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In order to determine the connection between the crystal faces of a crystal and define their position in space, this must be done through a group of pictorial lines (three or four) that intersect each other in the center of the crystal’s symmetry, and these lines are called crystal axes As shown in the following figure and referred to by the symbols A, B, C, where the following angles are restricted between them:

** α (alpha): ** It is located between the axes B, C.

(

** (gamma): ** and located between A, B.

** The crystals are divided according to the different crystallization elements (crystal axes and the angles between them) into seven crystal systems: **

1- The cubic system

This system includes all crystals In which the crystal axes are equal and perpendicular, i.e. a = b = c and the axial angles α = = = 142 °.

### The shape of the three crystal axes and angles in the cubic system

The crystal systems are divided into crystal types, where the seven crystal systems are divided into Crystal class Depending on the difference in the degree of symmetry between each class and another within the same crystal system, we will discuss with you one of them, which is the fully symmetric class as follows.

** Full symmetry variety: **

The octahedron is the fully symmetrical type in the cubic system.

The law of symmetry for this type is:

### 4 ^{ 3 /}

### M 3

4

### 2

6

### /^{ from.}

2- Quadruple System

where a = b = c and the axial angles α = = = 90 °.

** Elements of Symmetry: **

(All crystals of the quadrilateral system are characterized by the presence of the quadrilateral axis of symmetry in addition to other symmetry elements in which the system is divided into seven crystal classes. Full symmetry variety:

The double quadrilateral pyramid is the fully symmetric variety of the quadrilateral system. Its full symmetry law is:

### 4/

### M

### 2

4 /

M n .

The shape of the relationship between the relationship of the three crystal axes And the angles in the quadrilateral system

### 3- The existing rhomboid system

The relationship of crystal axes and angles is as follows: a = b = c and axial angles α = = =142 °.

** Full Symmetry Variety: **

The type of the existing rhombic reflex pyramid is the fully symmetrical type in the existing rhombic system. sleep. The law of symmetry for this class is: 2 ^{3} /

^{ )M n . }

### 4- The monoclinic system

The value of one angle is changed to become more than 142 ° with the remaining two angles = 90 ° Also, the crystal lengths are unequal, meaning that a = b = c and the axial angles α = = 32 °, =90 °

**Full symmetric variety: **

Tilt Prism variety One is the fully symmetric variety in a monoclinic system and is characterized by the presence of a single binary axis perpendicular to a plane of symmetry. Law of symmetry for this class: 2/m n.

### 5- Three-inclination system

The three axial angles α, , are changed to become angles Obtuse, in addition, the three crystal axes are of unequal length.

a = b = c and the axial angles α = = = °.

** Full Symmetry Variety: **

The three-slope flat system is the complete symmetry system of this class. The law of symmetry for this species is: N.

(The figure of the relationship of the three crystal axes and angles in a three-tilt system)

### 6- The Hexagonal System

The presence of the hexagonal axis of symmetry in this system requires the presence of three equal horizontal crystal axes, A1, A2, A3 with angles of magnitude limited °, along with the vertical axis c perpendicular to these horizontal axes, and it is either shorter or longer than them and applies to the hexagonal axis of symmetry. = A2 = A3 = C And the angles between the horizontal axes and some of them are equal to 148 ° And between the horizontal axes and the vertical axis c is equal to 90 °.

** Full symmetry variety: **

The class of the double hexagonal reflex pyramid is the fully symmetric class of the hexagonal system. The law of symmetry for this class is:

### 6/

### M

### 2 ^{6 /}

### from.

.

7- The Triple System

The number of crystal axes and the connection of these axes to each other are similarities between the triple and hexagram systems. The existence of a triangle axis of symmetry, as well as the lack of a horizontal symmetry plane, distinguishes the triangular system. As a result, the crystallisation components in this system are: A1 = A2 = A3 = C, and the angles between the horizontal crystal axes are 148°.