Symmetries of Cairo-Prismatic Tilings

We study and catalog isoperimetric, planar tilings by unit-area Cairo and Prismatic pentagons. In particular, in counterpoint to the five wallpaper symmetry groups known to occur in Cairo-Prismatic tilings, we show that the five with order three rotational symmetry cannot occur

A MODEL STUDY OF ADLAYER PATTERN FORMATION OF RIGID DI- TRI- AND TETRATOPIC MOLECULES ON SQUARE AND TRIANGULAR LATTICES

In this work we considered the adlayer self-assembly of three model molecules di-, tri- and tetratopic with different sizes and potential energy parameters on square and hexagonal (triangular) lattices. For each case, we carry out minimization using an analytical gradient to find the most stable minima. In all cases we use “coarse-grained” site-to-site pairwise additive potential. We have explored how the change in the size of the molecule affects the pattern formation in the molecular adlayer. A primary focus of this work restricts the exploration of the landscape to a “unit cell” of 2x2 angles, labeled [1, 2, 3, 4] and extrapolate this to an infinite lattice by the application of tessellation. The model we study represents a 2-dimensional surface with fully occupied lattice sites and with boundary conditions to resemble the infinitely occupied surface. To investigate the patterns we have used several order parameters that can distinguish between the adlayers. We have found several adlayers varied as the shape and the size of the molecules’ change. We also have reported the chirality of the adlayer by using the order parameters. We note that homochiral patterns can be formed by using achiral molecules, and comment on the areas of parameter space where this occurs. The molecular pattern hierarchy of the ditopic molecule on a square lattice distributed from highly ordered motifs such as a linear sheet “short stripe” geometry to fourfold achiral windmill structure and chiral windmill pattern. On the other hand we have reported a pinwheel chiral structure of ditopic molecule on a triangular lattice. On both square and triangular lattices we also found several herringbone structures. Depending on the shape of the molecule and the surface lattice, the porous shape and size of the adlayer change wildly. We note several porous shapes such as square, rectangle, hexagon and octagon with their sizes depend on the molecular distance parameters. For instance we note a honeycomb structure of tritopic molecule on a triangular lattice distorts to a semi-hexagonal pattern as the size of the molecule increases. We have also conducted Monte Carlo simulation for a range of molecular sizes of ditopic molecule on both square and triangular lattices. We note that the adlayer patterns of the simple minimization method and the Monte Carlo simulation are quite consistent

Metamaterial optical diodes for linearly and circularly polarized light

Total intensity of light transmitted at non-normal incidence thorough planar metamaterials can be different for forward and backward propagation. For metamaterial patterns of different symmetries we observe this effect for circularly or linearly polarized light

Laser-Driven Rayleigh-Taylor Instability: Plasmonics Effects and Three-Dimensional Structures

The acceleration of dense targets driven by the radiation pressure of high-intensity lasers leads to a Rayleigh-Taylor instability (RTI) with rippling of the interaction surface. Using a simple model it is shown that the self-consistent modulation of the radiation pressure caused by a sinusoidal rippling affects substantially the wavevector spectrum of the RTI depending on the laser polarization. The plasmonic enhancement of the local field when the rippling period is close to a laser wavelength sets the dominant RTI scale. The nonlinear evolution is investigated by three dimensional simulations, which show the formation of stable structures with "wallpaper" symmetry.Comment: 5 pages, 5 figures. New version includes 2D and 3D simulations. More details in the analytical calculation are given in the previous versio

Investigation Of Plane Symmetry In Lattice Designs

The purpose of this research project is to analyze the scholarly article The Plane Symmetry Groups: Their Recognition and Notation by Doris Schattschneider. In this article, Schattschneider discusses an application of abstract algebra which is useful in art as well as crystallography: frieze groups and wallpaper groups. I was interested in pursuing this topic because it combines mathematics with its applications, particularly with my own interest in chemistry. The article provides a compiled resource of terminology and rules of these groups, but not one which was easily accessible to undergraduate students. In my research, I elaborated on the descriptions of certain types of periodic patterns to add to the accessibility, and analyzed a few designs to prove their classification based on the rules from Schattschneider\u27s article. I found that this resource provided a good source of rules for which mathematical proofs could be based, and proved the classification of two different periodic plane designs

Computational Investigation of the Morphological Design Dimensions of Historic Hexagonal-Based Islamic Geometric Patterns

This dissertation examines the morphology of Islamic Geometric Patterns (IGP). Using mixed methods, including the simulation of historical designs and content analysis, this dissertation explores the question of how it is possible to mathematically describe the IGP. The study argues that the compositional analysis of geometry is not solely sufficient to investigate the design characteristics of the IGP, and the underlying mathematics and computational nature of the IGP should be considered when investigating historical IGP. The study presents a parametric description method that captures the reality of the IGP in numeric form and utilizes the form to derive representational codes that include the information necessary to construct a geometry. The representational codes are utilized to further investigate the actual and virtual design space of the IGP, aiming at identifying morphological similarities between historical designs. This research challenges the long-standing paradigm that considers compositional analysis to be the key to researching historical IGP. Adopting a mathematical description shows that the historical focus on existing forms has left the relevant structural similarities between historical IGPs understudied. The research focused on the historical, hexagonal-based IGP and found that hexagonal-based IGP designs correlate to each other beyond just the actualized dimension and that deep, morphological connections exist in the virtual dimension. Using historical evidence, this dissertation identifies these connections and presents a categorization system that groups designs together based on their ‘morphogenetic’ characteristics