illustration of grasshopper. [image: awesomedude.com] |

It’s a sunny afternoon in July and a grasshopper lands on your lawn. The lawn has an area of a square meter. The grasshopper lands at a random place and then jumps 30 centimeters. Which shape must the lawn have so that the grasshopper is most likely to land on the lawn again after jumping?

I know, sounds like one of these contrived but irrelevant math problems that no one cares about unless you can get famous solving it. But the answer to this question is more interesting than it seems. And it’s more about physics than it is about math or grasshoppers.

It turns out the optimal shape of the lawn greatly depends on how far the grasshopper jumps compared to the square root of the area. In my opening example this ratio would have been 0.3, in which case the optimal lawn-shape looks like an inkblot

From Figure 3 of arXiv:1705.07621 |

No, it’s not round! I learned this from a paper by Olga Goulko and Adrian Kent, which was published in the Proceedings of the Royal Society (arXiv version here). You can of course rotate the lawn around its center without changing the probability of the grasshopper landing on it again. So, the space of all solutions has the symmetry of a disk. But the individual solutions don’t – the symmetry is broken.

You might know Adrian Kent from his work on quantum foundations, so how come his sudden interest in landscaping? The reason is that problems similar to this appear in certain types of Bell-inequalities. These inequalities, which are commonly employed to identify truly quantum behavior, often end up being combinatorial problems on the unit sphere. I can just imagine the authors sitting in front of this inequality, thinking, damn, there must be a way to calculate this.

As so often, the problem isn’t mathematically difficult to state but dang hard to solve. Indeed, they haven’t been able to derive a solution. In their paper, the authors offer estimates and bounds, but no full solution. Instead what they did (you will love this) is to map the problem back to a physical system. This physical system they configure so that it will settle on the optimal solution (ie optimal lawn-shape) at zero temperature. Then they simulate this system on the computer.

Concretely, the simulate the lawn of fixed area by randomly scattering squares over a template space that is much larger than the lawn. They allow a certain interaction between the little pieces of lawn, and then they calculate the probability for the pieces to move, depending on whether or not such a move will improve the grasshopper’s chance to stay on the green. The lawn is allowed to temporarily go into a less optimal configuration so that it will not get stuck in a local minimum. In the computer simulation, the temperature is then gradually decreased, which means that the lawn freezes and thereby approaches its most optimal configuration.

In the video below you see examples for different values of d, which is the above mentioned ratio between the distance the grasshopper jumps and the square root of the lawn-area:

For very small d, the optimal lawn is almost a disc (not shown in the video). For increasingly larger d, it becomes a cogwheel, where the number of cogs depends on d. If d increases above approximately 0.56 (the inverse square root of π), the lawn starts falling apart into disconnected pieces. There is a transition range in which the lawn doesn’t seem to settle on any particular shape. Beyond 0.65, there comes a shape which they refer to as a “three-bladed fan”, and after that come stripes of varying lengths.

This is summarized in the figure below, where the red line is the probability of the grasshopper to stay on the lawn for the optimal shape:

Figure 12 of arXiv:1705.07621 |

The authors did a number of checks to make sure the results aren’t numerical artifacts. For example, they checked that the lawn’s shape doesn’t depend on using a square grid for the simulation. But, no, a hexagonal grid gives the same results. They told me by email they are looking into the question whether the limited resolution might hide that the lawn shapes are actually fractal, but there doesn’t seem to be any indication for that.

I find this a super-cute example for how much surprises seemingly dull and simple math problems can harbor!

As a bonus, you can get a brief explanation of the paper from the authors themselves in this brief video.