Maximal and minimal polyiamonds

Abstract

The minimum perimeter of an n-polyiamond in whichever of |?6n|or |?6n|+ 1 has the same parity as n. To prove this result, we first obtain a lower bound on the perimeter by considering maximal polyiamonds (i.e., polyiamonds with a given perimeter and a maximum number of triangles). We then show how to construct minimal polyiamonds that attain the perimeter lower bounds. The maximum number of triangles in a polyiamond with perimeter p is round (p^2/6)-?6, where ?6 is ) if p=0 (mod 6), and is 1 else.