We prove optimality of tilings of the flat torus by regular hexagons, squares, and equilateral triangles when minimizing weighted combinations of perimeter and number of vertices. We similarly show optimality of certain tilings of the 3-torus by polyhedra from among a selected candidate pool when minimizing w eighted combinations of interface area, edge length, and number of vertices. Finally, we provide n umerical evidence for the Log Convex Density Conjecture.
Li, Yifei; Mara, Michael; Plata, Isamar Rosa; and Wikner, Elena
"Tiling with Penalties and Isoperimetry with Density,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 13
, Article 6.
Available at: http://scholar.rose-hulman.edu/rhumj/vol13/iss1/6