Functional Geometry and the Determination of Pattern in Mosaic Receptors 377 



'The construction methods of greatest precision are ail primitive methods' (6). 

 The following are some examples. 



A surface of revolution is self-congruent for rotation about its axis. (One 

 degree of freedom : Strong method for thrust bearings.) 



A screw is self-congruent for simultaneous rotation about its axis and trans- 

 lation along it. (One degree of freedom: Rowland method of lapping.) 



A cylinder is self-congruent for all rotations about its axis and translations 

 along it. (Two degrees of freedom: Strong prescription for lapping a cylinder). 

 A cylindrical surface section is self-congruent for pure translations in the surface 

 with no component of rotation about a line normal to the surface. 



A gear of n identical teeth, 360^ /n apart in angle, is self-congruent for any 

 of n different angular displacements about its axis. (One continuous degree of 

 freedom plus one discrete. In the Strong method, the gear is polished within a 

 kind of open-ended squirrel cage of n lapping bars or pawls that slide between 

 the teeth. The cage is rotated by one bar after every stroke, and any initial 

 irregularity in either the gear or the cage is polished away.) The group operations 

 are those of the discrete group, C„. Functional geometry can therefore generate 

 perfect right angles or other angles. 



1 



Fig. 2. Self-congruence in translational periodicity. 



By analogy with the screw and the gear, a cylindrical surface with straight 

 parallel equally-spaced identical grooves (possibly helical) is self-congruent for 

 continuous translation in one direction in the surface and discrete translations 

 in the other, as indicated in Fig. 2. (In principle, the precision ruling of surfaces 

 might be accomplished in this way.) Perfect translational periodicities in two 

 or three dimensions might be generated in succession. 



There are more sophisticated possibilities on moving beyond ordinary group 

 theory: A plane is one of three surfaces of which any pair can be placed in 

 contact everywhere regardless of relative translations or rotations against each 

 other. (In making optical flats by the Whitworth method of lapping, three 

 flats are generated simultaneously by being polished against each other, with 

 frequent interchange of pairs to prevent development of concave or convex 

 surfaces). 



Restated in terms of displacement congruences: A plane is self-congruent 

 for all translations and rotations in the surface and for two-fold rotations about 

 an axis in the surface. Note that three-fold rotations about such an axis would 

 be impossible. This exemplifies a fundamental physical restriction on possible 

 generating processes, of a kind wc will encounter shortly in the biological cases. 



