…”

Section: Introductionmentioning

confidence: 99%

Procedural generation of aesthetic patterns from dynamics and iteration processes

Gdawiec

2017

International Journal of Applied Mathematics and Computer Science

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Aesthetic patterns are widely used nowadays, e.g., in jewellery design, carpet design, as textures and patterns on wallpapers, etc. Most of the work during the design stage is carried out by a designer manually. Therefore, it is highly useful to develop methods for aesthetic pattern generation. In this paper, we present methods for generating aesthetic patterns using the dynamics of a discrete dynamical system. The presented methods are based on the use of various iteration processes from fixed point theory (Mann, S, Noor, etc.) and the application of an affine combination of these iterations. Moreover, we propose new convergence tests that enrich the obtained patterns. The proposed methods generate patterns in a procedural way and can be easily implemented on the GPU. The presented examples show that using the proposed methods we are able to obtain a variety of interesting patterns. Moreover, the numerical examples show that the use of the GPU implementation with shaders allows the generation of patterns in real time and the speed-up (compared with a CPU implementation) ranges from about 1000 to 2500 times.

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“…From this representation, P2VP, PS, and PI occupy the square (red), hexagonal (blue), and decagonal (orange) sites, respectively, of a regular (4.6.12) Archimedean tiling pattern. It is noteworthy that the PS domain adopts a circular shape within the solvent‐annealed 3 µ ‐SPI thin film which differs from the hexagonal prism expected for an ideal SJC (4.6.12) tiling . Although such curvature of the PS domain increases the PI/PS and PS/P2VP interfacial areas as suggested by the minimal fundamental triangles drawn in Figure , it would allow for the minimization of the intermaterial dividing surface (IMDS) between the two components (i.e., PI and P2VP) of the structure that have the highest incompatibility parameter, χ(χ SI ≈ 0.088, χ SP ≈ 0.178, and χ PI ≈ 0.8 at 25 °C), as well as for a relaxation of the PS arms .…”

mentioning

confidence: 94%

Nanoscale Archimedean Tilings Formed by 3‐Miktoarm Star Terpolymer Thin Films

Antoine

Aissou

Mumtaz

et al. 2019

Macromol. Rapid Commun.

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3‐Miktoarm star terpolymer architecture (3µ‐ABC), consisting of three dissimilar polymer chains, A, B, and C connected at a junction point, provides a unique opportunity in the design of complex nanoscale patterns such as Archimedean tilings that are not accessible from linear ABC terpolymers. In this work, the synthesis and the self‐assembly of 3‐miktoarm star terpolymers, namely, polystyrene‐arm‐poly(2‐vinylpyridine)‐arm‐polyisoprene (3µ‐SPI) into Archimedean tiling patterns is described. Several 3µ‐SPI terpolymers are produced via a mid‐functional PS‐b‐P2VP, synthesized by sequential anionic polymerization, using a 1,1‐diphenylethylene bearing a tert‐butyldimethylsilyl–protected hydroxyl functionality as a core molecule. PI arms with different sizes are then linked to the deprotected hydroxyl function of the core molecule via a Steglich esterification. Solvent‐annealed 3µ‐SPI thin films exhibit nanoscale prisms arranged into a (4.6.12) Archimedean tiling pattern. Depending on the size of the low etch‐resistant PI arm and the solvent selected to promote the self‐assembly of 3µ‐SPI thin films, the voided columns occupy the square or decagonal sites of the (4.6.12) pattern. The use of this (4.6.12) tiling produced for the first time from self‐assembled 3µ‐ABC thin films can be a promising route to build 2D photonic crystals having complete photonic band gaps, where the light propagation is completely prohibited.

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Beautiful Math, Part 5: Colorful Archimedean Tilings from Dynamical Systems

Cited by 9 publications

References 14 publications

Pembangkitan Pola Simetri Rotasi 90◦ Dari Simulasi Sistem Dinamik

Pembangkitan Pola Simetri Rotasi 90◦ Dari Simulasi Sistem Dinamik

Procedural generation of aesthetic patterns from dynamics and iteration processes

Nanoscale Archimedean Tilings Formed by 3‐Miktoarm Star Terpolymer Thin Films

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