Functional Geometry and the Determination of Pattern in Mosaic Receptors 375 



elements ■ ■ • a • ■ ■ p ■ ■ • be detemiined, within the receptor network and solely 

 by its normal functional operations? And if so, how? 



The present paper aims to show that at least one simple method exists for 

 this functional detennination of addresses. It can be called the method of 

 functional geometry. It seems feasible for use at least in an artificial mosaic 

 system. It may or may not be the method used by the eye or by any other 

 biological system, although many of the results here strongly suggest that it is. 

 In any case, its existence removes a principal conceptual difficulty of non- 

 addressed mosaic receptors. And the examination of one particular method 

 can help sharpen up our experimental inquiries as to what methods of address- 

 determination and pattern-perception actually are used in biological systems. 



II. FUNCTIONAL GEOMETRY 



There is a class of geometrical operations that is of great importance in the 

 highest precision machine work and in anatomy, especially in the joints of 

 vertebrates. The operations are related to group theory but, as we shall see, 

 they might form the axiomatic basis of a separate systematic mathematical 

 discipline. If this discipline were ever created, an appropriate name for it 

 would be functional geometry. 



Generation of Perfect Surfaces 



An illustrative operation of this class is that by which an optician or an 

 amateur telescope maker grinds and polishes a spherical lens or mirror surface 

 (4). A rough blank of glass is placed against another rough blank of glass 

 or metal or pitch, with grinding or polishing powder between them. The blanks 

 are pressed and rubbed together by hand or by a rather crude and loose grinding 

 machine, as shown schematically in Fig. 1a. The operation continues with 



Fig. 1. Self-congruence of a sphere or a circular arc 

 under random translation. 



successively finer grades of powder. Finally each of the surfaces approaches a 

 perfectly spherical shape to a precision which may be one-tenth of a wavelength 

 of light, or better if desired. 



Theoretically, if edge effects are neglected, the method can approach infinite 

 precision. Its practical precision is limited only by the patience of the optician 

 and the accuracy of available testing methods. The accuracy of approximation 

 to a perfect sphere can be many orders of magnitude higher than the accuracy 



