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Math Science

The Mathematics of Lawn Mowing 514

Posted by timothy from the go-fast-turn-left dept.
Hugh Pickens writes "I enjoy mowing my six-acre lawn with my John Deere 757 zero-turn every week, and over the course of the last five years of mowing I have come up with my own most efficient method of getting the job done which takes me about three hours. While completing my task this morning, I decided after I finished to research the subject to discover if there is a method for determining the most efficient path for mowing, and found that Australians Bunkard Polster and Marty Ross wrote last summer about an elegant mathematical presentation of the problem of mowing an irregularly shaped area as efficiently as possible. First we simplify our golf course mowing problem by covering the course with an array of circles with each circle radius equal to the width of the mower disc. Connecting the centers of the circles produces an equilateral triangular grid, with vertices at the circle centers. Following a path consisting of grid edges, there will necessarily be a fair amount of overlap so the statement of the problem is to minimize the overlap by minimizing the number of vertices that are visited more than once which Polster and Ross say is easily achieved by well-known computer search algorithms. Any other tips from Slashdot readers?"
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The Mathematics of Lawn Mowing More

The Mathematics of Lawn Mowing

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  • Here's a tip... (Score:5, Funny)

    by nbetcher ( 973062 ) <nbetcher@nOSPAM.gmail.com> on Saturday August 06, 2011 @08:28AM (#37006238)
    ... hire someone to mow it for you. :)

    • Re: (Score:2, Insightful)

      by M. Baranczak ( 726671 )

      Here's another tip. More of a rhetorical question, actually: what the hell does anybody need a six-acre lawn for? Can you honestly say that it provides you with more enjoyment than, say, a half-acre lawn?

      • Re:Here's a tip... (Score:4, Insightful)

        by Aladrin ( 926209 ) on Saturday August 06, 2011 @08:42AM (#37006350)

        Absolutely. A half-acre is barely enough to do any sports-like activities. There's a very good chance that the ball/disc/boomerang/whatever will end up in the neighbor's yard.

        At best, it's a pain to go around and get it, and everyone waits while one person has a good run, unbalancing everything.

        At worst, you have a cranky neighbor or break something in your neighbor's yard.

        So yeah, having a big yard can make a difference.

        If all you do is barbecue, then a 1/2 acre yard is more than enough, though.

      • Absolutely. The biggest advantage to a larger lawn is that you are guaranteed to have more space between your house and your neighbor's houses. The second advantage that going and playing outside is much easier in a larger lawn. (I know, go outside and play, who does THAT any more?) A half-acre lawn generally means you have at best 20 yards or so between houses, since most half-acre lawns are in town. That's really not a lot of room and the houses are fairly densely packed together in that area. Try to go o
      • To reduce wifi interference from those pesky neighbours, duh.

      • what the hell does anybody need a six-acre lawn for? Can you honestly say that it provides you with more enjoyment than, say, a half-acre lawn?

        Yes, it would. Except for the mowing part.

    • Re: (Score:2, Redundant)

      by mabhatter654 ( 561290 )

      Sheep!

  • Believe it or not sometimes people are better at solving certain problems than computers. This is one of those fuzzy problems with lots of irregularities that a human is excellent at working out with just a little help from a stopwatch.

    • "Believe it or not sometimes people are better at solving certain problems than computers. "

      "Kill the heretic! Kill him! Persecute! Kill!"

    • Re: (Score:2)

      by Oswald ( 235719 )

      I second this post. There are too many options/variables when you mow a large irregular space and too many possibilities that won't be considered by one-algorithm-fits-all solutions. How much is your mower's maneuverability degraded as speed goes up--would you be better off with a solution that had more straight lines but let you go faster? How about that little patch that sticks off the northwest corner--are you better off fitting it into the big pattern, or do you leave it until last, then deadhead ove

      • You don't even have to deadhead to those edge cases (heh, in this case literally). When your more organized route takes you to where that odd corner is, go ahead and finish it separately and then return to the main pattern where you left off. At most you'll be wasting the distance from the place where your main pattern joins the corner section and the farthest spot in that corner.

