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.

Author Bio

Yifei Li is a math major from Berea College, class of 2012

Michael Mara is a math and computer science double major from Williams College, class of 2012.

Isamar Rosa Plata is a civil engineering major at the University of Puerto Rico at Mayag�ez, Class 2013.

Elena Wikner is a math major from Williams College, class of 2011.