170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 222 222 CH10 GEOMETRIC VISUAL HALLUCINATIONS In this chapter I present the results of some 35 years work with my students and collaborators,onthedevelopmentofamathematicalmodeloftheoriginoftheform constants in theneural networks ofthe visual brain. 10. 1. 1 The nature of the neocortex If one examines the histological section of the neocortex shown in Figure 10. 6, at first sight it appears spatially homogeneous. That is, any one region looks more-or-less the same as anyother. Weformulatethismathematicallyasfollows:let H[x]denotetheregionlocatedat someposition x. Thentheconditionforspatialhomogeneityisexpressed(partially) as H[x]=H[x+L]. Wesaythesectionis translationallyinvariant ortranslationally symmetric on the length-scale L. In similar fashion if we move from one location to another, in any direction, the result is the same. The section is also rotationally invariant orrotationallysymmetric. Wewritethisresultas H[x]=H[R𝜃[x]],where 𝜃is the direction relative to horizontal in neocortical coordinates. Finally we also havereflectionsymmetry H [x]=H[-x]. Thus,onthelength-scale L,theneocortex is invariant under translations, rotations and reflections. Mathematically it is said to have the symmetries of the group 𝔾of rigid body motions in the plane-the Euclidean Group 𝔼(2). Mathematically we represent this by the equation H[x]= H[𝔾[x]]. Figure 10. 6 Histological section of neocortex. Source: Reproduced with permission of John Wiley &Sons.