Maximal and minimal polyiamonds
Winston C. Yang
winston@cs.wisc.edu
Computer Science
University of Wisconsin-Madison
Robert R. Meyer
rrm@cs.wisc.edu
Computer Science
University of Wisconsin-Madison
The minimum perimeter of an polyiamond with n triangles is whichever of
ceil(sqrt(6n)) or ceil(sqrt(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) - delta, where delta is 0 if p is a multiple of 6, and is
1 else.
This research was supported in part by National Science Foundation grant
DMI-0100220.