CHAPTER VI. 



The Periodical Repetition of Identical Units in a Plane. 

 The Repeat as the Unit of an Endless Pattern. Homogeneity 

 in Periodical Arrangements. - - Homologous Points. The 

 Homogeneous Distribution of Points in a Plane. - - The Net- 

 Plane. - - The Netplane-structure in Endless plane Patterns. - 

 The Symmetry of a Pattern and of its Net-plane. - - Points 

 regularly distributed in Space. - - The Space-lattice. - - Some 

 general Properties of Space-lattices. The Symmetry of 



Bravais' Space-lattices. -- Elements of Pseudo-symmetry. - 

 General Symmetry- Relations in Endless Systems. The 



Problem of the General Deduction of All Homogeneous Arran- 

 gements in Space. - The Fundamental Domain in Endless 

 Homogeneous Structures. -- Enantiomorphism and Congruency 

 of Structural Units. The Theories of Sohncke, Von Fedorow, 

 Schoenflies. General Results. Endless Periodical Patterns 

 and Crystal-structure. - - The Crystal as a Tridimensional Sym- 

 metrical Pattern. -- Continuous and Discontinuous Properties 

 of Crystalline Matter. - - The Space-lattice as a Geometrical 

 Expression of Hauy's Law. - - The Chemical Structure of 

 Crystals: various Ideas. Barlow-Pope's Theory of the Unit- 

 stere. - - The Diffraction of Rontgen-rays in Crystals. 

 Some Remarks about the Investigations of Laue, Bragg, 

 Debije. - - The Deductions of Bragg concerning the Space- 

 lattices of Crystals. Some simple Examples. - - The Chemical 

 Molecule in the Crystalline State. - - Molecular and Atomic 

 Forces; Valency and Coordination. - - The Periodical Arran- 

 gements in Living Nature. - - Disposition of Leaves in Plants. 

 Views on Phyllotaxis. - - A Contrast between Inanimate and 

 Living Nature with respect to the Manifestation of Symmetry. 

 Some Final Remarks. 



1. Hitherto we have dealt exclusively with the symmetry 

 of limited systems. In such figures only a limited number of points 

 correspond to each given point ; the original point can successively be 



