Functional Geometry and the Determination of Pattern in Mosaic Receptors 385 



3". Decentered Circularity — During combined translational and rotational 

 scanning, if the conditions of theorem (3) or theorem (3') are satisfied, then (3a) 

 and (3b) or (3a') and (3b') are vaHd; but: 



(3c") the Z-axis of the rotation does not pass through the center of curvature 

 of the boundary or boundaries. 



The oculomotor motions that could be directed by this kind of analog 

 control correspond to the crude motions of the grinding machine in generating 

 spherical surfaces. They do not need to be exact. They need only to be capable 

 of making retinal displacements across the field sufficiently good that during 

 the tremor movements the retinal excitation will have an adequate chance to 

 signal if it is congruent with the original pattern. This signal could also interact 

 with the oculomotor system to make the scanning more stable and accurate. 

 In practice, visual acuity for moving patterns is considerably reduced (15), 

 presumably because of the increased tracking errors and decreased chance of 

 a congruence signal. 



We can now see the effects of combined motions on theorems (2) and (3). 

 Evidently Operation (3), the examination of circles, gives a functional self- 

 centering specification of the axis of rotation, even if it is off the Z-axis, whenever 

 congruence is maintained. Any tremor about other axes simply provides useful 

 scanning motions. 



But Operations (2) and (2'), the examination of collinearity and parallelism, 

 cannot discriminate perfect straight lines from perfect circular arcs of large 

 radius except by invoking the accuracy of the sensing and analogue control, a 

 discrimination of much lower accuracy than the mosaic self-congruence dis- 

 criminations. It seems that our perception of such differences is in fact small 

 unless there are known straight lines nearby permitting a self-congruence test 

 for parallelism. There is a familiar illusion in which a comparison straight line 

 appears curved in the opposite direction from a curved line. This shows an 

 uncertainty in the analogue system, which tends to scan along the bisector so 

 as to give the figure bilateral symmetry. 



The Z rotations of the eyeball during scanning of curved patterns evidently 

 deserve examination. In the classical ZoUner illusion (parallel lines appear to 

 be non-parallel when crossed by oblique converging lines) there might also be a 

 Z-rotation of the retinal coordinate system, perhaps in the sense of stabilizing 

 the local foveal pattern and the local bilateral symmetry axis as the fixation 

 point oscillates from one of the parallel lines to the other. 



With further combinations of difference cells and time-differentiation cells 

 and feedbacks, probably the tracing out of any pattern by scanning could be 

 converted at a high enough stage into a constant output from some subtractive 

 neuron. If adjustments of the scanning rates at various points in the pattern are 

 also introduced, probably changes of size and distortions of shape could even 

 be accommodated in a constant output at a still higher stage neuron. We can 

 dimly visualize how this might proceed by stages to a neuron capable of producing 

 a fixed output, or better, a total motion of advance or retreat, whenever so 

 specific an object as a particular person is scanned, regardless of aspect and 

 light. 



Even if in later life the proprioceptive tracing of patterns by scanning 

 becomes subordinate to direct mosaic pattern-perception, the long stages of 



