Slashdot recently linked to an article co authored…

Slashdot recently linked to an article (co-authored by a mathematician) about the most efficient method for mowing grass. I have no mathematical degrees to my credit, but I don’t find the method described by Polster & Ross to be useful. Go read it first, then come back and read why I think my method is better.

The problem with the Polster & Ross article is that it conflates two very different tasks: mowing a lawn and mowing a golf course. The illustrations all depict a golf course, and one with a very irregular shape. Most people interested in this topic are mowing lawns, which tend to be more or less rectangular. The method described by Polster & Ross may be very efficient for golf courses, but it’s useless for lawns. It also may be intended for riding mowers, but I use a push mower, and that makes a big difference.

The article gets one thing right: you want to mow in long straight lines as much as possible. Why? Because turns are inefficient. They slow you down, and they also require more effort, tiring you out sooner. So turns should be as infrequent as you can make them.

The best way to accomplish this is to break the lawn into rectangular blocks. Why rectangles? Well, you’re essentially tiling a plane, and there are only three polygons you can use to do that: triangles, squares, and hexagons.  Hexes are the worst choice because they have the most vertices, which means too many turns. Squares are better, and triangles would be even better, but you can’t use them because you’re trying to tile a rectangular area, and triangles aren’t ideal for that. So you use squares.  And in practice, you’ll end up with groups of squares that form a rectangle, and that’s what you mow.

Your property may be perfectly rectangular, but it gets chopped into more uneven shapes by your house, your driveway, trees and bushes, flowerbeds, and so on. Integral calculus tells us that we can fill irregular areas if we use progressively smaller rectangles, but this quickly becomes impractical for mowing.  If a rectangle is narrower or shorter than about three times the length of your mower, you can’t really mow around its edges, and are forced to use back-and-forth strokes. This means that you end up with several large and medium-sized rectangular blocks, and some small, irregular shapes in various corners.

End result: You mow around the edges of the rectangular blocks, and then finish off the leftover bits with back-and-forth patterns.

Looking at the golf course in Polster & Ross’s illustrations, I think I might still try my method on it. I would break it into one big rectangle in the center, one smaller rectangle in the lower right portion, and a bunch of irregularly shaped leftover pieces. However, Polster & Ross give no indication of how big that golf course is, and I’m not sure my method would scale well. What do you guys think?

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