By observing the crystals, you can come to a conclusion, that they are not only convex polyhedrons, limited by flat walls, but that they also exhibit symmetry phenomena. There are three types of symmetry about its three elements: straight, plane and point.
Simple, with respect to which the crystals are symmetrical, they are called axes of symmetry. The axis of symmetry is straight through the crystal, around which the crystal rotated 360 ° coincides with its original position n times.
There are axes of symmetry: twice, triples, fourfold and sixfold. A two-fold axis is called such an axis of symmetry, around which the crystal is rotated 360 ° 2 times it takes the same position. The angle of rotation of the double axis is 360/2 = 180°. The threefold axis is the axis of symmetry, around which the crystal, when rotated by 360 °, occupies the same position 3 times. The rotation angle of the triple axis is 360/3 = 120°, rotation angle of the quadruple axis - 360/4 = 90°, the angle of rotation of the six-fold axis 360/6 = 60°.
A characteristic cross-section perpendicular to the two-fold axis of symmetry is a rectangle or a rhombus, to the three-fold axis - an equilateral triangle, to the quadruple axis - square, and up to six - hexagon. The fivefold axis does not exist in the crystal world, Also, axes with a multiplicity greater than six are not known.
Crystals with different axes of symmetry: a - double axis, b - triple axis, c - quad axis, d - sixfold axis.
The crystals of some substances do not have any axis of symmetry, others only have one, finally there are crystals, with a greater number of equal or different symmetry axes. An example would be a cuboid, in which they occur 3 tandem axes perpendicular to each other, passing through the centers of opposite walls.
The plane of symmetry divides the crystal into two parts, which are so related to each other, like the object to be reflected in a flat mirror or the left hand to the right. Each point in the crystal has the same point on the other side of the plane of symmetry and the same distance from it. Crystals of some substances do not have any plane of symmetry, others only have one, and in some there are several of them. An example would be a cuboid, in which they occur 3 planes of symmetry perpendicular to each other. Most, because until 9 plane of symmetry has a cube.
Symmetry elements in crystals: a - a crystal with three double axes, b - plane of symmetry, c - three planes of symmetry perpendicular to each other.
The point of symmetry in a crystal is called the center of symmetry. Each point on the crystal having the center of symmetry corresponds to an analogous point on the other side of the center of symmetry. This point is on the straight line drawn through the selected point on the crystal and through the center of symmetry, at the same distance from the center of symmetry as the selected point on the crystal. If any level is to be considered, it is by the action of the center of symmetry that a plane parallel to it is obtained. The center of symmetry is therefore in the crystals, in which there are only pairs of parallel planes.
There are numerous examples of crystals, in which there is only one symmetry element, e.g.. center of symmetry, plane of symmetry or any axis of symmetry (twice, tricot, four times or six times). In many crystals, two or more symmetry elements are found, e.g.. axis of symmetry and center of symmetry, axis of symmetry and several planes of symmetry etc.. The greatest number of symmetry elements, namely 3 quad axles, 4 triple axles, 6 double axis, 9 symmetry planes and center of symmetry, they have such regular lumps, like a cube and an octahedron.
Planes of symmetry in a cube.
On the basis of the identified elements of symmetry, crystals of various substances are classified as crystallographic classes. Crystals of different substances may have the same elements of symmetry. The crystallographic class therefore includes different crystal forms having the same set of symmetry elements. Apart from the lowest class, without any symmetry elements, there is a class that has only a center of symmetry and classes that have only a double axis or only a plane of symmetry, or a tandem axle, plane of symmetry and center of symmetry. The classes with the higher symmetry are those that include crystals with three double axes, with one double axis and two planes, with three double axes, three planes and centers of symmetry, etc..