        • Re: (Score:2)

          by cynyr ( 703126 )

          that's what i always did, route out the main area, and each corner case separately. When you bump into "corner" you start in on it's route. when you finish go back to the big area.

    • It looks an awful lot like the solution in TFA is just:

      1. Follow the perimeter

      2. Make concentric, slightly overlapping passes until you reach the center

      That's about what I do with my 5 acre lawn.

      As a P.S., what's with all the people hating on big lawns? I'm not rich or wasteful, it's a rental. It was a big old farm in the 1800s.
      I have chickens and dogs, and a vegetable garden.
      My wife and kid love it. We do fun shit outside. Ease up /.ers. -Rico

      • Re: (Score:2)

        by cynyr ( 703126 )

        i would love some "yard" to grow some plants. namely some food so i could show my kids how that really works, and that it isn't magic pixies that make it show up in the store.

    • Believe it or not sometimes people are better at solving certain problems than computers. This is one of those fuzzy problems with lots of irregularities that a human is excellent at working out with just a little help from a stopwatch.

      Actually this is a perfectly normal problem where the results you get out of a computer will depend a lot on how well you define the problem. If you define the shape of the lawn, the size of the cutter, and the turning characteristics of the mower accurately, I have little doubt that a computer can come up with a more optimal solution than a human (even if only by a small amount). A human with a stopwatch is unlikely to try more than about 15 different routes while a computer in simulation can try millions

  • ... get off my lawn! :-)

    SCNR

  • In America, you mow the lawn!

  • Interesting Story! (Score:4, Insightful)

    by MyLongNickName ( 822545 ) on Saturday August 06, 2011 @08:35AM (#37006298) Journal

    First, let me say that I do like this type of story. Interesting, thought provoking, nerdy and mathematical in nature.

    I will also preface what I am about to say by noting that people are free to make whatever life tradeoffs they want.

    At the same time, I really wonder why anyone would want a property that takes three hours just to cut the grass. Life is short, why spend it maintaining a large property. I make low six figures now and could afford a lot more of a house than I have, and even when I upgrade to a nicer neighborhood next year will still way underbuy what the bank wants me to borrow.

    If you are stinking rich and want the large property, go ahead... but hire someone to do it for you. Your time is more valuable than the cost of having someone cut your grass. Give some teenager or out of work adult the opportunity to earn some money. That is the real win-win of capitalism.

    Finally, the article linked to seems light on the math itself, but seems very descriptive. I don't know that there is a purely mathematical solution to the problem but wonder if genetic algorithms would get you to where you want to be. I also wonder if you have a yard like mine with tree roots all over the place would change the outcome :)

    • Re: (Score:3)

      by brunes69 ( 86786 )

      For many people (especially with a ride on), mowing the lawn is not so much of a chore but an escape, a way to be out away from work, the wife, kids, noise... to have a couple of beers, listing to some tunes, basking in the sun - just in general relax for a few hours. It's stress relief. And the type of people with 6 figure incomes usually need it the most.

      Now, i am not one of those people.. I get my stress relief playing some 360. But everyone has to have their own outlet, and I know lots of guys who say

      • Re: (Score:2, Insightful)

        by Anonymous Coward

        Lawn mowing is a job that you can complete on your own that requires no meetings, team discussions or 300 e-mails before the project can even start. thenyou can step back and admire your work, knowing 37 other people will not take credit for it. ;o)

      • My couple of hours of mowing the lawn tends to be chock full of going over things in my head. Mowing the lawn is something that requires almost no brain power, so I am free to pontificate to my hearts content.
        Three hours would probably be a bit much, but he says he enjoys it. I would say, if he enjoys it that much, maybe he should start cutting other people's lawns. I've had several offers from people on cutting my lawn, which is an acre plot, but ends up being about half an acre of lawn thanks to the hou

      • It is only an escape if you use a scythe. Sitting on your arse in a tractor is a chore, and a rather boring one, too.

