Functional Geometry and the Determination of Pattern in Mosaic Receptors 381 



sequence depending on some function of the relative transverse cell displace- 

 m;nts, sensitivities and repetition rates, and on the image boundary gradient. 

 'Unchanged excitation' would mean successive recurrences of this same sequence, 

 and changes in the sequence could correspond to changes in the image 

 amounting to only a small fraction of a cell diameter. Jhc gradient-discriminating 

 power would then be limited essentially by signal-to-noise considerations rather 

 than by mosaic structure and it might be far higher than the static mosaic 

 resolving power, as numerous authors have suggested. The transmitted self- 

 congruence signal, whatever it is, need contain no trace of the static mosaic 

 structural irregularities. It is also independent of differences in the sensitivity of 

 different receptor cells and could remain unchanged even if a few of them should 

 fail completely (closure). These would be important biological advantages for 

 the self-congruence method of address-determination. 



2'. Parallelism — If the elements/^/? • • • A' of Operation 2 also are grouped 

 into /■ subsets each of whose excitations can be duplicated for r different trans- 

 verse displacements, with a different set of displacements for each subset of 

 elements, then: 



(2a') there are r parallel Hnear boundaries in the field ; 

 (2b') each subset lies on the image of one of these boundaries; and 

 (2c') the first drift movement is parallel to the boundaries, while the discrete 

 transverse displacements are not. 



It is typical of functional geometry that it simultaneously limits (2a) the 

 type of external pattern that can be interpreted (2b), the type of internal relation- 

 ship that can be organized, and (2c) the operational motions that can produce a 

 coincidence of the two. This situation for pattern structure is no different 

 from that for the eyeball, where functional geometry simultaneously limits the 

 shape of the external socket, the shape of the ball, and the possible movements 

 and musculatures. We shall see over and over that the functional geometry, if 

 it is the address-determining method, is neither experience nor structure but 

 stands outside them both, imposing an inescapably limited selection of forms 

 on the only experiences we can perceive and the only structures we can create. 



Point (2c), the establishment of retinal relationships and proprioceptive 

 oculomotor signals relative to each other, as suggested by Helmholtz, is not 

 the least important aspect of address determination, now that proprioceptive 

 muscle spindles are known to be present (13, 14). 



Note that it is the boundaries in the external f eld that are linear, and not their 

 retinal images, when self-congruence is the method of discrimination. Likewise 

 the projections on the cerebral cortex can have any kind of twist, distortion or 

 discontinuity— which they have — without destroying a functional definition of 

 collinearity and parallelism. 



The ambiguous word 'linear' is used in theorems (2) and (2') so as to postpone 

 for a moment the question of how well these procedures will distinguish a 

 perfectly straight line from a perfect circular arc of very slight curvature. But 

 aside from that question, a mosaic detector is seen to be in principle far more 

 accurate than a single-element detector. With the latter, straight lines could be 

 discriminated by tracing them out, perhaps using small hunting movements 

 superimposed on a long sweep, but the accuracy is limited by the accuracy of 



