Slashdot Mirror
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.
In America, you mow the lawn!
...is your friend!
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
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.
When the foot seeks the place of the head, the line is crossed. Know your place. Keep your place. Be a shoe.
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 mowed grass and never on the soon-to-be-mowed grass.
I wonder what effect this would have on the system.
No one ever had to evacuate a city because the solar panels broke!
The 'optimal' solution has the mower finishing in the middle of the lawn, which is usually not where you want to leave it parked.
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.
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.
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 that.
"If you make people think they're thinking, they'll love you; But if you really make them think, they'll hate you." - DM
What? Floodfill ftw.
Seven puppies were harmed during the making of this post.
When America collapses, people with 6-acre lawns will soon find very persuasive arguments that they should be sharing their land with the less lucky.
...for a tractor-haybine combination with an 80" swath and a 20' turning radius.
Warning: this article may contain humor, sarcasm, parody, and perhaps even irony. Read at your own risk.
It's a case of the Traveling Salesman Problem http://en.wikipedia.org/wiki/Travelling_salesman_problem On the one hand it is the special case with Euclidean 2d distances (more or less, depending on hills). But also, it is the special case where all point-point distances are equal, depending on what exactly you meant by 'grid', which is called Graphical TSP. "Computer search algorithms" is a little bit of a weasel word... but as far as I know TSP instances are exactly solved in practice on moderate-size instances using integer linear programming (ILP) techniques, the work of Bill Cook and co-authors is likely useful. From a theoretical (but not so important) perspective: the 2d Euclidean problem is NP-complete but admits an "approximation scheme." I am not sure about the doubly special case you present, but my gut feeling would be it's also NP-complete.
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.
The quickest method I've found for mowing my lawn is to hire someone else. It literally takes me only 3 to 5 minutes to write the check, and there is no geometry involved.
mowing 6 acres with a lawn mower? that should be a crime in its own. invest in 20-25 goats and a fence. not sheep. sheep are stupid and they smell. goats are smart. you'll find that looking at them is really relaxing. they'll also keep your lawn at golf-course grade length. price? sell your john deere and you should have enough to buy the lot.
It could be described as a variant of the Traveling Salesman problem, where each node is a mower-sized swath of grass and your object is to visit very node, returning to the starting one...
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"?
Yeah, but then if Americans built parks, they'd have to get to know their neighbours.
Rule of Slashdot #0: You and people like you are not representative of the larger population. - A.C.
The best solution: don't mow it.
Why the hell do you have 6 acres of grass? Plant some trees for christs sake.
DRTFA but assuming the circles are laid out to cover the entire lawn then it's an unconstrained 'Travelling salesman' problem, if not then it's constrained TS problem.
And did you exchange a walk on part in the war for a lead role in a cage? - Pink Floyd.
Anyone else looked at these patterns and thought "CNC milling"?
I agree that public parks are a much better solution than large yards. Part of the problem, however, is that when you're choosing a new home, access to public recreation can be just one of many factors to choose from. It might be more important to me to be close to public transportation, the grocery store, my children's school, my job, etc. Since Americans have previously NOT chosen to build spacious parks everywhere, that might leave me without adequate access. I might, however, be able to find a larger lot of land meeting my other criteria. It's not an ideal solution but the alternative is choosing a smaller lot and then immediately start rallying for land to be purchased or rededicated for a park. Even if that works, though, I wouldn't be able to enjoy a large, open space for a few years.
Virtue finds and chooses the mean.
Aristotle, Ethica Nichomachea
Even if the zero-turn is very impressive it is not the mathematical model. When turning the width of the combined blades and the average speed will be reduced, making each turn sub optimal. As a boy I used a(n unsafe, think of the children) mower on a rope turning perfect spirals with a distant supervisor and finished off the in betweens in direct supervision mode.
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.
My goats and sheep mow the lawn while I drink 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.
1. Buy beer. 2. Start mower. 3. Open beer. 4. Start "mowing and drinking" 5. Be amazed as time flies and the grass gets cut
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 apply for a circular lawn, is obviously a spiral pattern. For an irregular shape lawn one obvious heuristic would be to decompose the lawn area into a set of various sized circular blobs and do each of these in an expanding spiral pattern, then onto the next.
