Math 'Creating Measures' Compact-ness Task, Example #3 (solution)
Malcolm Swan
Mathematics Education
University of Nottingham
Malcolm.Swan@nottingham.ac.uk
Jim Ridgway
School of Education
University of Durham
Jim.Ridgway@durham.ac.uk
This problem gives you the chance to
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Over recent years, a number of geographers have tried to find ways of defining the shape of an area. In particular, they have tried to devise a measure of 'compactness'. You probably have some intuitive idea of what "compact" means already. Below are two islands. Island B is more compact than island A. "Compact-ness" has nothing to do with the size of the island. You can have small, compact islands and large compact islands.
One person has suggested the following way of measuring "compactness."
Shape | ||||||
Area | ||||||
Perimeter (cm) | ||||||
Area ÷ Perimeter |
This is dimensionless and does give measures in accord with intuition:
Staircase | ||||||
Area ÷ Perimeter2 |
Thus, in order of "compactness", we have that
B (most compact), D and F (tie), A and C (tie), E (least compact).
The ratio 4A/p2 is quoted in Selkirk (1982) as the "circularity ratio." One criticism of its use by geographers is that it is difficult to define and calculate the perimeter of a very large irregular boundary such as a country or a river basin. (Fractal geometry might suggest that such perimeters may even be infinite!) Instead, there are several other possibilities:
Form Ratio: | |
A = area L = length of line joining the area's two most distant points |
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Compactness ratio: | |
A = area R = radius of the smallest circle which will surround the shape |
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Radius ratio: | |
r = radius of the largest circle which will fit inside the shape R = radius of the smallest circle which will surround the shape |
These are all quoted in Selkirk (1982) as methods which geographers have used. For interest, Selkirk offers the following calculations for four countries, using these ratios:
Circularity ratio | ||||
Form ratio | ||||
Compactness ratio | ||||
Radius ratio |
Selkirk's results place the countries in a similar order, apart from the circularity ratio, which suggests that Czecholslovakia is less compact than Austria.