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Spatiotemporal Geometry 45

Figure 2.13. (a) Iso-covariance contours assuming a Euclidean metric (Eq.
2.12)—continuous lines—and an absolute metric (Eq. 2.13)—dashed
lines, (b) Minimum path length between two points at a Euclidean
distance r in Euclidean space (curve 1) and in a space with fractal
length dimension d 0 = 1.15 (curve 2).

d
cluster. Fractals are homogeneous functions such that £(br) = b °l(r), where
r is the appropriate Euclidean distance, d 0 is the fractal exponent for the specific
property, and & is a scaling factor. In two dimensions, e.g., the perimeter of
a cluster's hull scales as th oc a dh for a cluster of linear dimension a (dh is
the corresponding fractal exponent); in contrast, the perimeter of Euclidean
objects varies linearly with the length a.
The following example gives an illustration of how some of the metrics
considered above can lead to quite different geometric properties of space.

EXAMPLE 2.11: In the geostatistical analysis of spatial isotropy in R? (Olea,
1999), one needs to define iso-covariance contours. In Figure 2.13a it is shown
that, while in the case of the metric in Equation 2.12 these contours are cir-
cles of radius r, 2r, etc., in the case of the metric in Equation 2.13 the iso-
covariances are squares with sides -\/2r, 1^/2r, etc. In Figure 2.13b (Christakos
et al., 2000a) we show the minimum path length between two points sepa-
rated by a Euclidean distance r in Euclidean space and in a fractal space with
= 1.15. The path length in the Euclidean space is a linear function of the
d 0
distance between the two points, for all types of paths. The fractal path length
increases nonlinearly, because the fractal space is nonuniform and obstacles to
motion occur at all scales.
We should distinguish carefully a mere change of coordinates from an ac-
tual change of geometry. One can use, e.g., non-rectangular coordinates on
a flat Euclidean plane as well. Generally, what distinguishes a flat Euclidean