A similar heuristic would be to start by spiraling inwards around the entire lawn, and "recurse" into smaller areas when they (via having narrow "neck" entrances) are about to be cut off from the main spiral - specifically when the neck has been reduced to two tractor widths wide (one path in, one path out). In fact, this may well be the optimum strategy, particularly as it takes advantage of the specific problem topology rather than being a generic traveling salesman heuristic.
Please send cash to SpinyNorman c/o Slashdot if this makes you money!
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
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 pass the mower over, this would be a wall, tree, mailbox, or valuable planting. A soft edge would be a material that you can pas the mower over, this would be a driveway, patio, and certain planting bed edges.
I find that the best balance in real world finish and speed is when I mow 2 laps around the edges so I can turn with my old school garden tractor, then to create boxes that move across the yard. This allows for maximum velocity to be maintained, long orderly cuts which provide better finished appearance than irregular turnings, and adequate turning radius for my equipment, which is very different from the OP's.
Since my machine has a top mowing speed of about 5mph, and a 46 inch cut, this takes just under 1 hour per acre, for me. About 3 hours for 4 acres on a good day mowing neat growth. If my meadow has sat for several weeks, it will take 3 hours to mow and mulch that area alone, a common problem in the spring since it is river floodplain and will be under water a fair number of times.
With a zero turn cutting a 60 inch path at up to 9.5mph going in straight lines will outweigh minimizing recut because slowing down to turn reduces the area cut per unit time more significantly than cutting a small area a second or third time. That is a muscle car of a lawnmower the OP is using. He should take a whole lot less time to mow than he reports, probably because of all the turns he is taking.
Phil
Laugh, it's good for you!
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.
You are welcome on my lawn.
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.
Some of us mow the lawn to make it look good. If it takes a little longer to leave a nice pattern, that's what headphones and a good collection of MP3s are for.
(name withheld by request)
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.
Buy a goat.
Extra Added Advantage: At your convenience, Lawnmower Curry.
I've calculated my velocity with such exquisite precision that I have no idea where I am.
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.
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.
Why are we still growing grass that needs to be cut?
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.
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.
Oh, wait. Parks are probably too "socialist" at best, or "communist" at worst, for most Americans.
Having your own private 6 acre lawn is no different than companies creating their own private clouds. What if you had a family reunion and wanted to play a game of football or soccer? If you went to a park, there's a good chance you couldn't find enough space for your entire reunion plus whatever games you wanted to play.
There's nothing wrong with having a big yard.
Idiot. There are plenty of parks in the United States. Many people have large lawns because they like to garden, they like to have plenty of room between themselves and their neighbors, they want to have animals, or a pool, or any number of other reasons. Not everyone wants to live in an apartment complex or in a row of attached townhouses.
My parents live on 3 acres of a wooded hillside. I live in a suburban beighborhood on a 6,000 square foot lot.
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.
Still waiting on Serviscope_minor to wake up to fucking reality and realize that Jessica Price isn't going to fuck him.
I have a self propelled lawnmower so as long as i hold a lever an just calmly walk behind it mowing is not an issue.
That is until i try to turn which requires me to use a comparatively large amount of force, it also takes a lot of time as you sort of have to stop, turn then go.
And this is where i think the algorithm is flawed, as it doesn't consider turns but only area covered.
I find just keeping to the outer edge of the unmoved area works best while straightening out curves as best as possible, then just mowing it sector by sector (as ours is not exactly convex in shape)
- "There is nothing quite like an ineffective solution to an nonexistant problem"
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.
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.
sed -e 's/Chuck Norris/Rajnikant/g' joke > fact
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
I've got a better tip. Why not just get a few sheep and let them mow (and manure) the lawn? Better yet, make sure the animals are ewes and milk them. I'm a professional cheesemaker, so my choice for the product will be obvious, but ewe's milk is usually high in fat, so you can also have amazing cream (or butter).
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.
Why did you immediately assume less lawn meant less land? The area around the lawn could still be yours, but, for example, be used for growing useful stuff. (Food, biofuel...)
FRA: STFU GTFO
Wouldn't finding a 6 acre lot close to public transportation be a lot harder than finding a small lot close to public transportation *and* a public park? Public transport usually doesn't serve such sparsely populated areas very frequently... Same goes for grocery store, school and jobs, all are much more likely to be within walking distance of a densely populated area with a park nearby than in a sparsely populated area with 6 acre lots.
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. ;-)
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.
I come here for the love
Exactly. Bragging about how he enjoys to mow his lawn every single week with his John Deere 757 zero-turn makes makes him a yuppie and a complete tool. He obviously needs to move out of suburbia.
