Image Sampling with Quasicrystals

We investigate the use of quasicrystals in image sampling. Quasicrystals produce space-filling, non-periodic point sets that are uniformly discrete and relatively dense, thereby ensuring the sample sites are evenly spread out throughout the sampled image. Their self-similar structure can be attractive for creating sampling patterns endowed with a decorative symmetry. We present a brief general overview of the algebraic theory of cut-and-project quasicrystals based on the geometry of the golden ratio. To assess the practical utility of quasicrystal sampling, we evaluate the visual effects of a variety of non-adaptive image sampling strategies on photorealistic image reconstruction and non-photorealistic image rendering used in multiresolution image representations. For computer visualization of point sets used in image sampling, we introduce a mosaic rendering technique.Comment: For a full resolution version of this paper, along with supplementary materials, please visit at http://www.Eyemaginary.com/Portfolio/Publications.htm

Geometric realizations of two dimensional substitutive tilings

We define 2-dimensional topological substitutions. A tiling of the Euclidean plane, or of the hyperbolic plane, is substitutive if the underlying 2-complex can be obtained by iteration of a 2-dimensional topological substitution. We prove that there is no primitive substitutive tiling of the hyperbolic plane $\mathbb{H}^2$. However, we give an example of substitutive tiling of \Hyp^2 which is non-primitive.Comment: 30 pages, 13 figure

Hyperbolic tilings and formal language theory

In this paper, we try to give the appropriate class of languages to which belong various objects associated with tessellations in the hyperbolic plane.Comment: In Proceedings MCU 2013, arXiv:1309.104

An aperiodic hexagonal tile

We show that a single prototile can fill space uniformly but not admit a periodic tiling. A two-dimensional, hexagonal prototile with markings that enforce local matching rules is proven to be aperiodic by two independent methods. The space--filling tiling that can be built from copies of the prototile has the structure of a union of honeycombs with lattice constants of $2^n a$, where $a$ sets the scale of the most dense lattice and $n$ takes all positive integer values. There are two local isomorphism classes consistent with the matching rules and there is a nontrivial relation between these tilings and a previous construction by Penrose. Alternative forms of the prototile enforce the local matching rules by shape alone, one using a prototile that is not a connected region and the other using a three--dimensional prototile.Comment: 32 pages, 24 figures; submitted to Journal of Combinatorial Theory Series A. Version 2 is a major revision. Parts of Version 1 have been expanded and parts have been moved to a separate article (arXiv:1003.4279

Pictures worth a thousand tiles, a geometrical programming language for self-assembly

International audienceWe present a novel way to design self-assembling systems using a notion of signal (or ray) akin to what is used in analyzing the behavior of cellular automata. This allows purely geometrical constructions, with a smaller specification and easier analysis. We show how to design a system of signals for a given set of shapes, and how to transform these signals into a set of tiles which self-assemble into the desired shapes. We show how to use this technique on three examples : squares (with optimal assembly time and a small number of tiles), general polygons, and a quasi periodic pattern : Robinson tiling

An evaluation of building sets designed for modular machine tool structures to support sustainable manufacturing

The modularization of machine tool frames is an approach when designing new machine tool structures in a sustainable context. By integration of microsystem technology and designing lightweight modules, a smart alternative to conventional machine tool frames is developed. In previous studies, this concept has been evaluated along with a compilation of the possible use-case scenarios and the potential benefits from using modular electronics. In the presented paper, the geometric requirements from the selected use-case scenarios for machine tool structures are identified by dividing the structures in their ideal mechanic equivalents. A set of rules is developed driven by the generalized geometric requirements of the machine tool frames. Three different approaches of polyhedral building sets are shown and evaluated for their merits based on criteria of geometric functionality and sustainability. Finally, a prototypical modular portal frame is presented for the proof of concept

Optimal Monohedral Tilings of Hyperbolic Surfaces

The hexagon is the least-perimeter tile in the Euclidean plane for any given area. On hyperbolic surfaces, this isoperimetric problem differs for every given area, as solutions do not scale. Cox conjectured that a regular k-gonal tile with 120-degree angles is isoperimetric. For area π/3, the regular heptagon has 120-degree angles and therefore tiles many hyperbolic surfaces. For other areas, we show the existence of many tiles but provide no conjectured optima. On closed hyperbolic surfaces, we verify via a reduction argument using cutting and pasting transformations and convex hulls that the regular 7-gon is the optimal n-gonal tile of area π/3 for 3≤n≤10. However, for n\u3e10, it is difficult to rule out non-convex n-gons that tile irregularly