| This thesis investigates photonic quasicrystals and ellipsoid packings, two problems in which geometry plays an important role in condensed matter physics and where the use of stereolithographically fabricated macroscopic models helps to solve fundamental problems.; Part I. Photonic quasicrystals may have bandgap properties that make them well-suited for scientific and technological applications in which photonic crystals are normally considered. They are constructed from two or more types of dielectric material arranged in a quasiperiodic pattern whose rotational symmetry is forbidden for periodic crystals. Because quasicrystals have higher point group symmetry (are closer to spherical) than ordinary crystals, their gap center frequencies are closer and the gaps widths are more uniform, optimal conditions for forming a complete, more spherical bandgap. Our calculations on 2-D Penrose quasicrystals show very isotropic bandgaps. Analogous computations 3-D quasicrystals are challenging and have not been performed to date. Here we circumvent the 3-D computational problem by doing an experiment. Using stereolithography we constructed an icosahedral photonic quasicrystal with centimeter-scale cells and performed microwave transmission measurements. We show that the quasicrystal exhibits sizeable stopgaps and determine their effective Brillouin zones. As a check, we perform calculations and experiments on a similar photonic crystal with diamond symmetry. Our studies confirm that quasicrystals are excellent candidates for photonic bandgap materials.; Part II. The measurements of the overall density &phis; avr, &phis;(r), and the core density agree with simulations. Recent simulations indicate that ellipsoids can randomly pack more densely than spheres and, remarkably, ellipsoids with aspect ratio 1.25:1:0.8 can approach the densest crystal packing (FCC) of spheres, with packing fraction 74%. We demonstrate that such dense packings are experimentally realizable. We introduce a novel way of determining packing density for a finite number of particles that minimizes wall effects. We have fabricated ellipsoids using stereolithography and show that in a spherical container of radius R, the radial packing fraction, &phis;(r), can be obtained from V(h), the volume of added fluid to fill the sphere to height h. Additionally, we perform an MRI scan of our packing and obtain &phis;(r) from it. The measurements of the overall density &phis;avr, &phis;(r), and the core density &phis; 0 = 0.74 +/- 0.005agree with simulations. |
