Dedicating years of schooling to pursue higher math degrees may help solve certain problems, but does it make any difference for something as simple as cutting your grass? To find out, we spent the week discussing the mathematics of lawn mowing with colleagues at the University of Nevada, Las Vegas.
Our thoughts generally split into two directions: If your lawn is simple enough, then you can do some fairly specific calculations to figure out the most efficient way to mow it. But if your lawn is weird enough, it might resemble a famous mathematical allegory.
We’ve broken down how to mow your lawn using math.
First, you have to ask the question, “What is the topology of my lawn?” Topology is a branch of math that’s only officially existed for about a century. Some mathematicians call it “wiggly geometry” or “geometry without measuring.” Topology studies how regions and surfaces are similar or different, but not in terms of measurements like in geometry.
You can remember it like this: “What’s the volume of a sphere?” is a geometry question. “What’s the difference between a sphere and a donut?” is a topology question
Topologically, your lawn might be one connected region with no holes, or it could have one or more holes in it. If the ‘hole’ in your lawn is a small obstacle like a mailbox, or a large one like a bunch of bushes, those are topologically the same. We’re just concerned with how many holes you have to circumnavigate during your mow.
Two fun topology keywords are worth noting: All lawns are bounded and orientable. “Bounded” means no lawn extends infinitely in any direction. “Orientable,” meanwhile, means you can’t walk along a lawn and come back to the same spot upside down, like on a Möbius strip.
We checked to make sure there are no art installations out there with grass covering a Möbius strip, and so far it looks like nobody has managed it. It’s probably pretty tough to get grass to grow upside-down.
How to Mow a Simple Lawn
Let’s start with the simpler case. Suppose you have a nice rectangular lawn with no gaps, and your mower can’t make perfect turns, so you have to cut straight lines all the way across the lawn. Considering this will make those nice aesthetic parallel mower marks, it’s probably the most common situation. But you still probably want to know if you should mow lengthwise or widthwise.
To get more mathematical, let’s define optimized mowing. An optimal mow would be any route in which each blade of grass had the lawnmower pass over it exactly once.
Say your lawn has length L and width W, and your mower cuts a path with width M. Now, if either L or W is an exact multiple of M, then an optimal mow is possible. If not, then you’ll have some area of grass that gets hit more than once. Call this the “error area.”