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Old 25th November 2017, 08:01 PM   #1
Samson
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The grasshopper conundrum

A seemingly simple mathematical question has complex possibilities. I have nothing really to add, but the animation of the computer solving the problem is remarkable.

https://phys.org/news/2017-11-lawn-grasshoppers.html
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Old 25th November 2017, 08:19 PM   #2
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1. Why would I want grasshoppers on my lawn?
2. Do they know the buggers have wings?
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Old 25th November 2017, 10:34 PM   #3
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Now I have a headache. Why would it ever be anything other than a disk?
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Old 25th November 2017, 11:09 PM   #4
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Originally Posted by Loss Leader View Post
Now I have a headache. Why would it ever be anything other than a disk?
The first pattern, with short jumps, does have a disc, it's just it's surrounded by a wavy pattern that makes it look more like a doily. I can kind of see why that would be more forgiving than a perfect disc.

The stripes for long-haul grasshoppers seemed more mysterious to me.
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Old 25th November 2017, 11:14 PM   #5
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I feel like I'm missing information. Is this the probabilistic form for just one single jump? Is it for multiple jumps? Can the grasshopper choose to jump less than the maximum distance?

Every time I run my simulation, I get a donut. Of course, I'm running the simulation in a Krispy Kream so that might be at least part of the problem.
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Old 25th November 2017, 11:25 PM   #6
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Originally Posted by Minoosh View Post
The stripes for long-haul grasshoppers seemed more mysterious to me.
I think the idea is if your lawn area is "small" and the jump distance is "large", then a disc shape is inefficient because any jump larger than the diameter guarantees that the grasshopper lands off the lawn.

But if the spaces are discontinuous, then for the lucky grasshoppers that are pointed in the right direction, they can hop from one stripe to the next.
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Old 26th November 2017, 09:28 AM   #7
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Originally Posted by Loss Leader View Post
I feel like I'm missing information. Is this the probabilistic form for just one single jump? Is it for multiple jumps? Can the grasshopper choose to jump less than the maximum distance?

Every time I run my simulation, I get a donut. Of course, I'm running the simulation in a Krispy Kream so that might be at least part of the problem.
Yes, but it works for the Maximum Blood Sugar Conundrum.
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Old 26th November 2017, 09:48 AM   #8
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Originally Posted by Loss Leader View Post
I feel like I'm missing information. Is this the probabilistic form for just one single jump? Is it for multiple jumps? Can the grasshopper choose to jump less than the maximum distance?
The simulation progresses from what appears a uniform distribution to the resulting shape by many, many, many jump iterations. Based on the article "If it then jumps a certain distance", not 'average distance' or 'maximum distance', the jump distance appears fixed.



Quote:
Every time I run my simulation, I get a donut. Of course, I'm running the simulation in a Krispy Kream so that might be at least part of the problem.
That might make you more jumpy that the grasshoppers. What then is the optimal shape of the Krispy Kream to keep you inside?
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Old 26th November 2017, 11:47 AM   #9
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Originally Posted by Loss Leader View Post
I feel like I'm missing information. Is this the probabilistic form for just one single jump? Is it for multiple jumps? Can the grasshopper choose to jump less than the maximum distance?

Every time I run my simulation, I get a donut. Of course, I'm running the simulation in a Krispy Kream so that might be at least part of the problem.
I see it as the area is a constant.
The distance each individual grasshopper jumps is a constant.
Ther should be no minimum area for any hopper to land, eg backwards and forwards on a minimal width straight strip.
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Old 27th November 2017, 08:18 AM   #10
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Originally Posted by Loss Leader View Post
I feel like I'm missing information. Is this the probabilistic form for just one single jump? Is it for multiple jumps? Can the grasshopper choose to jump less than the maximum distance?

Every time I run my simulation, I get a donut. Of course, I'm running the simulation in a Krispy Kream so that might be at least part of the problem.
Isn't that Krispy Kreme? If not there may be a trademark problem....... For some reason I am thinking Millenially....
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Old 27th November 2017, 08:20 AM   #11
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Originally Posted by The Man View Post
The simulation progresses from what appears a uniform distribution to the resulting shape by many, many, many jump iterations. Based on the article "If it then jumps a certain distance", not 'average distance' or 'maximum distance', the jump distance appears fixed.





That might make you more jumpy that the grasshoppers. What then is the optimal shape of the Krispy Kream to keep you inside?
Krispy Kreme. THE dessert of my childhood!!!! And later!!

Last edited by fuelair; 27th November 2017 at 08:50 AM. Reason: dessert not desert!!!!!
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Old 27th November 2017, 10:49 AM   #12
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This might be like the Traveling Salesman problem in that it is theoretically solvable but not solvable in pratice. Perhaps the comment "... the shapes bear a resemblance to shapes seen in nature, including the contours of flowers, the patterns in seashells and the stripes on some animals" hints at Phi which may impose some constraints on the approach/solution to the problem. (You might easily guess my math is not up to these things that I still find amusing to play around with.)
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Old 4th December 2017, 09:10 AM   #13
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Follow-up article.

Grasshopper problem yields insight into quantum theory.

also

Quote:
And finally, the researchers suggest that it may be interesting to look into the origins of the lawn shapes themselves, as some of the lawn patterns resemble patterns that repeatedly emerge in nature, such as in flowers, seashells, and animal stripes. In accordance with the theory of morphogenesis proposed by Alan Turing, these patterns may arise as optimal solutions for chemical reasons, which may help explain the diverse and complex shapes of the lawns that appear in the grasshopper problem.
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Old 4th December 2017, 09:21 AM   #14
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Does chirality enter this picture at all? Example: mint & caraway have markedly different taste but their molecules are chemically exactly alike except they are mirror images of one another. Likewise, orange & lemon (say the bells of St. Clements*).

*Had to toss that in just for fun; the two are indeed mirror molecules.
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Old 5th December 2017, 11:59 AM   #15
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A couple of observations:

Since (for a random set of hops) the angular orientation of the pattern must be arbitrary (which is one reason for intuitively expecting a disc), the derived patterns must somehow be dependent on the initial conditions of the simulation. That this is not addressed seems very odd.

Even more peculiar, "Often, but not always, these optimal shapes possess some type of symmetry."
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Old 5th December 2017, 04:31 PM   #16
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Originally Posted by WhatRoughBeast View Post
A couple of observations:

Since (for a random set of hops) the angular orientation of the pattern must be arbitrary (which is one reason for intuitively expecting a disc), the derived patterns must somehow be dependent on the initial conditions of the simulation. That this is not addressed seems very odd.

Even more peculiar, "Often, but not always, these optimal shapes possess some type of symmetry."

From the video the initial conditions appear to be a fairly even distribution over the entire viewing space. As such, I would expect a reflective or rebounding boundary condition at the edges of the viewing space.
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