    • Re: (Score:2)

      by Shimbo ( 100005 )

      Your time is more valuable than the cost of having someone cut your grass. Give some teenager or out of work adult the opportunity to earn some money. That is the real win-win of capitalism.

      You missed the part where he said he enjoyed doing it. Not doing what the hell you like because your time is too valuable would be my idea of hell.

  • I can't speak to mathematical models of efficiency but I can tell you about landscapers models of gas and employee efficiency.
    Use trimmers and small mowers to shape up the irregular areas until they reach a common area or edge.
    Allow the riding mowers to tackle the larger squared zones
    Of course none of this accounts for the grass, which must be hauled with an attached trailer on the riding mowers and regularly emptied regardless of the efficiency pattern.
    • Agreed. Even at home, my mantra is, "Minimize turns."

      Do 1 or 2 laps around the outer edge to give you a buffer so you don't spray grass all over the sidewalks, etc., and then choose the longest dimension of the ~rectangular shape you have left. Make long passes back and forth along this edge, alternating left, right, left, right. (If you're not bagging, this means you will run over some of your discharge sometimes, but that's OK as long as the grass isn't too tall.)

      Using this method reduced the tim

      • Re: (Score:2)

        by MagicM ( 85041 )

        This is my method as well, for the same reasons.

        However I do feel like taking "minimize turns" to the extreme would be even more efficient. Starting at the outer edge and going inward in more or less a spiral should eliminate turns completely. You just end up with an odd pattern for your neighbors to talk about.

        • When there are six acres between you and your neighbors, nobody gives a crap what patterns you leave in your lawn. Not everybody is a homeowners association jerk who has to compulsively measure each blade of grass to ensure uniform height.

  • May we suggest ... (Score:4, Insightful)

    by lysdexia ( 897 ) on Saturday August 06, 2011 @08:37AM (#37006316) Homepage
    Sheep?

  • Because I live in a US county that publicly owns two hydro-electric dams our electric power rates are low enough to make it much more economical to use an electric-powered lawn mower instead of a gasoline-powered lawn mower. The safest method of mowing the grass would be to ensure that the power cord always stays out of the way of the grass-cutting head of the mower. This complicates the efficient mowing technique because, in general, it's better to simply mow so that the power cord is always on the freshly

    • electric-powered lawn mower [...] ensure that the power cord always stays out of the way

      I thought the power cord would stay in the garage and charge the mower's battery overnight.

      • Batteries increase by two the number of conversions before the electricity is applied. Every conversion costs power, and as a bonus, the battery also slowly loses power just sitting there.

  • Doesn't look good (Score:5, Funny)

    by Arlet ( 29997 ) on Saturday August 06, 2011 @08:43AM (#37006368)

    The 'optimal' solution has the mower finishing in the middle of the lawn, which is usually not where you want to leave it parked.

    • Re: (Score:3)

      by bgarcia ( 33222 )
      Unlike the "painting the floor" task, it's not that much of an issue here.

      • Re: (Score:2)

        by Arlet ( 29997 )

        Not much, but assuming you want to move the mower back, the proposed solution isn't optimal because your traveling twice over the same strip of grass.

        • Which is completely irrelevant, because every mower I've ever used or known has the capability of being turned off. Push mowers you could turn off and push to wherever. Riding mowers, I don't know of a single riding mower that does not allow you to turn off the blades while still keep travelling. I also don't know of a single person who just parks their mower in the exact spot they finished mowing the lawn.

          Long story short, turn off the blades, move mower to wherever you want to store it.

          • Re: (Score:2)

            by Arlet ( 29997 )

            You missed the point. If you're trying to optimize for the shortest path, you don't want to travel the mower across the lawn with the blades turned off. Instead, put the blades down, and find a path that starts/ends at the same place (or at least along the edge). That way, the total path will be shorter.

