Pure Nerd Fun: The Grasshopper Problem

illustration of grasshopper.
[image: awesomedude.com]

It’s a sunny afternoon inward July together with a grasshopper lands on your lawn. The lawn has an surface area of a foursquare meter. The grasshopper lands at a random house together with thus jumps xxx centimeters. Which cast must the lawn convey thus that the grasshopper is most probable to Earth on the lawn i time to a greater extent than afterward jumping?

I know, sounds similar i of these contrived but irrelevant math problems that no i cares most unless yous tin dismiss acquire famous solving it. But the answer to this interrogation is to a greater extent than interesting than it seems. And it’s to a greater extent than most physics than it is most math or grasshoppers.

It turns out the optimal cast of the lawn greatly depends on how far the grasshopper jumps compared to the foursquare root of the area. In my opening illustration this ratio would convey been 0.3, inward which instance the optimal lawn-shape looks similar an inkblot

From Figure 3 of arXiv:1705.07621

No, it’s non round! I learned this from a newspaper past times Olga Goulko together with Adrian Kent, which was published inward the Proceedings of the Royal Society (arXiv version here). You tin dismiss of course of didactics rotate the lawn about its middle without changing the probability of the grasshopper landing on it again. So, the infinite of all solutions has the symmetry of a disk. But the private solutions don’t – the symmetry is broken.

You mightiness know Adrian Kent from his run on quantum foundations, thus how come upward his abrupt involvement inward landscaping? The argue is that problems similar to this appear inward for sure types of Bell-inequalities. These inequalities, which are usually employed to position genuinely quantum behavior, oftentimes halt upward existence combinatorial problems on the unit of measurement sphere. I tin dismiss merely imagine the authors sitting inward front end of this inequality, thinking, damn, in that place must locomote a way to calculate this.

As thus often, the work isn’t mathematically hard to Earth but dang hard to solve. Indeed, they haven’t been able to derive a solution. In their paper, the authors offering estimates together with bounds, but no total solution. Instead what they did (you volition dearest this) is to map the work dorsum to a physical system. This physical organisation they configure thus that it volition settle on the optimal solution (ie optimal lawn-shape) at zippo temperature. Then they copy this organisation on the computer.

Concretely, the copy the lawn of fixed surface area past times randomly scattering squares over a template infinite that is much larger than the lawn. They permit a for sure interaction betwixt the piffling pieces of lawn, together with thus they calculate the probability for the pieces to move, depending on whether or non such a motion volition better the grasshopper’s direct chances to remain on the green. The lawn is allowed to temporarily drib dead into a less optimal configuration thus that it volition non acquire stuck inward a local minimum. In the figurer simulation, the temperature is thus gradually decreased, which way that the lawn freezes together with thereby approaches its most optimal configuration.

In the video below yous run across examples for dissimilar values of d, which is the to a higher house mentioned ratio betwixt the distance the grasshopper jumps together with the foursquare root of the lawn-area:





For real pocket-sized d, the optimal lawn is almost a disc (not shown inward the video). For increasingly larger d, it becomes a cogwheel, where the issue of cogs depends on d. If d increases to a higher house about 0.56 (the inverse foursquare root of π), the lawn starts falling apart into disconnected pieces. There is a transition arrive at inward which the lawn doesn’t seem to settle on whatsoever detail shape. Beyond 0.65, in that place comes a cast which they refer to equally a “three-bladed fan”, together with afterward that come upward stripes of varying lengths.

This is summarized inward the figure below, where the cherry-red work is the probability of the grasshopper to remain on the lawn for the optimal shape:

Figure 12 of arXiv:1705.07621

The authors did a issue of checks to brand for sure the results aren’t numerical artifacts. For example, they checked that the lawn’s cast doesn’t depend on using a foursquare grid for the simulation. But, no, a hexagonal grid gives the same results. They told me past times e-mail they are looking into the interrogation whether the express resolution mightiness shroud that the lawn shapes are really fractal, but in that place doesn’t seem to locomote whatsoever indication for that.

I detect this a super-cute illustration for how much surprises seemingly wearisome together with uncomplicated math problems tin dismiss harbor!

As a bonus, yous tin dismiss acquire a brief explanation of the newspaper from the authors themselves inward this brief video.