Why does mowing the long way need fewer U-turns?

Stripe a rectangular lawn along its long side or along its short side, and the finished grass looks the same. The trip to get there is not. This guide walks through one small experiment in Lawn Stripes, an ExplainerTools Original. It also says what the model cannot tell you.

Predict first

The model lawn is a flat rectangle, 12 units long and 8 units wide. The cutter is an ideal disk 2 units across, so each straight pass clears a strip 2 units wide. Before touching anything, make a guess. Does the stripe direction change how far the mower travels? If it does, which way is shorter, and by about how much?

Try it

1. Open the tool with Stripe direction on Along the long side. Let it drive, or press Drive the route. Count 4 lanes and 3 U-turns.
2. When it stops, read Whole path: 57.42 units. Lawn cut now says 100% · checked.
3. Choose Along the short side, press Start of the route, then Drive the route. This time you count 6 lanes and 5 U-turns, and the whole path is 63.71 units.
4. Press In the first U-turn. The explanation card says how little new grass that turn reaches.

What you should see

Both routes drive exactly 48 units of straight lanes: 4 × 12 one way and 6 × 8 the other. Both cut every point of the 96-square-unit lawn. The tool checks that with an exact distance test, not by counting pixels. The only difference is the U-turns. Each is a half-circle of radius 1 that loops past the edge and adds π, about 3.14 units. Three add 9.42 units; five add 15.71. That gap of 2π, about 6.28 units, is the whole difference between 57.42 and 63.71.

Why it happens

A U-turn joins one lane to the next, so a route has one fewer turn than it has lanes. Running along the long side takes fewer lanes to cross the lawn, so it takes fewer turns. Here the turns swing out over the paving, where the cutter clears almost nothing. The small patch of grass a turn does reach is grass the next lane would cut anyway. The turns are travel spent getting back into position.

The tool also has a Turn cost knob. It adds a declared, hypothetical charge for each U-turn, in distance-equivalent units. At 5 units per U-turn, the scores become 72.42 and 88.71. That score is not seconds, fuel or energy. It only shows how the gap grows, by twice the charge, if you decide turns should count extra.

What the model leaves out

This is an ideal round cutter on a flat, empty rectangle, and it may drive past the edge to turn. There are no trees, slopes, trimmed edges, steering limits or overlapping passes. Nothing here measures time or fuel. Neither pattern is claimed to be the best possible route; the tool compares two specified routes. Back-and-forth stripes are the classic way to cover a simple area, as in Choset's boustrophedon decomposition. Arkin, Fekete and Mitchell showed that finding the best mowing route in general is hard. Fekete and colleagues compute near-optimal tours for this kind of disk cutter. Arkin and colleagues study covering tours in which turns carry their own cost.

The next question

What should happen on a square lawn? Count the lanes each way before you decide, then explain your answer to someone else. A real yard, with flower beds and a tree, would need its own model. Try it together: one person predicts the score gap at 10 units per U-turn, and the other moves the knob.

Watch it, then try it

The 35.1-second Short drives the same two routes in the same tool.

Try it yourself