Celtic art contains mysterious and fascinating aesthetic elements including complex knot work motifs. The problem is that creating and exploring these motifs require substantial human effort. One solution to this problem is to create a process that collaboratively uses interactive and procedural methods within a computer graphic environment. Spline models of Celtic knot work can be interactively modeled and used as input into procedural shaders. Procedural shaders are computer programs that describe surface, light, and volumetric appearances to a renderer. The control points of spline models can be used to drive shading procedures such as the coloring and displacement of surface meshes. The result of this thesis provides both an automated and interactive process that is capable of producing complex interlaced structures such as Celtic knot work within a three-dimensional environment.

Celtic art contains mysterious and fascinating aesthetic elements including complex knot work motifs. The problem is that creating and exploring these motifs require substantial human effort. One solution to this problem is to create a process that collaboratively uses interactive and procedural methods within a computer graphic environment. Spline models of Celtic knot work can be interactively modeled and used as input into procedural shaders. Procedural shaders are computer programs that describe surface, light, and volumetric appearances to a renderer. The control points of spline models can be used to drive shading procedures such as the coloring and displacement of surface meshes. The result of this thesis provides both an automated and interactive process that is capable of producing complex interlaced structures such as Celtic knot work within a three-dimensional environment.

This thesis consists of three articles all relatedin some way to eigenvalues of random matrices and theirprincipal minors and also to tilings of various planar regions with dominoes or rhombuses.Consider an $N\times N$ matrix $H_N=[h_{ij}]_{i,j=1}^N$ from the Gaussian unitary ensemble (GUE). Denote its principal minors (submatrices in the upper left corner) by $H_n=[h_{ij}]_{i,j=1}^n$ for  $n=1$, \dots, $N$. We show in paper A that  all the $N(N+1)/2$ eigenvaluesof $H_1$, \dots, $H_N$ form a determinantal process on $N$ copies of the real line $\mathbb{R}$. We also show that this distribution arises as a scaling limit in tilings of an Aztec diamond with dominoes.We discuss a corresponding result for rhombus tilings of a hexagonwhich was later proved by Okounkov and Reshtikhin. We give a new proof of that statement in the introductionto this thesis.In paper B we perform a similar analysis for the Anti-symmetric Gaussian unitary ensemble (A-GUE). We show that the positive eigenvalues of an $N\times N$ A-GUE matrix andits principal minors form a determinantal processon $N$ copies of the positive real line $\mathbb{R}^+$.We also show that this distribution of all these eigenvalues appears as a scaling limit of tilings of half a hexagon with rhombuses. In paper C we study the shuffling algorithm for tilings of an Aztec diamond. This leads to the study of an interacting set of interlacedparticles that evolve in time. We conjecture that the diffusion limit of thisprocess is a process studied by Warrenand establish some results in this direction. / QC 20100804

An innovative approach for characterization of one dimensional Photonic Band Gap structures comprised of metallic inclusions (i.e. subwavelength dipole elements or resonant ring elements) is presented. Through an efficient S- to T-parameters conversion technique, a detailed analysis has been performed to investigate the variation of the dispersion characteristics of 1-D PBG structures as a function of the cell element configuration. Also, for the first time, the angular sensitivity of the structure has been studied in order to obtain the projected band diagrams for both TE and TM polarizations. Polarization sensitivity of the subwavelength cell element is exploited to propose a novel combination of elements which allows achieving PBGs with simultaneous frequency and polarization selectivity. The proposed approach demonstrates that the dispersion characteristic of each orthogonal polarization can be independently adjusted with dipole elements parallel to that same polarization. Generally, the structure has potential applications in orthomode transducer, and generally whenever the polarization of the incoming signal is to be used as a means of separating it from another signal in the same frequency band that is of orthogonal polarization. The current distribution and the resonance behavior of the ring element is studied and the effect of resonance on dispersion characteristics of 1-D PBGs composed of rings is investigated for the first time, for both individual and coupled rings. Interestingly, it is observed that 1-D PBG composed of resonant elements consistently has a bandgap around the resonant frequency of the single layer structure.

An innovative approach for characterization of one dimensional Photonic Band Gap structures comprised of metallic inclusions (i.e. subwavelength dipole elements or resonant ring elements) is presented. Through an efficient S- to T-parameters conversion technique, a detailed analysis has been performed to investigate the variation of the dispersion characteristics of 1-D PBG structures as a function of the cell element configuration. Also, for the first time, the angular sensitivity of the structure has been studied in order to obtain the projected band diagrams for both TE and TM polarizations. Polarization sensitivity of the subwavelength cell element is exploited to propose a novel combination of elements which allows achieving PBGs with simultaneous frequency and polarization selectivity. The proposed approach demonstrates that the dispersion characteristic of each orthogonal polarization can be independently adjusted with dipole elements parallel to that same polarization. Generally, the structure has potential applications in orthomode transducer, and generally whenever the polarization of the incoming signal is to be used as a means of separating it from another signal in the same frequency band that is of orthogonal polarization. The current distribution and the resonance behavior of the ring element is studied and the effect of resonance on dispersion characteristics of 1-D PBGs composed of rings is investigated for the first time, for both individual and coupled rings. Interestingly, it is observed that 1-D PBG composed of resonant elements consistently has a bandgap around the resonant frequency of the single layer structure.

An innovative approach for characterization of one dimensional Photonic Band Gap structures comprised of metallic inclusions (i.e. subwavelength dipole elements or resonant ring elements) is presented. Through an efficient S- to T-parameters conversion technique, a detailed analysis has been performed to investigate the variation of the dispersion characteristics of 1-D PBG structures as a function of the cell element configuration. Also, for the first time, the angular sensitivity of the structure has been studied in order to obtain the projected band diagrams for both TE and TM polarizations. Polarization sensitivity of the subwavelength cell element is exploited to propose a novel combination of elements which allows achieving PBGs with simultaneous frequency and polarization selectivity. The proposed approach demonstrates that the dispersion characteristic of each orthogonal polarization can be independently adjusted with dipole elements parallel to that same polarization. Generally, the structure has potential applications in orthomode transducer, and generally whenever the polarization of the incoming signal is to be used as a means of separating it from another signal in the same frequency band that is of orthogonal polarization. The current distribution and the resonance behavior of the ring element is studied and the effect of resonance on dispersion characteristics of 1-D PBGs composed of rings is investigated for the first time, for both individual and coupled rings. Interestingly, it is observed that 1-D PBG composed of resonant elements consistently has a bandgap around the resonant frequency of the single layer structure.