Tilings, tiling spaces and topology

To understand an aperiodic tiling (or a quasicrystal modeled on an aperiodic tiling), we construct a space of similar tilings, on which the group of translations acts naturally. This space is then an (abstract) dynamical system. Dynamical properties of the space (such as mixing, or the spectrum of the translation operator) are closely related to bulk properties of the individual tilings (such as the diffraction pattern). The topology of the space of tilings, particularly the Cech cohomology, gives information on how the original tiling can be deformed. Tiling spaces can be constructed as inverse limits of branched manifolds.Comment: 8 pages, including 2 figures, talk given at ICQ

Rigidity percolation on aperiodic lattices

We studied the rigidity percolation (RP) model for aperiodic (quasi-crystal) lattices. The RP thresholds (for bond dilution) were obtained for several aperiodic lattices via computer simulation using the "pebble game" algorithm. It was found that the (two rhombi) Penrose lattice is always floppy in view of the RP model. The same was found for the Ammann's octagonal tiling and the Socolar's dodecagonal tiling. In order to impose the percolation transition we used so c. "ferro" modification of these aperiodic tilings. We studied as well the "pinwheel" tiling which has "infinitely-fold" orientational symmetry. The obtained estimates for the modified Penrose, Ammann and Socolar lattices are respectively: $p_{cP} =0.836\pm 0.002$, $p_{cA} = 0.769\pm0.002$, $p_{cS} = 0.938\pm0.001$. The bond RP threshold of the pinwheel tiling was estimated to $p_c = 0.69\pm0.01$. It was found that these results are very close to the Maxwell (the mean-field like) approximation for them.Comment: 9 LaTeX pages, 3 PostScript figures included via epsf.st

Fractals Generated by Modifying Aperiodic Substitution Tilings

This study proposes a method for producing an endless number of fractals using aperiodic substitution tilings, as illustrated by the Ammann Chair tiling. Higher order substitutions of aperiodic tiling are utilised in relation to the Siepinski carpet concept. The fractal dimensions of the fractals generated by the Ammann Chair tiling are calculated.Comment: 7 page

Towards aperiodic tesellation: a self-organising particle spring system approach

The derivation of novel programming methods for the generation of aperiodic tiling patterns, predominantly in 2d space, has attracted considerable attention from both researchers and practicing architects. So far L-Systems and quasicrystals are the only tools which can be used for the creation of aperiodic tiling patterns. This project attempts to create a self organizing particle spring system for aperiodic tiling formation on a 2d surface. The proposed method simulates natural dynamic procedures and applies a generative particle spring system for tiling formation. The initial inspiration of the thesis is the realization of tiling patterns for non-planar geometries, by using the previously stated method. The architectural reasoning behind that would be to use a minimal number of types of prefabricated units (e.g. Penrose rhombuses) to create an irregular and complex pattern or geometry