FUNCTIONAL GEOMETRY AND THE 

 DETERMINATION OF PATTERN IN MOSAIC 



RECEPTORS 



John R. Platt 



University of Chicago, Chicago, Illinois 



Abstract — Every visual pattern element — straight lines, curved lines, parallel lines, angles, 

 periodicities — shows some self-congruence under translations or rotations. A random mosaic 

 of detector cells, like the 10* cells of the human eye, can be used as a null detector to indicate 

 this self-congruence during scanning operations. This operational definition of pattern is called 

 functional geometry. It underlies the generation of precision optical and machine surfaces by 

 the Whitworth, Rowland and Strong methods and theoretically can approach infinite precision, 

 starting from rough materials. It converts a space pattern into time pattern repetitions whose 

 accuracy is not limited by the mosaic structure. The spherical eyeball shape is generated by 

 functional geometry, and its almost perfect rotation operations can establish among the 

 retinal cells an external Euclidean metric of perception-space which is independent of the 

 distortions of mapping on the retina or the cortex. 



A variety of second-stage and higher-stage neuroanatomical structures would have to be 

 grown for tracking and detecting pattern repetitions. These would almost certainly include 

 delay lines and null-transmitter cells to transmit only the identical parts of multiple input 

 patterns. 



Such pattern-perception in the mature network is equivalent to determination of the 

 initially unknown space relationships or addresses of the random detector cells. A non- 

 addressed mosaic requires much less initial assembly information than a pre-addressed mosaic, 

 but requires a long learning and growth time for address-determination after operation begins. 

 It has other quasi-human characteristics, since to determine addresses it consumes information 

 in abstracting properties, draws analogies, shows closure, may^ts symbols, learns from experience, 

 incorporates functional memories in the network structure, and apparently might even need 

 to sleep. But the self-congruences of functional geometry would impose certain paradoxical 

 and Kantian restrictions on the learning process, such that only certain congruent types of 

 experience can be learned at all, and only certain congruent types of address-connections can 

 be formed, regardless of what the experiences are. 



This paper revolves around the problem of visual pattern perception by the 

 human eye and brain. It is an attempt to generalize the problem; to restate 

 it in a language suitable for electrical networks; and to see what basic physical 

 principles might be involved, what detailed neural relationships might be 

 required, and how these principles and relationships restrict and determine the 

 general properties of such networks. 



The eye has millions of simultaneously active photodetectors. The theory of 

 connections in such a system is still in a primitive state. It is therefore necessary 

 to begin by introducing and explaining a number of new terms which will be 

 needed in the analysis. 



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