THE PROPERTIES OF MINERALS. ()71 



mirror reflections. All faces of a crystal are symmetrical about a 

 point, and in some crystals are grouped in accordance with certain 

 definite planes of symmetry ; that is, a plane which is capable of dividing 

 the crystal in halves, whose internal and external foi'nis are to each 

 other as a reflection, and also with axes of symmetry. Axes of sym- 

 metry may be defined as lines passing through the center of a crystal 

 about which it can be revolved through an angle of <;0 degrees (hex- 

 agonal axis), J)0 degrees (tetragonal axis), 1-0 degrees (trigonal axis), 

 or 180 degrees (digonal axis), without its i)osition as a whole in space 

 being changed. All crystals maybe grouped under seven systems in 

 thirty-two types in accordance with their symmetry, as follows : 



Isometric .sij.stem. — Types: Holohedral, tetrahedral-hemihedral, 

 pentagonal -hemihedral, plagiohedral-hemihedral, and tetartohedral. 

 The most general form possesses nine planes of symmetry, of which 

 three (S planes) are at right angles to one another, and six ( 2i' planes) 

 which bisect the angles formed by the S planes. There are thirteen 

 axes of symmetry, of which three are tetragonal ( □ ) and are per- 

 pendicular to the S planes and parallel to the intersection of two 

 S and two ^ planes; four are trigonal (A) and are parallel to 

 the intersections of three 2 planes; six are digonal (O) and are 

 l)erpendicular to the 2 planes and parallel to the intersection of an 

 S and 2:' plane. 



The types are illustrated by nu)dels which may be described as 

 fi)llows : 



Holohedral type. — The hexoctahedron. Form having four trigonal (A), three 

 tetragonal (D), and six digonal (O) axes, and nine planes of symmetry, three S and 

 six 3 planes. 



Tetrahedral-hemihedral type. — The ti^trahedron. Form having four trigonal (A) 

 and three digonal (D) axes, with six 2 planes of symmetry. 



Pentayonal-hemihedral type. — The pentagonal dodecahedron. Form having four 

 trigonal (A) and three digonal (J) axes, with three S planes of symmetry. 



Playiohedral-hemihedral type. — The pentagonal ikositetrabedron. Form having 

 four trigonal (A), three tetragonal (D), and six digonal (J) axes, with no plane of 

 symmetry. 



Tetartohedral type. — The tetrahedron. Form having four trigonal (A) and three 

 digonal (Ojaxes, with no plane of symmetry. 



Hexagonal system. — Types: Holohedral, hemimorphic, pyramidal- 

 hemihedral, trapezohedral-hemihedral, and hemimori)hic-hemihedral. 

 The most general form possesses seven i)lanes of symmetry, of which 

 six (S planes) intersect each other at angles of GO degrees, and one 

 (C plane) which is at right angles to these. 



There are seven axes of symmetry, of which one, the hexagonal (O) 

 is perpendicular to the C plane and parallel to the intersections of the 

 S planes; six are digonal (O) and lie in the C plane at right angles to 

 the S i^lanes. 



The types are illustrated by models which are described as follows: 



Holohedral type. — The dihexagoual bipyramid. Form having one hexagonal (O) 

 and six digonal (O) axes, with one C and six S planes of symmetry. 



