Madison Howard, Angela Vichitbandha, Kate Ponto
The University of Kentucky Math Lab aims to encourage undergraduate math interest across a range of projects. These include research projects and outreach as well visualization projects. We interpret visualization broadly making room for both 3d printing of polyhedra, simplicial complexes, and string diagrams and more traditional textile crafts such has quilting and crochet.
The Poincare disk admits many interesting tilings. This octahedral tiling corresponds to the universal cover of the two-holed torus, the colors were chosen to reflect this interpretation of the tiling. If the large colored regions of the same color were identified, the central white region would form a two holed torus. If we identify the edges between the smaller colored regions with elements of the fundamental group of the two holed torus the colors of the smaller regions are determined by the fundamental group elements crossed as we move out from the central white region.