I have about three acres of actual yard. Not lawn, but yard. It's a mix-match of various grasses and is anything but flat. The other four acres of my property is hardwoods, and beyond that is nationally protected forest. It has also been my goal to outright eliminate the lawn immediately surrounding my house. I've taken a somewhat similar path by planting trees and laying down stone walkways everywhere. I've been pleased to see that while the grass doesn't fair very well, I have a lot of soft moss coming up instead. It's been too hot to really plant any younglings this year, but I'm hoping to get a bunch of ferns in next spring. They're provide even better ground cover for the moss, and also make the ground a little more varied and interesting.
"He who can destroy a thing, controls a thing." --Paul Atreides, Dune
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.
A perfect mowing mows at every vertex exactly once. The perfect mowing exists if there is a hamiltonian path in the triangular grid graph on the lawn. In general the hamiltonian path problem is NP-complete even on the triangular grid graph. However [1] states:
A hamiltonian cycle in a connected, locally connected triangular grid graph (not isomorphic to D) can be found in polynomial time.
D is the linearly-convex hull of the Star of David. A polynomial time algorithm which is not exactly simple is available in [2]. It can be applied to solid grid graphs.
This approximately means if your lawn is not shaped like the Star of David and does not enclose any trees, bushes or ponds, you can implement the algorithm from [2] and get an perfect mowing path in polynomial time.
[1] Gordon, Orlovich, Werner. COMPLEXITY OF THE HAMILTONIAN CYCLE PROBLEM IN TRIANGULAR GRID GRAPHS
[2] W. Lenhart and C. Umans. Hamiltonian Cycles in Solid Grid Graphs
First of all, the vast majority of americans live crunched together like sardines, we know our neighbors. Second, we have parks, but parks have a lot of rules and regulations about how they are to be used, what is and is not allowed, etc. Some of the rules are good, some are set up by the hive-mind's most insidious members (the people who block efforts to remove alchohol purchase on sunday laws, and push for violence in video games laws). If you want to fly remote controlled planes, for example, you're all but out of luck in most parks, even though a good many of them do not have enough people for this to actually be dangerous.
Finally, it is the preference of many not to have to live so close to their neighbors that they can see their nose hair. Most of us cannot afford to do so: the places were there is land are not places where there are jobs. But that does not reduce our desire to make it happen in our lifetime.
Travelling Salesman Problem, sorry. The most efficient path is learning to chip better ;) Oh, and start gathering the ones which are far away first, so you don't have to carry a full bucket around to get them.
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?
Rule of Slashdot #0: You and people like you are not representative of the larger population. - A.C.
im[sic] using my scythe
and havent found the perfect way to mow my lawn.... any suggestions?
Definitely not. If you can use a scythe without fucking up your back, you are a hero and need no help from a bunch of nerds. However, if you insist, the scythe effectively eliminates the travelling salesman problem, since there are (theoretically) zero redundant passes, since anyone who really knows how to use the instrument knows that you only cut once on a given area.
Unless you live in the US Midwest, where grasslands are natural habitat, you should plant some trees on that six acre plot of earth.
In most of the rest of the country, the natural habitat is NOT grassland, its either woodlands, desert, or wetlands.
If you simply must have grass, don't mow it a large portion of the six acres, let it mature into a semi-natural meadow.
If the Government becomes a lawbreaker, it breeds contempt for law;
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.
Goats.
"If any question why we died, Tell them because our fathers lied."
Having your own private 6 acre lawn is no different than companies creating their own private clouds. Plus, what if you had a family reunion and wanted to play a game of football or soccer? If you went to a park, there's a good chance you couldn't find enough space for your entire reunion plus whatever games you wanted to play. There's nothing wrong with having a big yard.
But this is slashdot so the geek solution would be more appropriate. Robotics. A "roomba" lawn mower. :-)
And in some places, "their neighbors" in the parks are crazy homeless guys (who may or may not be doing drugs or litter playgrounds with biohazard sharps).
The World Wide Web is dying. Soon, we shall have only the Internet.
I've been reading rational and articulated posts defending people's big lawns for about 10 minutes now... Then I come across this.
You sir summed it up better than everyone else and make me wish I had mod points.
Mind the frickin' laser...
I spent my teenage years mowing my parents 3 acre lawn with a tractor and gang mower and always trying to figure out the optimum method. My favorite was the random swaths, but maybe not the most efficient. I think that got me interested in computer programming, algorithms, and graphic design. Remember, Philo Farnsworth invented television by looking at corn fields.
