6/8 AN INTRODUCTION TO CYBERNETICS 



ISOMORPHIC MACHINES 



6/8. Study of a Black Box can thus give the experimenter informa- 

 tion up to a certain amount; and, if the inputs and outputs are given, 

 cannot possibly be made to give more. How much information 

 will be discussed in S.13/I5 (especially its last Ex.). Here it is 

 sufficient if we notice that the canonical representation specifies 

 or identifies the mechanism "up to an isomorphism". 



"Isomorphic" means, roughly, "similar in pattern". It is a 

 concept of the widest range and of the utmost importance to all who 

 would treat accurately of matters in which "pattern" plays a part. 

 Let us consider first a few examples merely to illustrate the basic ideas. 



A photographic negative and the print from it are, so far as the 

 pattern of the picture is concerned, isomorphic. Squares in the 

 negative appear as squares in the print; circles appear as circles; 

 parallel lines in the one stay as parallel lines in the other. Thus 

 certain relations between the parts within the negative appear as 

 the same relations in the print, though the appearances so far as 

 brightness is concerned are different, exactly opposite in fact. Thus 

 the operation of changing from negative to print leaves these 

 relations unaltered (compare S.5/2). 



A map and the countryside that it represents are isomorphic 

 (if the map is accurate !). Relationships in the country, such as that 

 towns A, B and C form an equilateral triangle, occur unchanged 

 on the map, where the representative dots for A, B and C also 

 form an equilateral triangle. 



The patterns need not be visual. If a stone is thrown vertically 

 upwards with an initial velocity of 50 ft. per second, there is an 

 isomorphism between the set of points in the air such that at time 

 / the stone was h feet up and the set of those points on a graph that 

 satisfy the equation 



y = 50x - 16.v2. 



The lines along which air flows (at sub-sonic speeds) past an 

 aerofoil form a pattern that is identical with the lines along which 

 electric current flows in a conducting liquid past a non-conductor 

 of the same shape as the aerofoil. The two patterns are the same, 

 though the physical bases are different. 



Another isomorphism is worth consideration in more detail. 

 Fig. 6/8/1 shows two dynamic systems, each with an input and an 

 output. In the upper one, the left-hand axle / is the input; it can 

 be rotated to any position, shown on the dial u. It is connected 



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