## 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?

## Bartholomew Xerxes Ogilvie, Jr. 11:15 am

on8 August 2011 Permalink |Well, the authors do acknowledge that most lawns are rectangular and that “a little trial and error usually suffices” before turning to golf courses — which I interpret as saying “this is only an interesting problem with a large and complex shape.” For a smaller and mostly rectangular lawn, I think the approach you describe is the only one that makes sense — assuming you care about efficiency (more on that in a moment).

My front lawn is actually quite irregular as lawns go (since most of our land is wooded); the front is roughly triangular, but with strips along the driveway and frontage. I mow most of the main section as a series of long strips, each shorter than the one before, but there are irregular spots all along where I use lots of back-and-forth motions. This is one reason why a riding mower, or even a self-propelled push mower, would be impractical for me to use; there just aren’t enough long straight sections.

But efficiency is far from my top priority. For one thing, the total area I have to mow is rather small, so any time saved by using an optimally efficient method would be negligible. More importantly, mowing stands in for my daily walk on days that I mow, and when you’re exercising, efficiency is exactly what you want to avoid.

## Pat 1:16 pm

on8 August 2011 Permalink |Efficiency matters to me because most of my mowing takes place during the months when North Carolina is unbearably hot. To avoid the worst of the heat, I mow as late in the evening as possible. The faster I mow, the later I can start and the lower the temperature I have to endure.