  • How does the circular mower cut corners? Don't most people have a corner of smaller radius that their imaginary circular lawnmower?

    Shouldn't the problem be how to sweep a straight line of some given width to cover an area? I'm guessing the circular mower is some sort of simplifying assumption. Never had a lawn before, so no idea.

    • Re: (Score:2)

      by Arlet ( 29997 )

      TFA has a picture of the lawnmower. It is circular, and not at all imaginary.

    • The typical lawn mower is like an upside-down helicopter. The blades sweep out a circular path. At a corner, you typically mow past the edge, reverse-and-turn so you're lined up for the next edge, then proceed. For a typical lawn mower there would be very little difference between your straight line sweep and the circle sweep.

      The person who posted the article, however, has a zero turn radius mower. They don't have to do the reverse-and-turn bit. The mower can turn on a dime. This stop and pivot tu

  • This neglects the reality that even with zero turn mowers, there is some cost to turning.
    You can't make a right angle turn at full speed.
    There isn't a mathematically correct solution unless you correctly model the costs of turning.
    If you're doing it 'by hand' - then you also need to model the cost of screwing up.
    It may be that comparatively simple schemes - such as an interleaved raster scan may be
    in practice optimal for a human to mow it.

  • IME (Score:2)

    by Aladrin ( 926209 )

    In my experience (with 1-acre and 4-acre sections to mow) there is a little you can do to optimize the route, but in general, you want to end up with the clippings shooting toward the center of the lawn so it's easy to rake. (The bags on the mowers are a pain because you have to empty them so often.) So the perfect path in the article is marred by the fact that you then have to either re-mow some of it to shoot the clippings in the right direction, or get out a blower and spend just as much time doing th

    • If he mows every week he could be mulching and leaving the clippings. One week grass growth us pretty minimal no matter where you live - maybe an inch or two.

    • if you hate bags, maybe you should try getting mowers with a "mulching" setting. No point in throwing away all that free fertilizer.

      There's really no point in bagging unless you're trying to gather up vegetative waste for a purpose, like perhaps a compost pile or something. "Filling up the county dump with yard waste" is not a worthwhile purpose.

  • What? Floodfill ftw.
    • FTA: "Puzzle to ponder: Suppose you have a lawn that is 200 square meters in area and your mower disc is 70 cm in diameter. Show that any mowing path for this lawn must be at least 280 metres in length."

      Mower is .7m. Imagine the simplest lawn, one with no turns, the width of the mower. So x*.7m = 200m => x = 285m.

  • ...for a tractor-haybine combination with an 80" swath and a 20' turning radius.

  • If you are using your wet RAM thinking about about lawn care you can possibly take it as a sign that your youth and the more interesting times of your life are over.

  • Pay someone to mow it for you.

  • Lawn? (Score:5, Insightful)

    by Rising Ape ( 1620461 ) on Saturday August 06, 2011 @09:34AM (#37006732)

    Six acres isn't a lawn, it's a field... anyone else get the impression this guy just wanted a reason to say "I have a six acre lawn"?

  • Best solution (Score:5, Insightful)

    by Charliemopps ( 1157495 ) on Saturday August 06, 2011 @09:37AM (#37006752)
    The best solution: don't mow it.
    Why the hell do you have 6 acres of grass? Plant some trees for christs sake.

  • CNC (Score:2)

    by Yvan256 ( 722131 )

    Anyone else looked at these patterns and thought "CNC milling"?

  • from a mathematician with a few dozen published papers and half a dozen published books on mathematics.

    So now you ask slashdot just to make sure???

  • And retire to the porch with your beer.

  • Examination of the example in the article suggests a heuristic algorithm that should provide near-optimal solutions and is suitable for real-time execution on neural wetware.

    1. Start by mowing around the outside border.
    2. Proceed going around, from the outside in.
    3. When you reach a strip <= 3 mowers wide, clear it with short back-and-forths.

    Proof of an upper bound on excess mowing vis-a-vis the optimal solution is left as an exercise for the reader.

