As a second example consider adaptation to the environment. Adapt (from Webster) means: “to change (oneself) so that one’s behavior, attitudes, _etc._, will conform to new or changed circumstances. Adaptation in biology means a change in structure, function or form that produces better adjustment to the environment.” These statements suggest a simulation because adjustment to the environment implies survival by exposing the organism to the beneficial rather than the inimical effects of the environment. If we represent the environment (or portion thereof) as a relation as shown in Figure 2, we note that the ability to predict what effect a given disturbance will have is due to a simulation of the cause-effect relation which characterizes the environment. It would be a mistake to infer from these examples that simulation preserves the appearance of the causes and effects which characterize a relation. We clarify this situation by examining a relation and its simulation. Consider the relation between two mothers and their sons as pictured in Figure 4. Observe that if symbols (points) are substituted for the actual physical objects (mothers and sons), the relation is not altered in any way. This is what we mean by simulation and this is how a SOM simulates. It is not even necessary that the objects, used to display the relation, be defined; _i.e._, these objects may be primitive. (If this were not so, no mathematical or physical theory could model the environment.) The main prerequisite is sufficient resolution to distinguish the objects from each other. [Illustration: Figure 4—A relation of objects—displayed and simulated] MATHEMATICAL MODEL The mathematical model must represent both the environment and the SOM and for reasons given in the companion paper each is represented as a metrizable topology. For uniqueness we factor each space into equal parts and represent the environment as the channel W ⟶ X. (Ref. 10a) Consider now the SOM to be represented by the cascaded channels X ⟶ Y ⟶ Z where X ⟶ Y is a variable which represents the reorganization of the SOM existing input-output relation represented by Y ⟶ Z. The solution of the three channels-in-cascade problem W ⟶ X ⟶ Y ⟶ Z, where p(W) (11), p(X), p(X|W), p(Y), p(Z), p(Z|Y) are fixed, yields that middle channel p₀(Y|X), from a set of permissible middle channels {p(Y|X)}, which maximizes R(Z,W). Then the resulting middle channel describes that reorganization of the SOM which yields the optimum simulation of W ⟶ X by the SOM, within the constraints upon Ch(Z,Y). The solution (the middle channel) depends of course on the particular end channels. Obviously the algorithm which is used to find the solution does not. It follows that if some physical process were constrained to carrying out the steps specified by the algorithm, said process would be capable of simulation and would exhibit self-organization.