While I appreciate the mathematics and the problem involved, why the heck does the guy have a lawn so large that it take a full three hours to cut? This feels more like a "Hey look guys, I can afford a lawn that is so big, it take three hours to cut!" more than any sort of real request for help with the math.
Oh... for some mod points right now.
In fact, if I can see them through a rifle scope, they're too close.
For them, maybe. :)
What's that? You have an irrational fear of public places that you wish to transfer to your children?
Who the hell has time to get up in the morning and mow the fucking six-acre lawn? Don't these people have jobs? Night life? Anything?
Welcome to the joy of Daylight Saving Time.
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.
To minimize the amount of time spent mowing, get someone else to do it, e.g. hire some neighbor's kid.
To minimize the amount of resources spent mowing, forego mowing entirely.
I have had both and its much nicer to sit on your back porch and not have to deal with the 100 shit head kids on the entire block all fighting over what equates to an acre across everyones back yard, or listening to your next door neighbor take a dump
I presume your in a city in a city, you guys would rather be touching total strangers that haven't bathed in a month, all day every day for your entire lives while living in a shoe-box that cost more than all than you have listed, its amazing
its not socialist, or communist, we have tons of parks in America, we just don't worship them as the last chunk of unpaved land in our entire country as you obviously do
Would this be similar to a polygon flood fill algorithm?
http://en.wikipedia.org/wiki/Flood_fill
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.
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.
Oh, wait. Parks are probably too "socialist" at best, or "communist" at worst, for most Americans.
^ jealous of our cheap land
"I have a 6 acre lawn and I'm trying to take less than 3 hours to mow it."
A friend has a self-propelled mower tethered by a rope to a pipe located in the middle of his yard. The mower propels forward and slowly spirals in as the rope wraps around the pipe. He drinks a beer and enjoys mowing his lawn too.
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.
"Trespassers will be shot!"
"Survivors will be shot again!"
Vintage computer adverts: http://www.vintageadbrowser.com/computers-and-software-ads
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
Plant trees on the 6 acres and you will have less lawn to worry about. And less visibility towards your neighbors.
If you are worried about trespassers - make sure that you have holly and poison ivy growing. Anyone stupid enough to trespass will get punished in an appropriate way.
If you just want something nice - plant some beech, oak, birch and pine. Mixed growth usually looks better, and it is great to dampen storm winds.
If builders built buildings the way programmers wrote programs, then the first woodpecker would destroy civilization.
Junkies and drug dealers can be gotten rid of by families actually using the parks; nobody wants to do a deal or shoot up where there are a dozen respectable folk with cell phones hanging around.
As for the homeless and crazy, maybe it's not such a bad thing to expose your children to these people, even encourage your kids to talk to them; might make your kids wonder why the richest country in the world even has homeless and crazy people wandering around. It might even make them wonder if there's anything they can do about it.
Rule of Slashdot #0: You and people like you are not representative of the larger population. - A.C.
The economy was good, but the poor were still getting poorer, and the middle class were getting increasingly more self-centred and scared of their own shadows.
But don't worry, in a year or two, if we don't reverse the trend toward a more community-oriented attitude, you might even get to experience what it's like to have to sleep in a park yourself, along with me and the rest of the middle class. See you there.
Rule of Slashdot #0: You and people like you are not representative of the larger population. - A.C.
I see goats all the time here taking down high grass. This works well in CA because you only need to take the grass down for the dry season.
In climates with year-round rain, the goats would have to be trucked in more often.
If you don't want barnyard animals running around, my other suggestion is "plant trees". Mowing them is much less frequent, and much more profitable.
For all intensive purposes, "whom" is no longer a word. That begs the question, "who cares"?
Sorry, I can't hear you from way over there!
For large sets, this will be our guide even unto death, for the LORD will work for each type of data it is applied to...
+1 FTW!
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?
I think the gist of the grandparent post was that if everybody didn't waste so much space, time, and money on individual huge yards then the park wouldn't have to be a couple miles away, but rather a couple blocks away (or closer). Where I live the lots are 1/4 acre, which is still plenty big for barbecuing, gardening, and keeping outdoor pets (most of my neighbors have dogs, and one even has chickens), and I've got 3 pretty big parks within a 1-mile radius.
