SUMMARY OF DEVELOPMENT 377 



Crystals of this class possess an alternating axis about which 

 the faces of the crystals alternate, and are hence named Ame- 

 baxial (amoibos = alternating, axon = an axis). 



The above manifestly exhausts all possible combinations of axes and 

 planes of symmetry with periods of 2, 3, 4, 6, producing thirty groups of 

 symmetry. Two other types of symmetry are possible : 



1. Forms possessing a plane of symmetry only. They may be viewed as 



containing a one-fold axis parallel to the plane, and hence referred 

 to the Hedral class — one-fold. 



2. Forms without symmetry, the asymmetric group of Groth. They are 



reproduced by a rotation of 360 degrees about a one-fold axis, and 

 hence may be referred to the Axial class — one-fold. 



This manifestly exhausts all possible types of symmetry produced by 

 rotation and reflection in forms having periods 1, 2, 3, 4, 6. Thirty-two 

 groups of symmetry are thus seen to be developed, occurring in seven 

 classes. 



Summary of development. — The results reached and the periods of 

 each class are exhibited in the following table : 



Period of Axis. 

 Singular axis. No singular 

 I. Symmetry by rotation about an axis. axis. 



1. One axis— Axial 1, 2, 3, 4, 6 



2. Many axes — Poly axial 2, 3, 4, 6 2, 4 



II. Symmetry by reflection about planes. 



A. One axis: 



3. Plane normal to axis — Orthoaxial. ... 2, 3, 4, 6 



4. Planes parallel to axis — Hedral 1, 2, 3, 4, 6 



B. Many axes : 



5. Planes coincident with axes — Ortho- 



hedral 2, 3, 4, 6 2, 4 



6. Planes alternating with axes — Ameba- 



Jiedral -, -, 4, 6 -, 4 



III. Symmetry by combined rotation and reflection. 



7. Alternating axis only — Amebaxial. ... 2, -, 4, 6 



REFERENCE OF THE CLASSES TO THE ACCEPTED SYSTEMS OF CRYSTALS 



We may inquire what relation the preceding development bears to the 

 crystal systems. An examination of the above table shows that the 

 groups fall into natural assemblages, based upon the character and period 

 of their axes of symmetry. These are the accepted systems, as follows : 



1. Crystals possessing no singular axis of symmetry, characterized by 

 three equal axes, constitute the Isometric system, which develops in two- 

 and four-fold divisions. 



2. Six-fold groups — Hexagonal system. 



3. Four-fold groups — Tetragonal system. 



4. Three-fold groups — Trigonal system. 



