We provide a list of conjectured surface-area-minimizing n-hedral tiles of space for n from 4 to 14, previously known only for n equal to 5 and 6. We find the optimal "orientation-preserving" tetrahedral tile (n=4), and we give a nice new proof for the optimal 5-hedron (a triangular prism).

Author Bio

Paul Gallagher graduated in 2014 from the University of Pennsylvania. He was part of the 2013 Geometry Group at the Williams College SMALL REU. He is pursuing a PhD in mathematics.

Whan Ghang is a 2013 graduate from MIT with a B.S. in mathematics. As an undergraduate he attended SMALL program in 2012 Summer with Steven and Zane under the supervision of Professor Morgan. He is currently a graduate student in mathematics at Harvard.

David Hu graduated from Georgetown University's McDonough School of Business with a BS in finance and a minor in mathematics in May of 2014. In the fall, he plans on continuing his studies at SUNY Stony Brook, where he will pursue a PhD in Mathematics. His primary areas of interest are differential geometry and global analysis. This paper is the result of a collaboration between the 2012 and 2013 Geometry Groups of the SMALL REU at Williams College, advised by Professor Frank Morgan.

Zane Martin graduated from Williams College in 2013, where he majored in mathematics. His undergraduate work focused on geometry, though he is also interested in topology and applied math. His work on this paper began at the 2012 "SMALL" REU at Williams College, and continued during the 2013 SMALL program. He currently works at a civil rights firm in New York City, and hopes to attend graduate school in mathematics this fall.

Maggie Miller is a third-year math major at the University of Texas at Austin. She is interested in knot theory and differential geometry, and hopes to pursue these interests in graduate school in 2015. In her free time, she plays guitar and crochets

Byron Perpetua graduated in 2014 from Williams College, where he majored in mathematics and music, conducted research in geometry and commutative algebra, and directed the student orchestra. He is currently working as a data analyst at Achievement First.

Steven Waruhiu is a 2013 graduate from the University of Chicago with a B.S. is mathematics. As an undergrad, he participated in the Williams College Small REU where research for this paper began. He presented a talk on this paper at the 2012 MAA Mathfest in Madison and at the 2013 AMS/MAA Joint Mathematics Meeting in San Diego. At the moment, Steven is a software developer for a company in Chicago.