Also, my neighborhood has a "parents network" that organizes playgroups; there's a very decent chance that the park would have a parent on premises.
"[Regarding the 'cloud,'] ownership was what made America different than Russia." -- Woz
Amen, brother.
Yes, I would. If all he's doing is wasting resources maintaining it, it's better off returning to its natural state. At least then it would become a habitat for wildlife (something we need definitely need more of). Plus, unless he lives in an arid area, it would eventually produce timber.
"[Regarding the 'cloud,'] ownership was what made America different than Russia." -- Woz
As a kid I enjoyed trying all different kinds of patterns:
- back & forth stripes (at various angles)
- going round the outside & spiraling in
- starting in a circle in the middle & spiraling out.
- randomly cruising around...
Usually, though, it seemed like spiraling in was the fastest way. There was always the issue of pocket areas that get cut off from the rest, and exactly how do you finish up that irregular part left in the middle of each area (without a zero-turn radius mower).
Could you share the goats with your neighbors?
Uh, "if it looks roughly mouse-shaped according to my infra-red sensitive pit, eat it"? --Chris Burke 09-08-10
I'm not buying your strawman argument. In reality either the park is much bigger than a soccer field or there are a whole bunch of tiny parks and you could just go to the next one a block over to find space to play frisbee.
If your parks really are tiny and far apart, then it means you live in an utter shithole that's worse even than US cities at funding parks.
"[Regarding the 'cloud,'] ownership was what made America different than Russia." -- Woz
Optimal satisfaction with your zero-radius mower, and the fact that you bought it, requires that the number of U-turns be maximized.
Parks??? You mean like Central Park (NYC)
NOW GET OFF MY LAWN !!!
In typical American parks you can expect to be harassed by the police for such things as being there at the wrong time of day, walking your dog in the wrong area, parking during the wrong hours, or straight up arrested for bringing a beer to your barbecue. Not to mention the random extended closures, or the super weird people who make up their own park rules and go to crazy lengths to try to enforce them.
I'm not saying that I don't enjoy our parks because I do, but they are not a perfect replacement for having your own space.
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.
Amen.
I really need to get me one of those.
But I suppose I'd need my own house first though, huh?
Umm, what? You can just reserve the park for a reunion, including the playing fields. I don't even think it costs money, at least not significant money, for public parks.
Since it uses electricity, it will also be better for your environment compared to your 2 stroke mower that doubles as a gasoline evaporator. They are available from several companies in several prices.
If you got like the poster have to spend $3500 + gas and maintenance + salary to get your lawn mowed, there are really big ones available that will probably get ROI in 2-3 years. There's some mathematics for you.
Custom electronics and digital signage for your business: www.evcircuits.com
No it doesn't Mr AC; I'm Canadian.
Of course, those same crazy Americans insist on getting married and sharing their garden with their spouse and possibly children, at an immediate 1/2 loss in personal area, shrinking to 1/4 over time on average.
If you ask me, there's something weird going on...
So these Americans want to partake in some outdoor activity that requires a bit of open grass. Their solution? Buy a 6 acre lawn
If your kids want to spend all day playing outside, as they should, and the nearest park is 6 blocks away across some busy intersections in a neighborhood not considered particularly safe for little kids to wander alone, a back yard is a great idea. 6 acre's is overkill obviously, but the poster never said it was for "partaking in some outdoor activity", it's just land he has to maintain. Even if you don't really use the area, you still have keep the grass fairly short otherwise the snakes move in and it becomes a problem in bushfire season.
Parks are great, but they aren't the answer to everyone's problems.
Yes, I would. If all he's doing is wasting resources maintaining it, it's better off returning to its natural state.
And how do you know what the natural state of his land? Maybe grassland is the natural state of his area. As for wasting resources, maybe maintaining a lawn is his hobby so it's not wasting resources to him.
At least then it would become a habitat for wildlife (something we need definitely need more of). Plus, unless he lives in an arid area, it would eventually produce timber.
Grasslands produce little timber. Some swamplands produce no timber. Besides I don't know where the poster lives. If he lives near a national park, there is plenty of land around for wildlife.
Well, there's spam egg sausage and spam, that's not got much spam in it.
Maybe you could reserve the park. But the rest of my statement still applies. Besides, the original post didn't say where the person was from. Have you ever been to the Midwest? I come from an area where it's commonplace for people to have several acres... that's what happens when there's a low population density, public parks or not.
