One thing that complicates the answer to this question is the variables. You have the speed of the mower, the width of the mower, the dimensions of the yard, the amount of time it takes to turn, etc. There's also the issue of things like trees and shrubs in the yard, which might make one or the other pattern better. Let's ignore trees and shrubs, assume that the mower is 2 feet wide and assume a square-shaped yard that is 100 feet on a side (making it 0.25 acres or so). Here are the details of the paths:
- You could walk 100 feet up one edge of the yard, make a 180-degree turn, walk back another 100 feet, make a 180-degree turn, and so on, 50 times.
- You could walk 100 feet up one edge, turn 90 degrees, walk 98 feet, turn 90 degrees, and so on, spiraling in toward the center. You would have to make 25 spirals, with each spiral getting progressively smaller.
If you go ahead and calculate the distance you have to travel for each path, it is the same -- you have to walk 5,000 feet or so (nearly a mile!). The big difference comes in the turns. In the first option, you have to make 49 180-degree turns. In the second option, you have to make 99 90-degree turns. If you assume that it takes twice as long to turn 180 degrees than it does to turn 90 degrees, then the two paths are approximately equal again.
If one type of turn is significantly more involved than another, then that could shift the equation. For example, on a riding mower, a 90-degree turn is probably much easier than a 180-degree turn. With a walk-behind mower, a 180-degree turn and a 90-degree turn may differ only slightly.
So the key is to get out there with a stop watch and see which type of turn is more efficient for you. Or get a goat, or one of those robot lawn mowers to do the work for you.