378 John R. Platt 



The sophistication of this definition of a plane is that it is antecedent to 

 the definition of a straight fine in this geometry and requires no definitions of 

 fines or axes or coordinate systems or rectifinear translations. 



Other surfaces can be generated by grinding and lapping operations that 

 maintain only a line of contact between two self-congruent surfaces, such as 

 two surfaces of revolution rotating about skew-perpendicular axes. 



Biological Examples 



Any two physiological surfaces that are pressed and rubbed together con- 

 tinuously must exhibit displacement congruences approximating mathematical 

 perfection. 



The familiar chicken drumstick has at its lower end a perfect surface of 

 revolution sweeping through an angle of about 270°. (One degree of freedom. 

 It may be slightly helical, since the revolution is not complete.) The helical 

 grooves on the narwhal tusk may be generated by displacement congruences as 

 it grows from its socket. Ball-and-socket joints are likewise famifiar in anatomy, 

 with accurately spherical surfaces. (Three degrees of freedom.) 



The eyeball-and-socket is perhaps the most perfect instance of this type. 

 The spheres must be very precise if there are not to be considerable changes of 

 pressure during normal rotations. The oculomotor musculature provides all 

 three rotations, about the Z-axis (vertical axis), the 7-axis (transverse horizon- 

 tal), and the Z-axis (longitudinal horizontal). Functional geometry provides a 

 precise self-centering specification of the center of the spheres and therefore of 

 the reference point about which all the operations of the three-dimensional 

 continuous rotation group can be carried out. 



What is more important, these motions provide the necessary displacements 

 by which the displacement congruences of any pattern in the external field may be 

 detected by the retina. 



To anticipate the results of the next section, if an arc in the external field 

 produces an excitation pattern on the retina (Fig. 1b), the pattern can remain 

 unchanged during a displacement of the fixation point along the arc if, and only 

 if, the arc as seen from the eyeball is either a straight fine or the arc of a perfect 

 circle, with constant curvature. This is the two-dimensional analogue of the 

 functional definition of a perfect sphere given above. 



Likewise a set of lines in the field is parallel and equidistant if and only 

 if the excitation pattern can remain unchanged as the fixation point moves from 

 one line to the next or moves along the lines (Fig. 2). This is the analogue of 

 the functional definition of a surface with parallel equally-spaced grooves. 



These are indeed the kinds of pattern judgment that the human eye makes 

 most precisely. Our peculiar sensitivity to changes of curvature and to non- 

 parallelism and non-periodicity is well known in model-making and in pattern 

 tracing and analysis. 



An extreme case is the curvature-continuity judgment involved in 'vernier 

 acuity'. If two ends of a line join imperfectly in the middle, the eye can still 

 perceive the break when the lateral displacement is as small as 2 seconds of arc — 

 l/30th the diameter of a retinal cone (7). Regardless of what neural connections 

 might be needed to make such a discrimination, it is obvious that the judgment 

 must depend on a physical operation of inherently high precision, inherently 