TFA says "Clearly what is optimal for a lazy Maths Master is to push the lawnmower the shortest distance possible." and goes on from there.
While an interesting NP problem in itself, the things you're more likely to be trying to minimize are time, fuel, cost. Hence, turning matters significantly.
I know, I've spent lots of time thinking on this but through experience come to the conclusion that using long parallel strips as often as possible is easiest on mind and body because, finally, acceleration is also quite important.
You agree with me.
LawnBott LB3510 costs about $4,000. Assuming your time is worth $50/hour, at 3 hours a week that means you are spending $150 a week. You make your money back in 6 months.
excitingthingstodo.blogspot.com
who cares how long it takes......
The math is fun here, thanks for the enjoyable distraction. Do you have a ZT or do you have to compensate for turning radii as well?
If you don't enjoy mowing, but like the big outdoor space there are some alternatives.
This year tilled up and reseeded the "grass" around my gardens with dwarf dutch clover. Best idea ever. Before It was a huge PITA to string-trim between the raised beds. I've only mowed it twice this year, and it looks great. In the spring we're going to replace the rest of the lawn with a dwarf clover and dwarf fescue mix.
Now I can spend my mowing time in the garden instead.
I live adjacent to a grass runway, and part of my responsibilities include mowing my section of the runway. Some of my neighbors are particular and for safety reasons demand the mowing occur lengthwise along the runway so you can see an airplane coming at you and get out of the way. This is a constraint that could be mitigated if I had 360 degree vision.
How do the authors address constraints like this?
Ross Youngblood
It's probably too late to get modded up so you can see this, but:
The technical term you are looking for is "coverage path planning", and there are well known algorithms to solve it efficiently such as Choset's Boustrophedon. I don't think any of them are optimal for a nontrivial shape, but they will probably beat most human heuristics.
In addition to lawns, this is important for machining (material removal), de-mining (completeness is essential, but overlap is expensive), and large floor cleaning.
http://scholar.google.com/scholar?q=coverage+path+planning
Goats are best, Sheep eat around stubborn grasses; horses make loblollys and cows are just trouble. Six acres is stupid huge for a lawn. Let 5 of it go to field and hay the thing. For the fastest time, mow the same pattern every single time; this is bad for your lawn. You should change up the pattern once in a while.
Dynamic Programming (http://en.wikipedia.org/wiki/Dynamic_programming) find the optimal path in any problem like this.
Assuming you set up the problem well, the DP solution is indisputably the best (proven mathematically).
It requires the problem to be modelled in discrete time and space i believe, but it looks like they're doing something like that already.
Land is a finite resource, and it's division and use is the interest of everyone.
larger or more regularly (e.g. rectangular) shaped fields. It essentially guarantees an overlap of approx 13% (1-sqrt(3)/2) of the diameter of the lawnmower cutting circle, merely in an effort to reduce the overlap in the number of turns. If the same field was made larger, the 13% cost would far outweigh the gains made in reducing the turn overlap. A rectangular path would easily eliminate the 13% waste and then the problem could be reduced to minimizing the number of turns.
Yes, I totally agree. I used to mow a mere 1.5 acres and it was all about minimizing turns of the 90 or 180 degree variety. I once tried to start in the middle and just mow a big spiral. It started out great except the mostly rectangular lawn then had some problematic corners to attend to and that was just a pain in the ass because if you wanted to clear a corner you suddenly had turns > 90 degrees and on my tractor that was really hard.
Being a hockey player I modeled my technique on what a zambonie does on the ice. the zamboni doesn't really like turns - it's on freaking ice- but it can do a 90 degree turn ok but never a 180. So it first goes down the middle and then across one end and up the far side, across the far end and up the middle again, but one-zamboni width over. Trying to guess the exact middle of your lawn is a bit tough and your efficiency largely rested on that initial choice.
At any rate, some overlap in lawn mowing is always preferably to never having to stop and turn around.
spiral doesn't work. i tried it. imagine a square lawn, which most are, or at least rectangular. a spiral starts out great. you carve a big circle out of the square quickly. now you have 4 corners to mow and the remaining shape is such that you have some really difficult turns to make as you try to clear them. you end up having to go backward and forward in your tractor and it just sucks.
i found a modified zamboni approach to work very well.
It may be the most optimal solution in terms of not covering the same place twice, but I'm not sure it is the quickest. I'm not bothered if I go over the same area twice, so long as I get the job done as fast as possible - making a turn takes time and there are plenty of those in the proposed solution. It may still be the quickest as some of those turns are quite small angles, but I think a 'turn penalty' is required to truly find the quickest solution.
