The Leigh projectSweeping Pennington Flash: 66.2 hectares, one robot boat
Give an agent a lake and a job that is only finished when every part of it has been visited.
The learner takes the outline of Pennington Flash from OpenStreetMap, where it is drawn as a ring of 179 points, and converts it to metres on the British National Grid. The shoelace formula gives its area as 662,045 square metres and its shoreline as 5,197 m; the water fits inside a rectangle 1,605 m east to west and 893 m north to south. Then comes the agent. It is an imagined robot boat whose sensor sees 10 m to either side, so each pass it makes covers a strip 20 m wide. Its task is to pass over the whole lake. We test two kinds of boat.
The planned boat uses the pattern farmers use in a field, called a boustrophedon: straight parallel lanes 20 m apart, turning at the shore and coming back along the next lane. The only decision is the direction of the lanes. We tried 36 directions, every 5 degrees round a half circle, and measured each one in Python.
Straight-lane sweeps of Pennington Flash with a 20 m swath (our Python geometry on the OpenStreetMap outline)| Lane direction | Lanes | Separate straight runs | Total lane length | Water covered |
|---|
| Bearing 120 degrees (fewest runs) | 47 | 55 | 33.1 km | 98.8% |
| Bearing 35 degrees (most runs) | 82 | 105 | 33.1 km | 99.4% |
| All 36 directions tried | 44 to 84 | 55 to 105 | 33.0 to 33.2 km | |
The surprise is the fourth column. Whichever way the lanes run, their total length stays close to 33.1 km, because it is simply the area divided by the swath width. What the direction really changes is the number of straight runs, and each run ends in a turn, a slow-down and a small overlap. Lanes laid along the long side of the lake need 55 runs; lanes across it need 105, nearly twice as many turns for the same water. Where the shore bends inward, a single lane is broken into two or three runs, which is why runs outnumber lanes. The small gaps left along the shore, about one percent of the water, are the price of keeping to straight lines.
The wandering boat: distance travelled before a share of the lake had been seen, 20 runs (our simulation)| Share of water seen | Median distance | Quickest run | Slowest run |
|---|
| 50% | 21.5 km | 19.7 km | 28.8 km |
| 90% | 74.2 km | 68.4 km | 88.0 km |
| 95% | 95.3 km | 88.6 km | 117.3 km |
| 99% | 159.4 km | 135.5 km | 208.7 km |
The second boat has no plan. It moves forward in steps of 2.5 m and, whenever the next step would take it ashore, it picks a fresh random heading. Early on this looks fine: half the lake is seen after about 21.5 km, not far behind the planned boat. Then the returns collapse. Most of each new step passes over water it has already seen, so reaching 99% takes a median of 159.4 km, almost five times the planned sweep, and one unlucky run needed 208.7 km. Going from 90% to 99% cost the wanderer more distance than getting to 90% in the first place.
Ages 8 to 11
Shade a paper lake with a crayon as wide as a lane, once in rows and once by scribbling, and count the strokes each way.
Ages 11 to 15
Program a grid robot that sweeps a rectangle in lanes, then give it an L-shaped pond and fix the gaps it leaves.
Ages 15 and up
Load the real outline, test all 36 lane directions, and simulate the random boat against the plan.
Sources and limits
Lake outline © OpenStreetMap contributors, available under the Open Database Licence. The boat, its 20 m swath, the lanes and every distance above come from our own code. A real vessel would need to allow for wind, depth, wildlife and the rules of whoever manages the water, none of which we modelled. A different swath or a newer outline would change the figures.
