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The Mathematics of Lawn Mowing
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?"
... hire someone to mow it for you. :)
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.
A bullet may have your name on it but splash damage is addressed "To whom it may concern."
... get off my lawn! :-)
SCNR
The Tao of math: The numbers you can count are not the real numbers.
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 :)
See my journal for slashdot ID's by year. Mine created in 2005. http://slashdot.org/journal/289875/slashdot-ids-by-year
Sheep?
The 'optimal' solution has the mower finishing in the middle of the lawn, which is usually not where you want to leave it parked.
Pay someone to mow it for you.
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"?
The best solution: don't mow it.
Why the hell do you have 6 acres of grass? Plant some trees for christs sake.
Cover your yard in asphalt and paint it green. It also doubles as a tennis/basketball court. I hate "mowing lawns".
Gotta be careful with this one. We bought one goat. He did a great a great job. A guy from the ASPCA came and told us that single goats get depressed and lonely and that we had to buy him some friends or they would confiscate him. There's apparently a 3 goat minimum here. Now the lawn looks pretty sparse and we have to buy more food for the goats. I was able to get them a temp job taking care of a local open space till they ate all of the weeds. Goats don't understand the whole "slow down...you're gonna get us all laid off" thing. I have them posted on Craigslist looking another job but for now they're still eating my lawn.
We really need your help
http://www.gofundme.com/help-sherry
This is not quite equivalent to the TSP. TSP tries to find a minumum weight hamilton cycle, which does not allow repeated vertices. In the problem here, they are allowing vertices to be repeated, but they are trying to minimize the number of repeats. Also, the graph obtained by overlaying the triangular grid might not be hamiltonian.
That said, I suppose you could translate an instance of this problem to the TSP by doing something like adding weighted edges between nonadjacent vertices, and letting the weight of each new edge uv be the distance between u and v in the original triangular graph.
I believe what you are interested in is called a Eulerian Path: http://en.wikipedia.org/wiki/Eulerian_path
Also the similarly related Hamiltonian Path: http://en.wikipedia.org/wiki/Hamiltonian_path
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
Good Luck.
It's called peace and quite. Some crazy Americans believe that having enough space around you so that you cannot hear or see your neighbours is a good thing. Then, when YOU are ready to interact with people you simply go to some public space and interact. Just because some Americans don't want to live like rats in a city does not make them crazy. Some people don't mind the screaming kid next door, or the barking dog down the street but some do. Also, it's not a question of how much resources each person uses, it's a question of how many people are using them. No amount of conservation will offset unrestrained population growth.
Get off my damn lawn!
Asking people to think is like asking them to buy you a new car
Yeah, but then if Americans built parks, they'd have to get to know their neighbours.
Well, there's something to that, but it's not exclusive to Merkins. I have a largeish property (>100 acre) in Tasmania which might be regarded as a de facto park in the sense that I do absolutely nothing to discourage the occasional bushwalker from scrambling across, but I also do not have to worry about installing curtains or blinds in my windows. After all, anyone coming past my house after dark is likely to break their necks, assuming they don't get eaten by drop-bears. ;-)
So what? One day, unless someone does us a favour (;-)) you'll be as decrepit as we are, and you will find equally asinine and rewarding pursuits. Don't knock it until you've tried it...
What the fuck do you care how big someone's yard is? Petty little shit.
Let's see -- let little Johnny run off to play all day in an unsupervised parka couple miles away. Or . . . have a nice big backyard where a boy can be a boy and have fun doing kid stuff all day, with a parent on the premises (even if not directly observing). Yeah, gee. A park sounds perfect.
Especially considering how shitty a lot of parks are. Whether we're just talking idiots who let their dogs crap all over them to thugs hanging out causing trouble.
Well let's see: when I was a little boy we went unsupervised all the time. Violent crime has gone steadily down since I was a kid so why are people are more afraid of letting their kids play unsupervised? Just because we've been trained to fear danger more and more doesn't mean that we or our kids are actually in more danger.
And also, one of the perks of living in a neighbourhood where the families are acquainted with each other is that even when your eyes are not on your kid, other neighbours' eyes are, and if something goes wrong you can count on them to help your kid if needed (with the understanding that you do the same for your neighbour's kid). I'm not a Christian by any means, but "love thy neighbour" was a good practical piece of advice.
The people using the parks will act much less like assholes if they know each other. This is what community is all about. People behave better to those they know than those they don't know. It's the Golden Rule in action.
As for the condition of your parks, might I suggest a modest tax increase, sufficient to allow the municipality to maintain the parks? And failing that, a neighbourhood park maintenance co-op group?
Using your shitty parks as an excuse not to get to know your neighbours, when getting to know them is the best, cheapest solution to your shitty-park problem betrays a shocking lack of reasoning on your part. And yet you're not stupid (presumably). Have we come so far in our hyper-individualistic culture that we can't even see neighbourliness as a possible solution to our problems? Even the problems that were created by hyper-individualism in the first place?
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I hate to intrude on your little fantasy with reality but there are plenty of public parks in the USA. When folks are referencing 6 acre plots of land for their home they are generally in rural areas. Such areas commonly have natural fields, woods, etc nearby so public parks are less of a necessity.
Perhaps you should consider that many people outside the US are misinformed regarding life in the US, just as many in the US are misinformed about life outside the US.
This is a fantastic example of how Americans take a simple problem and absolutely fuck up the solution.
So these Americans want to partake in some outdoor activity that requires a bit of open grass. Their solution? Buy a 6 acre lawn, pay property taxes on this land, buy a lawnmower, buy fuel for the lawnmower, buy fertilizer for the lawn, and waste hours each week mowing the lawn. Even if they pay somebody to maintain it for them, it's still a huge waste of money, time, and effort.
I didn't read anywhere in the poster's comments that he bought 6 acres for outdoor activity. He has 6 acres; he wants help in finding the best way to mow it. Would you rather he bought the land and not maintained it? Then you would be griping about how lazy Americans are.
What do people in sensible countries do? They build parks, and everybody in the vicinity contributes a small amount of money towards its upkeep, without the burden falling directly on their shoulders. They can go use it whenever they want, and such parks are large enough that thousands of people can partake in all sorts of sports or other activities at the same time, from barbecuing, to playing catch, to even playing golf, without interfering with one another.
In some parts of America, 6 acres is cheap. Not in New York or San Francisco where land is at a premium but places like Montana, Texas, etc, you can buy 6 acres easily. So you would rather have the person donate 6 acres as a park instead of owning land as he might wish? For some sparsely populated areas of the country, there's little point in donating 6 acres to a park because no one is going to use it. As a comparison, France has about 116 persons /sq km. Texas has 37.5 persons /sq km and Montana has 2.5 persons /sq km.
Frankly you don't know his situation but somehow you know what the "sensible" thing to do is in this situation. He made a choice to own land; he made a choice to upkeep as he sees fit. I think it's downright arrogant for you to tell him how to handle his property.
Well, there's spam egg sausage and spam, that's not got much spam in it.
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. 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.
I don't see what aspect of his claim was irrational. That's the reality of public places these days, and frankly it was that bad when the economy was in good shape, I dread seeing what it will be like in a year or two.
"Who is the Journal of Quantum Physics going to believe?" --Stephen Hawking
I'd rather not have neighbors. If I can see them, they're too close. In fact, if I can see them through a rifle scope, they're too close.
Of course, because if you can see them through a scope, it means they can see you the same way. That's why everybody should have large enough security perimeter around their home, as well as over-the-horizon offensive capability. Why put one's family into unnecessary risk which can be easily avoided?