Non-periodic Tiling of Procedural Noise Functions

2018 ◽  
Vol 1 (2) ◽  
pp. 1-15
Author(s):  
Aleksandr Kirillov
Keyword(s):  
Periodic Tiling

Varieties of 12-fold rosettes and their applications in Moroccan geometric ornament

2019 ◽  
Vol 52 (3) ◽  
pp. 605-617
Author(s):  
Youssef Aboufadil ◽  
Abdelmalek Thalal ◽  
My Ahmed El Idrissi Raghni ◽  
Abdelaziz Jali ◽  
Amane Oueriagli
Keyword(s):  
Construction Method ◽  
Geometric Shape ◽  
Elementary Geometry ◽  
Islamic Art ◽  
Crystallographic Groups ◽  
Different Types ◽  
Periodic Tiling

This work is devoted to the study of the 12-fold rosettes frequently encountered in Islamic art, especially in Moroccan ornamentation. Several types of 12-fold rosettes are described according to the geometric shape of their petals. They have a large number of variants and offer the possibility of building new, previously unknown rosettes, while respecting the construction method Hasba used by master craftsmen. Artisans developed a technique for combining different types of 12-fold rosettes to construct infinite periodic tiling belonging to the 17 crystallographic groups. This technique enabled them to diversify the repeated patterns based on 12-fold rosettes. An analysis of their tiling suggests a method based on elementary geometry to build new patterns with different types of 12-fold overlapped rosettes and their variants. A procedure based on combination of the distances between two overlapped rosettes is then proposed, which enables generation of new periodic and quasiperiodic patterns.

scholarly journals Aperiodic 3D tiling by self-similar substitution method using solid unit

2014 ◽  
Vol 70 (a1) ◽  
pp. C1431-C1431
Author(s):  
Yasunari Watanabe
Keyword(s):  
Study Author ◽  
First Generation ◽  
The Body ◽  
The Other ◽  
Substitution Method ◽  
Left Hand ◽  
Aperiodic Tiling ◽  
Periodic Tiling ◽  
Self Similar ◽  
3D Solid

"There are two important aperiodic tiling methods. One is a projection method and the other is self-similar substitution method. Author applied projection and substitution method to several quasi-periodic tiling [1a] [1b] [1c] [2a] [2b] [2c]. Quantity of the information for research of aperiodic tiling except quasi-periodic tiling is not so large as that of quasi-periodic tiling. Author shows one of the substitution example. that is a ""pentomino tile"" which is made of five unit squares. First generation of a ""pentomino tile"" is composed of two original pentominos and chiral two original pentominos [3]. Author considered 3D solid body unit similar to exterior view of car-body as shown in figure, then succeeded in first generation tile using the body unit. In this study, author consider a relation between substitution and crystallographic rotation matrix and translation matrix and discuss general formulation of self-similar substitution. Caption of Figure: (left-hand) original generation, (center) First generation, (right-hand) Second generation"

Estimating the Size of Context-Free Tiling Languages

1987 ◽  
Vol 39 (6) ◽  
pp. 1413-1433
Author(s):  
Kenneth Holladay
Keyword(s):  
Difficult Problem ◽  
General Question ◽  
Periodic Tiling ◽  
Context Free ◽  
Special Case

The problem of counting polyominoes motivates this paper. We will develop a general question for study that has counting polyominoes as a special case. We generalize in two ways. Polyominoes are shapes on the tiling made of square tiles. We will consider shapes on other tilings. The set of all polyominoes can be generated by a context-free array grammar, but the size of this set is estimated by counting the words of certain subsets and supersets that are generated by more convenient grammars. Our general question is the problem of counting the words of a context-free array language on a periodic tiling.Counting polyominoes is a difficult problem that has not been completely solved yet. There are various techniques for roughly estimating the number of polyominoes of a given size. We will extend some of these techniques to our general question.

scholarly journals Complex tiling patterns in liquid crystals

2011 ◽  
Vol 2 (5) ◽  
pp. 669-680 ◽  
Author(s):  
C. Tschierske ◽  
C. Nürnberger ◽  
H. Ebert ◽  
B. Glettner ◽  
M. Prehm ◽  
...  
Keyword(s):  
Self Assembly ◽  
Liquid Crystalline ◽  
Periodic Structures ◽  
Three Dimensional ◽  
Lateral Position ◽  
Molecular Structures ◽  
Rod Bundles ◽  
Geometric Frustration ◽  
Periodic Tiling ◽  
Tiling Patterns

In this account recent progress in enhancing the complexity of liquid crystal self-assembly is highlighted. The discussed superstructures are formed mainly by polyphilic T-shaped and X-shaped molecules composed of a rod-like core, tethered with glycerol units at both ends and flexible non-polar chain(s) in lateral position, but also related inverted molecular structures are considered. A series of honeycomb phases composed of polygonal cylinders ranging from triangular to hexagonal, followed by giant cylinder honeycombs is observed for ternary T-shaped polyphiles on increasing the size of the lateral chain(s). Increasing the chain size further leads to new modes of lamellar organization followed by three-dimensional and two-dimensional structures incorporating branched and non-branched axial rod-bundles. Grafting incompatible chains to opposite sides of the rod-like core leads to quaternary X-shaped polyphiles. These form liquid crystalline honeycombs where different cells are filled with different material. Projected on an Euclidian plane, all honeycomb phases can be described either by uniformly coloured Archimedean and Laves tiling patterns (T-shaped polyphiles) or as multi-colour tiling patterns (X-shaped polyphiles). It is shown that geometric frustration, combined with the tendency to segregate incompatible chains into different compartments and the need to find a periodic tiling pattern, leads to a significant increase in the complexity of soft self-assembly. Mixing of different chains greatly enhances the number of possible ‘colours’ and in this way, periodic structures comprising up to seven distinct compartments can be generated. Relations to biological self-assembly are discussed shortly.

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