The conventional view holds that girih (geometric star-and-polygon, or strapwork) patterns in medieval Islamic architecture were conceived by their designers as a network of zigzagging lines, where the lines were drafted directly with a straightedge and a compass. We show that by 1200 C.E. a conceptual breakthrough occurred in which girih patterns were reconceived as tessellations of a special set of equilateral polygons ("girih tiles") decorated with lines. These tiles enabled the creation of increasingly complex periodic girih patterns, and by the 15th century, the tessellation approach was combined with self-similar transformations to construct nearly perfect quasi-crystalline Penrose patterns, five centuries before their discovery in the West.

Penrose tiling is an example of an aperiodic tiling. Here, a tiling is a covering of the plane by non-overlapping polygons or other shapes, and a tiling is A Penrose tiling is an example of an aperiodic tiling. Here, a tiling is a covering of the plane by non-overlapping polygons or other shapes, and a tiling is aperiodic if it does not contain arbitrarily large periodic regions or patches. However, despite their lack of translational symmetry, Penrose tilings may have both reflection symmetry and fivefold rotational symmetry. Penrose tilings are nam Penrose tilings are self-similar: they may be converted to equivalent Penrose tilings with different sizes of tiles, using processes called inflation and deflation. The pattern represented by every finite patch of tiles in a Penrose tiling occurs infinitely many times throughout the tiling. They are quasicrystals: implemented as a physical structure a Penrose tiling will produce diffraction patterns with Bragg peaks and five-fold symmetry, revealing the repeated patterns and fixed orientations of its tiles. The study of these tilings has been important in the understanding of physical materials that also form quasicrystals. Penrose tilings have also been applied in architecture and decoration, as in the floor tiling shown. The 2011 Nobel Prize in Chemistry was awarded to Dan Shechtman for "The discovery of quasicrystals." Penrose tiling was mentioned for having "'helped pave the way for the understanding of the discovery of quasicrystals.'" P… Many of the common features of Penrose tilings follow from a hierarchical pentagonal structure given by substitution rules: this is often referred to as inflation and deflation, or composition and decomposition, of tilings or (collections of) tiles. The substitution rules decompose each tile into smaller tiles of the same shape as those used in the tiling (and thus allow larger tiles to be "composed" from smaller ones). This shows that the Penrose tiling has a scaling self-similarity, and so can be thought of as a fractal, using the same process as the pentaflake. Penrose originally discovered the P1 tiling in this way, by decomposing a pentagon into six smaller pentagons (one half of a net of a dodecahedron) and five half-diamonds; he then observed that when he repeated this process the gaps between pentagons could all be filled by stars, diamonds, boats (or "justice caps", as he puts it) and other pentagons. By iterating this process indefinitely he obtained one of the two P1 tilings with pentagonal symmetry. In 1996, German mathematician Petra Gummelt demonstrated that a covering (so called to distinguish it from a non-overlapping tiling) equivalent to the Penrose tiling can be constructed using a single decagonal tile if…

FINDING: Penrose tiling and quasicrystals demonstrate that 5-fold rotational symmetry, long considered impossible for periodic crystals, is realized in aperiodic structures governed by the golden ratio. | MATH: Golden ratio φ = (1+√5)/2 ≈ 1.618; its reciprocal φ⁻¹ ≈ 0.618; 5-fold symmetry axis; inflation/deflation scaling factor φ; Penrose tiling uses two rhombi with acute angles 36° and 72° (multiples of 360°/5). | CONNECTION: Direct geometric harmony — φ appears in the ratio of diagonal to side in a regular pentagon, and in the self-similar scaling of Penrose tilings. The 5-fold symmetry is a crystallographic "forbidden" axis in periodic lattices but emerges in quasicrystals via φ-based aperiodic order. | DEPTH: 9 FINDING: Quasicrystals exhibit self-similarity under scaling by φ, linking fractal-like properties to non-periodic long-range order. | MATH: Inflation/deflation operation on Penrose tiling scales lengths by φ; diffraction patterns show sharp Bragg peaks at positions indexe Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Quasiperiodic crystals (QCs) are a new class of materials that exhibit long-range aperiodic translational order and high rotational symmetries. Unlike periodically arranged photonic crystals (or photonic band-gaps), PQCs possess unique light localization and transport properties related to their complex, multi-fractal energy spectra. Advances in 2D photonic structures are expected in the introduction of non linear and/or active functionality into a 2D PQC. One-dimensional semiconductor nanostructures are likewise promising materials both in fundamental research and in practical applications. CdSe/CdS rods present the appealing characteristics of strong and tunable light emission from green to red, are highly fluorescent and show linearly polarized emission. These characteristics open the way to a new class of hybrid devices based on polymers and colloidal NRs in which the unique optical properties of the inorganic moiety are combined with the processability of the host matrix to develop new high performing optical devices such as organic light-emitting diodes, ultra-low threshold lasers and non-linear devices. One of the challenges of these applications is the incorporation of inorganic nanoparticles into organic polymer matrices, since this is usually accompanied by phase separation, aggregation of nanoparticles, loss of transparency and luminescence quenching due to exciton energy transfer. In this paper two-dimensional (2D) hybrid PQCs which consist of air rods in a nanoco

Abstract Quantum transport can distinguish between dynamical phases of matter. For instance, ballistic propagation characterizes the absence of disorder, whereas in many-body localized phases, particles do not propagate for exponentially long times. Additional possibilities include states of matter exhibiting anomalous transport in which particles propagate with a non-trivial exponent. Here we report the experimental observation of anomalous transport across a broad range of the phase diagram of a kicked quasicrystal. The Hamiltonian of our system has been predicted to exhibit a rich phase diagram, including not only fully localized and fully delocalized phases but also an extended region comprising a nested pattern of localized, delocalized and multifractal states, which gives rise to anomalous transport. Our cold-atom realization is enabled by new Floquet engineering techniques, which expand the accessible phase diagram by five orders of magnitude. Mapping transport properties throughout the phase diagram, we observe disorder-driven re-entrant delocalization and sub-ballistic transport, and we present a theoretical explanation of these phenomena based on eigenstate multifractality.