We study the presumably unnecessary convexity hypothesis in a theorem of Chung et al. on perimeter-minimizing planar tilings by convex pentagons. We prove that the theorem holds without the convexity hypothesis in certain special cases, and we offer direction for further research.

Author Bio

Whan Ghang is a 2013 graduate from MIT with a B.S. in mathematics. As an undergraduate he attended the SMALL program in Summer 2012 with Steven and Zane under the supervision of Professor Morgan. He is a graduate student in mathematics at Harvard currently working at the National Institute for Mathematical Sciences (NIMS) which is located at Daejeon, Republic of Korea.

Zane Martin graduated with honors from Williams College in 2013, where he majored in mathematics. His work on this paper began at the 2012 SMALL REU at Williams College, and continued as a thesis under Professor Frank Morgan during his senior year. He currently works at a civil rights firm in New York City, and will enter into graduate studies in applied mathematics at Columbia University this fall.

Steven Waruhiu is a 2013 graduate from the University of Chicago with a B.S. in mathematics. As an undergrad, he participated in the Williams College SMALL REU where research for this paper began. At the moment, Steven is a software developer for a company in Chicago.