Functional Geometry and the Determination of Pattern in Mosaic Receptors 383 



for the retinal elements: an operation for each eye that can translate elements 

 with fixed relations from one part of the field to another — that is, scan — during 

 the intercomparison process. 



A figure with bilateral symmetry, whenever its median line is defined, has 

 translational periodicity for lateral scanning. This might be the basis for 

 whatever accuracy the human eye has in judging such symmetry. 



5. Angular Comparison— During pure rotational scanning about the Z-axis 

 in fields containing sharp boundaries, if a certain time pattern of excitation of 

 elements bed • • • is repeated after a certain fixed time delay by elementsy^/? • • •, 

 then : 



(5a) there is a stable pattern in the field, fixed or rotating about the Z-axis; 



(5b) there is a constant angular separation, with respect to the Z-axis, 

 between elements b and/, c and g, d and h; and so on. 



5. Angular Periodicity — If a relation like 4' is satisfied for pure rotational 

 Z displacements, then: 



(5a') there is a stable angularly periodic pattern in the field, with /• repetitions ; 

 and so on. 



Because of the limited range of Z-rotation in the human eye (about 20°), 

 our perception of angular periodicities may lose precision rapidly for larger 

 angles. Some of this acuity may be recovered by treating the angular periodicity 

 as a bilateral symmetry, converting the judgment into one of lateral translational 

 periodicity. 



The metric of the 'space'' of the addresses established by these operations is 

 that of the rotation-space of the eyeball and not that of the retina or cortex surface. 



All these operations have been internal operations, specified so as to depend 

 only on the internal properties of the decision net and its scanning system, 

 and to be as independent as possible of the object and properties of the external 

 field except for the minimum requirement that there is some variety of structure 

 and that at least sometimes its changes and motions are slow compared to those 

 of the eyeball. 



Displacements and motions in the external field could lead to another 

 similar list of theorems which would establish similarly an external metric and 

 an expected external behavior, whose familiar translational and other con- 

 stancies — comparable to the congruences produced by the internal operations — 

 we might finally interpret as invariant 'objects' in the field, uniform motions, 

 and so on. The external metric may or may not be consistent with the eyeball- 

 rotational metric. Probably the external metric is the primitive one, with the 

 scanning metric providing a sophisticated refinement. Inconsistency between the 

 two may be the source of many optical illusions. 



But since theorems (1) to (5) suffice to establish in several different ways 

 that consistent address determination within the network is at least physically 

 possible it seems more important to go on now to see how it would be anatomi- 

 cally possible. 



C. Possible Types of Neural Connections Required 



Proprioceptive Coordinate Specif cation 



What anatomical connections are needed for proprioceptive sensing and 

 control? The requirements and some possible ways of solving them, at least 



