3/10 AN INTRODUCTION TO CYBERNETICS 



In II of the same figure are shown enough arrows to specify 

 generally what happens when any point is transformed. Here the 

 arrows show the other changes that would have occurred had other 

 states been taken as the operands. It is easy to see, and to prove 

 geometrically, that all the arrows in this case are given by one rule: 

 with any given point as operand, run the arrow at 45° up and to the 

 left (or down and to the right) till it meets the diagonal represented 

 by the line y = x. 



Fig. 3/10/1 



The usefulness of the phase-space (II) can now be seen, for the 

 whole range of trajectories in the system can be seen at a glance, 

 frozen, as it were, into a single display. In this way it often happens 

 that some property may be displayed, or some thesis proved, with 

 the greatest ease, where the algebraic form would have been obscure. 



Such a representation in a plane is possible only when the vector 

 has two components. When it has three, a representation by a 

 three-dimensional model, or a perspective drawing, is often still 

 useful. When the number of components exceeds three, actual 

 representation is no longer possible, but the principle remains, and a 

 sketch representing such a higher-dimensional structure may still 

 be most useful, especially when what is significant are the general 

 topological, rather than the detailed, properties. 



(The words "phase space" are sometimes used to refer to the 

 empty space before the arrows have been inserted, i.e. the space 

 into which any set of arrows may be inserted, or the diagram, such 

 as II above, containing the set of arrows appropriate to a particular 

 transformation. The context usually makes obvious which is 

 intended.) 



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