Page 50 - Modern Spatiotemporal Geostatistics

Spatiotemporal Geometry 31

To visualize £ externally is to view it from a higher dimensional space that
includes it.
EXAMPLE 2.4: Non-Euclidean geometries are more common than one might
think. When our existence is confined to the surface of the Earth, we have
a two-dimensional world view (e.g., points on the surface are unambiguously
located by two coordinates—longitude and latitude). The surface can be inter-
nally represented by a non-Euclidean geometry of the Riemannian type (which
violates certain major assumptions of the Euclidean geometry; see Table 2.2).
If we are able to leave the surface of the Earth and move to outer space, we can
view Earth's surface from a three-dimensional standpoint that can be exter-
nally described in terms of Euclidean geometry (Table 2.2). However, we can-
not step outside of the three-dimensional world into a four-dimensional space
in order to visualize externally either a three-dimensional Euclidean space or
a three-dimensional non-Euclidean space. Instead, our visualization of three-
dimensional space can only be internal, i.e., from the standpoint of beings
confined within that space.

The epistemic conclusions to be drawn from the above analysis are sum-
marized by the final postulate of this section, as follows.

POSTULATE 2.4: The choice of an appropriate geometry to describe
space/time continuum £ depends on whether one adopts an intrinsic
or an extrinsic visualization of £.
The four fundamental postulates underlying the concept of space/time
continuum £ are illustrated in Figure 2.3. The continuum concept is para-
mount in representing the evolution of natural variables which assume values
at any point in space/time, thus requiring continuously varying spatiotemporal
coordinates. The operational importance of £ is its bookkeeping efficiency
that permits an ordered recording of physical measurements and the estab-
lishment of links between measurements by means of physical theories and
mathematical expressions.

POSTULATE 2.1: Events in Points on "E
nature
POSTULATE 2.2: Relationships Relationships

between events between points

Physical
POSTULATE 2.3: knowledge Spatiotemporal
geometry on £
POSTULATE 2.4: Visualization

Figure 2.3. The four fundamental postulates of space/time continuum £.