Regarding search and rescue, the object there is not to cover the whole search area as fast as possible. It is to cover the most likely locations for whatever is being sought as quickly as possible, and then cover the more unlikely places. For a static location that might mean starting in the middle and tracing a spiral. However, for someone falling overboard at sea, retracing the ship's path at close to it would be a higher priority than a simple spiral from any point.
I've got a fair sized lawn that takes me 6 hours to mow with a push mower so I think about this a lot... What I like to do is mow the edges and around obstacles first getting unmown rectangles, rounding off the sides of the rectangles, and then going around the rounded shape non-stop. The rounding off of the edges saves a lot of time that would otherwise be spent stopping and turning around and lets me just go continuously.
Plant wild flowers, or no mow grass(yes it exists, and works great)! -You will save $ on gas -pollute a lot less! (mowers are some of the least efficient motors around) -Make more fresh air -looks a lot better -get some nice birds, butterflies and such. -Gain 3 hours a week back. You can do a ton with that. And If you must mow, make a solar powered lawnmower with a server battery and starter mower like I did!
In most places, an acre is about the largest area that anybody actually mows. Beyond about an acre, you don't have a lot; you have a field. Thus, if you have six acres, you don't have a lawnmower; you have a tractor. There are exceptions, but it's a good rule of thumb.
I you own a parcel that's over an acre, chances are, you mow somewhere between a third of an acre and an acre around your house, and you either let the rest of it grow naturally or you lease it out to farmers (depending largely on whether it is arable land).
That said, for the rare individual who actually wants a six acre lawn, there are lawn tractors that can mow it in about forty-five minutes as long as you don't have very many trees.
Check out my sci-fi/humor trilogy at PatriotsBooks.
I think a quarter acre might be a little on the small side, depending on the size of your home. I've always found a third of an acre to be just about right for a comfortable lot, assuming that your lot backs up against a forest, a corn field, or some similar piece of largely uninhabited space.
If you're backed up against another home, I think I'd want at least twice that, with all of the extra space added in the form of additional depth behind the house. That way you can have reasonable privacy without having too many trees to mow around. Doing this doesn't reduce the sense of community, but still gives you a sense of privacy.
In fact, the ideal situation would be to alternate between streets and long park areas between subsequent rows of houses. That way, you aren't far from your neighbors, your kids don't have to go very far away to play, and there isn't a house immediately behind yours. Take your pick whether it's part of the lots or not, based on whether you'd rather mow it yourself or pay the HOA to pay someone else to mow it en masse.
Check out my sci-fi/humor trilogy at PatriotsBooks.
I don't know the natural state of his land. The point is that whatever it is, it requires fewer resources to maintain than a manicured lawn (and provides more habitat, etc.). Lawns have to be mowed; grasslands do not.
A lawn is more of a desert in terms of biodiversity than an actual desert is!
Hence, "unless he lives in an arid area." Grasslands are arid.
Some swamplands produce no timber.
If he lived in a swamp, we wouldn't be arguing about what he should do with his damn lawn, now would we?!
There's no such thing as "plenty of land around for wildlife" anywhere on Earth.
"[Regarding the 'cloud,'] ownership was what made America different than Russia." -- Woz
What is "quite"? :P
Ant(Dude) @ Quality Foraged Links (AQFL.net) & The Ant Farm (antfarm.ma.cx / antfarm.home.dhs.org).
Ahhh, Quite is the Babylonian god of being quiet. Spell check failed me again. Thanks for catching that.
Hehe. :)
Ant(Dude) @ Quality Foraged Links (AQFL.net) & The Ant Farm (antfarm.ma.cx / antfarm.home.dhs.org).
I don't know the natural state of his land. The point is that whatever it is, it requires fewer resources to maintain than a manicured lawn (and provides more habitat, etc.). Lawns have to be mowed; grasslands do not.
Do you actually live in a grassland or are you just speculating? You have to manicure grasslands otherwise they overrun everything. Unless the guy wants to be helicoptered into his house, the has to keep at least a driveway and the area around his house clear. Also in the advent of a wildfire, his house is pretty much destroyed if he doesn't keep a large enough perimeter. With any natural vegetation, to live in a residence which is not natural, there will be maintain unless he doesn't care his house is slowly destroyed by nature.
