ALEXANDER, Christopher: Definition of wholeness

446

Appendix 1

Supplement to chapter 3

Definition of wholeness

The wholeness, W, is a feature of physical space which appears everywhere, in every part of matter/space. It is, I believe, susceptible to a clear mathematical definition and is characterized by a well-defined mathematical structure.

Consider any region of space, R. We may, for convenience, impose a grain or mesh on the space, so that the number of points is considered finite, not infinite. Let us say that R contains n points. In cases which model the real world, there is usually some “coloring” or differentiation of type or character among the n points of R, so that the region has a visible or identifiable structure. The simplest coloring which produces a structure is a coloring in which some points are black, others white. In the two-dimensional case, R would then be a drawing in which we see some particular object. In the case where the coloring is not abstract, but material, points may be assigned labels corresponding to actual physical materials; for example, they might include solid and void, or various physical or chemical attributes. The region R is thus intended to represent a part of the real world in its overall geometric form and organization.

I shall now explain how to construct a wholeness W on the region R. Within the region of space R, which contains n points, there are 2ⁿ distinguishable subregions. Call a typical one of these subregions S_i. In what follows, we construct W by recognizing that there are different relative degrees of coherence which may be observed in the different subregions S_i.

It is common fact of experience that we see regions of space which have different degrees of coherence. For example, we consider an apple to be coherent If we consider the set of points that consists on half the apple, we shall probably consider it less coherent than the apple as a whole (…)

We do recognize coherence in the world. This coherence, is just that attribute which I have referred to throughout Book I as life. The structure of the wholeness W relies on the fact that we shall make such distinction of life explicit, and use them to erect a structure.

To make the idea of different degrees of life explicit, we introduce a measure of life c, on the subregions of R. Call each possible subregion of R, S_i, where i ranges from 1 to 2ⁿ. The life of the i-th subregion S_i is the to be c_i. Each c_i is a number between 0 and 1, and every subregion of R is to be given its measure of life. The most coherent regions have a c_i which is 0 or close to 0.

447

In all that follows, regardless the specific definition of c, c_i, is simply to be understood as some measure of relative life, in which the most coherent regions S_i have a life 1, the least coherent have a measure 0, and others have intermediate values.

I call the most coherent subregions of R, centers. A region will be considered more or less centered according to its life. The most coherent subregions S_i, which have a c_i close to 1, will be called the centers of R. Even among the centers, there are still degrees of relative life -some are more coherent than others- but all of them establish, through their life, a phenomenon of centeredness in space.

(…) the wholeness W is to be understoond as a system of center, containing the most coherent regions in R, rank-ordered according to their relative degree of life.

(…)

I define the wholeness W as the system which is created by the region R, together with the measure c and all those subregions which have measure more than some threshold and thus qualify as centers. For some practical purposes, the wholeness W is created by the interaction of the geometry of the region R and the rank order which is created on the centers of R by c.

(…) the character of a configuration is given by the particular system of subregions that are connected, and by the way these connected subregions overlap and lie within each other.