A Method of Measuring Shape

G EOGRAPHERS can analyze the many objects on the earth's surface according to a multitude of properties, of which one of the most significant is shape. The geographical literature abounds with comparisons of various objects on the landscape with geometric shapes. Settlement geographers may indicate that the objects of their study are linear, or perhaps star-shaped; Christaller suggested hexagons for idealized trade areas; political geographers employ a lexicon of descriptive terms to classify countries by shape. Boyce and Clark have reviewed a number of studies in geography in which descriptive references to shape were made.I One major problem of the shape concept in the literature is that the measurement of shape has been imprecise and has rested primarily on the subjective judgment of the geographer. Unlike area, shape cannot be expressed as a number. Ideally a method might be devised by which such descriptive terms as roughly circular, nearly rectangular, elongated, L-shaped, and the like would be replaced by a precise index of measurement in which each shape would be assigned a number. Such an assignment function should have three properties: (1) each shape is assigned a unique number; (2) no two shapes are assigned the same number; (3) two shapes that are similar are assigned numbers that are close together (thus shapes could be distributed along a continuum on which similar shapes are close together and dissimilar ones are far apart). A proposal for such a function was made by Boyce and Clark,2 but unfortunately their method does not satisfy the second requirement. This can easily be seen in Figure 1, in which a rectangular-shaped figure and a figure whose shape is that of a four-pointed star have identical index numbers. Desirable as it might be to have the assignment function described, it can be proved that no such function can possibly exist. In mathematical terms this is stated as follows.