If he lived in a swamp, we wouldn't be arguing about what he should do with his damn lawn, now would we?!
Maybe his developer turned swampland into a house in the middle of nowhere. But the area around him may still be swampland.
There's no such thing as "plenty of land around for wildlife" anywhere on Earth.
You seem to advocate taking away this man's right to do with his land as he sees fit in order to pursue a course of land usage you would prefer. Bear in mind, he's doing nothing illegal nor unlawful. But he is doing something you personally disagree with. By extension why don't we force everyone out of the cities and level them to nothingness because it seems most of the world disagrees with you.
Well, there's spam egg sausage and spam, that's not got much spam in it.
I'm not a Christian by any means, but "love thy neighbour" was a good practical piece of advice.
"Fences make good neighbors" is a better piece of advice.
People behave better to those they know than those they don't know. It's the Golden Rule in action.
There are two versions of the golden rule. The first is, "Treat others as you would expect to be treated". This has implications depending on whether or not you wish to be left alone. The other golden rule is a snarky version, "He who has the gold, makes the rules." YOU DO NOT KNOW WHAT THE GOLDEN RULE IS!
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?
Now we have two solutions to the problem,
1) Buy the house with the large lot (BTW, these lots are usually established and already zoned and not an act of social planning/forsight/SimCity on the part of the eventual owner)
2) Fight/change city hall
Given that changing how a city works means I have to deal with people who don't even know what the golden rule is (!), I know which solution I would prefer.
I've always mowed my lawn in a spiral that starts at the outer edge and works in a spiral towards the center. It's irregularly shaped, but there is little overlap.
I see the article proposes a solution which cuts (pun intended) about 1-2% of the time off of a spiral. When I'm cutting my lawn, I want to get it done, not do complex math during the process. A contracting spiral is obvious and effortless.
Wow, nothing but strawman arguments!
I never said he shouldn't maintain his driveway and (if applicable) fire break. It's highly unlikely that the driveway and firebreak take up the entire (or even a significant portion of) the six acres he was talking about.
It's still irrelevant; if the area was swampland then he wouldn't be mowing it. If it used to be swampland and got filled in to make the lawn, then leaving it natural would turn it into whatever sort of thing the uplands adjacent to the swamp are made out of (in south Georgia, for example, that would be pine forest).
Go fuck yourself, because I did not advocate anything of the sort and I resent you for your filthy lie claiming I did! All I said was that he (and the land, and the environment) would be better off if he left it alone instead of wasting his time and other resources mowing it for no good reason. I never said anything about forcing him to do so; he can be as stupid as he wants to be.
"[Regarding the 'cloud,'] ownership was what made America different than Russia." -- Woz
A quarter acre is on the small side for the suburbs (with HOAs and whatnot), maybe, but I live in the city in an area with a rectangular street grid, designed before all the post-WW2 suburbanization.
I think the main issue regarding privacy is not the distance between houses, but rather the elevation of the neighbor's windows/deck/yard in relation to the height of the fence. The house behind mine comes within 10 feet of my fence (and within 40 feet of my house... I have more side yard than back yard), but I have no problem with that because that house doesn't have any rear-facing windows. My privacy problems come from the houses on either side: the house on my right is 2 stories tall and has windows that overlook my yard, and the fence on my left is at the bottom of a hill so those neighbors can look up the hill over it. If the land were flat and the house on my right was a bungalow like the rest of the houses in the neighborhood, neither problem would exist.
"[Regarding the 'cloud,'] ownership was what made America different than Russia." -- Woz
If you really care about privacy while you are outdoors, this is what tree rows are for. :-)
Check out my sci-fi/humor trilogy at PatriotsBooks.
Go fuck yourself, because I did not advocate anything of the sort and I resent you for your filthy lie claiming I did! All I said was that he (and the land, and the environment) would be better off if he left it alone instead of wasting his time and other resources mowing it for no good reason. I never said anything about forcing him to do so; he can be as stupid as he wants to be.
What you seem to be forgetting is this is not your land. If you have 6 acres do with it as you wish. He wants to maintain a huge lawn. Yet in your judgment, that is wasting resources; in your judgement he's being stupid. Who the hell are you to judge him? Some people waste resources by playing video games for virtual currency. Some people waste resources by trimming tiny bushes into smaller bushes. I'm not the one who is telling him what he's wasting resources.
Well, there's spam egg sausage and spam, that's not got much spam in it.