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 Netplane. - - Points 

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

 general Properties of Space-lattices. The Symmetry of Bravais' 

 Space-lattices. -- Elements of Pseudo symmetry. -- General Sym- 

 metry-Relations in Endless Systems. The Problem of the 

 General Deduction of All Homogeneous Arrangements in Space. - 

 The Fundamental Domain in Endless Homogeneous Structures. 

 Enantiomorphism and Congruency of Structural Units. The 

 Theories of Sohncke, Von Fedorow, Schoenflies. General 

 Result. - - Endless Periodical Patterns and Crystalstructure. - 

 The Crystal as a Tridimensional Symmetrical Pattern. -- Con- 

 tinuous and Discontinuous Properties of Crystalline Matter. - 

 The Space-lattice as as Geometrical Expression of Hauy'sL^w. 

 The Chemical Structure of Crystals: various Ideas. - - Barlow- 

 Pope's Theory of the Unitstere. -- The Diffraction of Ront gen- 

 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 

 Arrangements in Living Nature. Disposition of Leaves in 

 Plants; Views on Phyllotaxis. -- A Contrast between Inanimate 

 and Living Nature with respect to the manifestation of Sym- 

 metry. Some Final Remarks. 



1. Hitherto we have dealt exclusively with the symmetry 

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

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



