Maximal and Minimal Polyhexes
Winston C. Yang
Computer Sciences Department
University of Wisconsin-Madison
A polyhex is a connected collection of congruent regular hexagons.
The minimum perimeter of a polyhex with n hexagons with unit edges is shown to be 2*ceil(sqrt(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).
The maximum number of hexagons in a polyhex with perimeter p is shown to be round(p^2/48)).
We then use this result to construct minimal polyhexes that attain the perimeter lower bounds.
The minimum perimeter results are useful in domain decomposition problems.
This research was supported in part by National Science Foundation grant DMI-0100220.