Maximal and minimal polyexes

Abstract

The minimum perimeter of a polyhex with n hexagons is 2|?(12n-3)|. To prove this result, we first obtain a lower bound on the perimeter by considering maximal polyhexes (i.e., polyhexes with a given perimeter and a maximum number of hexagons). We then show how to construct minimal polyhexes that attain the perimeter lower bounds. A polyhex has even perimeter. If p is even, then the maximum number of hexagons in a polyhex with perimeter p is round (p^2/48).