Comparing perimeters of arrays of hexagons vs. squares
April 25, 2006 8:42 AM Subscribe
Let's say I have an array of 100 adjoining hexagons, 10 hexagons per side. Let's also say I also have an array of 100 adjoining squares, 10 squares per side. Each single hexagon and each square encloses the same area. Is the perimeter of the array of 100 hexagons longer than the perimeter of the array of 100 squares?
posted by Rumple to Science & Nature (17 answers total)
It seems to me that necessarily the lumpy exterior of the hexagonal array will be longer than the perimeter of the (square) 10 X 10 array of squares. However, each individual hexagon has a more efficient perimeter to area ratio, and so the overall array of hexagons should be smaller in diameter, perhaps compensating for the lumpiness. I know I could just draw a bunch of squares and hexagons but I'd like to know if there is a logical or pure geometric principles answer to this.
(It stems from a student expressing surprise at a result in a GIS simulation she was running - the result, which found arrays of squares gave better results than arrays of hexagons for a perimeter-minimizing objective function, seemed to me self evident but neither of us are geometers. Googling just gave me endless discourses on the wisdom of honeybees.)