The Kirkby projectScoring 137 neighbourhood shapes, two ways
Census boundaries, two formulas, and a check on how much the drawing itself matters.
Output areas are the smallest areas the census reports on. The learner downloads the full-resolution boundaries of Kirkby's 137 from the ONS: 38,657 corner points in British National Grid metres. Their areas range from about 1.7 hectares to 449, with a median near 5.1. Python computes each area with the shoelace formula and each perimeter by adding up the edges.
Score one is the Polsby-Popper score, four pi times the area divided by the perimeter squared. A circle scores 1 and a square about 0.785; long, ragged shapes score low. Score two is the convex hull ratio: stretch an imaginary rubber band round the shape, and divide the shape's area by the area inside the band. A shape with no dents scores 1. Both are standard; they simply measure different things. Polsby-Popper punishes a long or wiggly edge, while the hull ratio punishes bites taken out of the outline.
Two compactness scores for 137 Kirkby output areas, our Python run| Measure | Lowest | Median | Highest |
|---|
| Polsby-Popper, full-resolution boundaries | 0.131 | 0.350 | 0.764 |
| Convex hull ratio, full-resolution boundaries | 0.432 | 0.730 | 0.998 |
| Polsby-Popper, generalised boundaries | n/a | 0.412 | n/a |
| Convex hull ratio, generalised boundaries | n/a | 0.757 | n/a |
Mostly the two scores agree: ranked from least to most compact, their rank correlation is 0.913. But they crown different winners. The oddest shape by Polsby-Popper, output area E00032611 with a score of 0.131, is only third by hull ratio. The oddest by hull ratio, E00032982 at 0.432, is sixth by Polsby-Popper. Of the ten least compact areas on each list, seven appear on both. The sharpest disagreement is E00172034, 125th of 137 by one score and 80th by the other.
Then the learner repeats everything on the generalised boundaries the ONS also publishes, drawn with 2,331 corner points instead of 38,657. The median Polsby-Popper score jumps from 0.350 to 0.412, because a smoother edge is a shorter one, while the median hull ratio moves only from 0.730 to 0.757. E00032611 is still the oddest, now scoring 0.159. So a single number reported without its formula and its boundary file is not really a result.
Ages 8 to 11
Make shapes from the same length of string and find which one holds the most squares.
Ages 11 to 15
Code the shoelace formula in Python and test it on shapes whose areas you already know.
Ages 15 and up
Load the ONS boundaries, compute both scores at both resolutions, and compare the rankings.
Boundaries and credits
Output area boundaries and the output area to built-up area lookup are from the Office for National Statistics under the Open Government Licence. The convex hulls use SciPy. The areas, perimeters, scores and rankings are all from our own run.
