Functional Geometry and the Determination of Pattern in Mosaic Receptors 373 



input element is only partly specified by its network-address. It is incomplete 

 unless the location in space, or space-address, is also given, at least relative to 

 the other elements, since both address components efTect the kinds and combina- 

 tions of messages sent through the network. 



Mosaic receptors may be pre-addressed or non-addressed. In a pre-addressed 

 system, each receptor element has a specified space address and network 

 address. It is completely connected in a unique and permanent way to the 

 decision net before the net begins to operate. In a non-addressed system, the 

 space address of an element, or its network address, or both, may need to be 

 determined after operation begins. 



This may be the main difference between the insect eye and the human eye. 

 The insect eye consists of a close-packed array of uniform receptor elements. 

 Because of their uniformity, they lie in long parallel lines. Absolute genetic 

 determination of the connections from each element to its neighbors and to 

 the decision net might be easy: a pre-addressed system. 



Straight lines in the field of view that fire all elements on one of the principal 

 lines of such an array should be easy to distinguish from curved lines, if such a 

 distinction were biologically useful. But straight lines in any other general 

 direction would be hard to distinguish from curved, without very elaborate 

 inter-connections in the network; and therein might lie the limitations of a 

 pre-addressed system. 



The human retina escapes this impasse. It appears to make no such dis- 

 tinction between straight lines in different directions. And indeed under a 

 microscope the cones in our foveas appear to be close-packed but sufficiently 

 non-uniform that no straight line arrangements are more than a few cones 

 long (2). 



Assembly Information 



This useful randomness seems inevitable from assembly considerations. A 

 non-random biomechanical assembly of 10^ elements distributed over several 

 square centimeters of the retina with individual tolerances of better than 1 

 micron would be almost inconceivable. Even if this could be achieved, the 

 complexity of a non-random wiring diagram for any system of 10^ input elements, 

 geometrically regular or irregular, would be almost impossible for the chromo- 

 somes to specify, as Pitts has emphasized (3). 



And so the randomness, if it has solved one dilemma, has evidently created 

 another. The addresses of the retinal elements are now uncertain. All straight 

 lines have been made equal by a device which appears to make it impossible 

 for the eye to identify straight lines at all! 



On the other hand, if this new problem could be solved — and the present 

 paper aims to show that it can — non-addressed systems would evidently have 

 one tremendous advantage over comparable pre-addressed systems: their 

 economy of assembly information. In pre-addressed systems, if the inter- 

 connections among /;; elements are to be specified in advance, the assembly 

 information must increase with a power of/;? at least as large as two and perhaps 

 much larger. 



This elaboration of initial design specification and mechanical assembly 

 detail is what makes our artificial electronic networks slow and expensive to 



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