7/14 AN INTRODUCTION TO CYBERNETICS 



Ex. 3: (Continued.) How many degrees of freedom has the vector? (Hint: 

 Would removal of the minute-hand cause an essential loss?) 



Ex. 4: As the two eyes move, pointing the axes in various directions, they define 

 a vector with four components: the upward and lateral deviations of the 

 right and left eyes. Man has binocular vision; the chameleon moves his 

 two eyes independently, each side searching for food on its own side of the 

 body. How many degrees of freedom have the chameleon's eyes ? Man's ? 



Ex. 5: An arrow, of fixed length, lying in a plane, has three degrees of freedom 

 for position (for two co-ordinates will fix the position of its centre, say, and 

 then one angle will determine its direction). How many degrees of freedom 

 has it if we add the restriction that it must always point in the direction of a 

 given point P? 



Ex. 6: r is a given closed and single- valued transformation, and a any of its 

 operands. Consider the set of vectors, each of three components, 



{a, T{a), T2{a)), 



with a given all its possible values in turn. How many degrees of freedom 

 has the set? 

 Ex. 7: In what way does the ordinary graph, of >' on x, show constraint? 



Ex. 8 : How many degrees of freedom has an ordinary body — a chair say — in 

 three dimensional space? 



IMPORTANCE OF CONSTRAINT 



7/14. Constraints are of high importance in cybernetics, and will 

 be given prominence through the remainder of this book, because 

 when a constraint exists advantage can usually be taken of it. 



Shannon's work, discussed chiefly in Chapter 9, displays this 

 thesis clearly. Most of it is directed to estimating the variety that 

 would exist if full independence occurred, showing that constraints 

 (there called "redundancy") exist, and showing how their existence 

 makes possible a more efficient use of the channel. 



The next few sections will also show something of the wide 

 applicability and great importance of the concept. 



7/15. Laws of Nature. First we can notice that the existence of 

 any invariant over a set of phenomena implies a constraint, for its 

 existence implies that the full range of variety does not occur. The 

 general theory of invariants is thus a part of the theory of constraints. 

 Further, as every law of nature implies the existence of an invariant, 

 it follows that every law of nature is a constraint. Thus, the New- 

 tonian law says that, of the vectors of planetary positions and 

 velocities which might occur, e.g. written on paper (the larger set), 

 only a smaller set will actually occur in the heavens; and the law 

 specifies what values the elements will have. From our point of view, 

 what is important is that the law excludes many positions and 

 velocities, predicting that they will never be found to occur. 



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