  • I have a simpler solution... (Score:4, Funny)

    by CosaNostra Pizza Inc ( 1299163 ) on Saturday August 06, 2011 @10:01AM (#37006956)
    Cover your yard in asphalt and paint it green. It also doubles as a tennis/basketball court. I hate "mowing lawns".

  • Once you've represented the lawn area as a tessellation of (slightly overlapping) lawnmower-sized patches, then isn't this just the traveling salesman problem - visit all patches with the least distance traveled?

    This is a classic NP problem... if the problem size (N) is too large to fully evaluate (in this case 6 acres = 29,000 square yards, tractor area = 1 square yard, so N = 29,000 which is rather large for this type of problem), then heuristics are you're friend.

    The optimal solution, which would only ap

  • I have 4 acres with trees, and get sick of mowing. (I also have a 1 acre woodlot) I see that the solution given is flawed for several reasons. In the real world it will produce a poor looking finish, but that is tolerable. The lawn will be one height and neat, but not well finished. Also the mowing modeled as a circular area is a built in inefficiency, because only the edge of the circle is cutting.

    I did not get out of the setup whether edges were treated as "hard" or "soft". A hard edge you cannot pas

  • Any other tips from Slashdot readers?

    I have found that the most efficient way to mow the lawn is to call a couple of guys, Manny and Angel, who leave the place looking great.

    The only downside, is they flirt with my daughter, my wife, my mother-in-law and probably when I'm not looking, my 8 year old border collie.

  • Eulerian Path (Score:3)

    by shellster_dude ( 1261444 ) on Saturday August 06, 2011 @10:23AM (#37007110)
    I believe what you are interested in is called a Eulerian Path: http://en.wikipedia.org/wiki/Eulerian_path [wikipedia.org]
    Also the similarly related Hamiltonian Path: http://en.wikipedia.org/wiki/Hamiltonian_path [wikipedia.org]
    As others have mentioned the actual method of solving the problem is probably best defined as "The traveling salesman" problem: http://en.wikipedia.org/wiki/Traveling_salesman_problem [wikipedia.org]

    Good Luck.

  • Buy a goat.

    Extra Added Advantage: At your convenience, Lawnmower Curry.

  • Lawns do not need to be mowed weekly. Dropping to every other week will save 3 hours weekly. The optimal solution will not be able to gain that much.

  • Seriously, you don't need math to know that you mow in a circle, with the ejecting side of the lawnmower facing towards the center of the yard at all times.

  • I am a mesh maker and work with tessellations, (mostly in 3D but also 2D). You don't need this heavy duty math to arrive at the solution. Any area you mow more than once represents wasted effort. The simplest non intersecting path is, start at the outer edge and follow the boundary of the unmowed area and progressively you move inwards. Additional brownie points to choose the direction so that clippings are discharged into unmowed area.
  • I just drive around the edge until it all disappears. You simply have to not care about the track pattern. If you're going for track pattern: one can't help the obsessive.
  • So the long winded blah blah blah page concludes, at the bottom, that you follow the perimeter, moving the mower in one row at a time, until you get to the center. Exactly like any 9-year-old kid does without anyone telling them. Amazing!

    What is next, analyzing the fastest way to brush our teeth? I'll try not to give away the best approach.

    Next week, kids, we move on to tire rotation.

  • This is a standard problem in CAM systems (Score:4, Insightful)

    by Animats ( 122034 ) on Saturday August 06, 2011 @01:58PM (#37008732) Homepage

    All modern computer-aided machining systems have solvers for this problem. When you tell a CAM system to machine an arbitrary area, it computes a tool path to do the job. Here's MasterCam doing it. [thomasnet.com] Even low-end 2D CAM systems can solve the lawnmower problem. High-end systems can solve much tougher problems, automatically deciding what tool to use, clearing big areas with big tools and finishing up the tight spots with small ones. The most advanced CAM tools can do that in 3D on very complex objects. [youtube.com]

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