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

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.

Periodic tiling of triangular and square nanotubes in a cationic metal–organic framework for selective anion exchange

A unique cationic metal–organic framework formed by connecting the neutral chain-like secondary building units with positively charged ligands shows charge- and size-dependent ion-exchange of anion dyes.

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

"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"

Complex tiling patterns in liquid crystals

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.