Modelling geometric constraint

In this chapter we shall see also that the analytical techniques of theoretical morphology allow us to take a spatial approach to the concept of evolutionary constraint. To explicitly define the concepts of evolutionary constraint considered here, I shall use Venn diagrams and set theory. A given biological form, f, may be described by a set of measurements taken from that form. Each type of measurement, such as length, width, or height, can be considered as a form dimension.

Form Hyperdimension

Form Hyperdimension

Figure 7.1. A theoretical hyperdimensional space of possible form Each dimension of the space represents a morphological trait that may be measured on a given biological form, f. All possible coordinate combinations within the hyperdimensional space represent the set of all possible biological forms. Although only eight dimensions are shown in this schematic diagram, the dimensionality of an actual hyperspace of form will be much larger.

The total set of the possible dimensions of form can be used to construct a hyperdimensional space of possible form (Fig. 7.1). All coordinate combinations within this space represent a universal set of form, U. Some of the coordinate combinations within the total hyperspace of form represent the set of geometrically possible forms: GPF = {f | f = geometrically possible forms} (Fig. 7.2). Other coordinate combinations within the total hyperspace represent the set of geometrically impossible forms: GIF = {f | f = geometrically impossible forms}.

The regions of impossible and possible form within the total hyperspace of form do not overlap one another; that is, a given set of coordinates within the hyperspace cannot simultaneously represent a possible geometry and an impossible geometry (Fig. 7.2). The sets of form GIF and GPF are thus compliments of each other:

The boundary between the sets GIF and GPF is here designated as the geometric constraint boundary (Fig. 7.2).

GIF U GPF = {f | f 2 GIF or f 2 GPF} = U, GIF n GPF = {f I f 2 GIF and f 2 GPF} = 0.

Figure 7.2. Geometric constraints. Geometric constraint as the boundary between the set of geometrically impossible forms (GIF) and the set of geometrically possible forms (GPF) in the total hyperspace of form dimensions (the universal set of form, U). Source: From McGhee (2006).

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