The symmetry of crystals. Representation of Bravais lattices

The repetition modes by translation in crystals must be compatible with the possible crystallographic point groups (the 32 crystal classes), and this is why we find only 14 types of translational lattices which are compatible with the crystal classes.

These types of lattices (translational repetiton modes) are known as the Bravais lattices...

The lattice points (black and gray points) do not represent atoms. They represent places in space that are indistinguishable from each other. An observer at any of these points would not be able to distinguish in which of them he is located.

Symbols  P   C    I    F   R   refer to the different lattice types:
• P = primitive, there is only 1 reticular point inside the cell. It has 1 point in each of the 8 corners, but only 1/8 of point inside the cell, that is, 8/8=1 complete lattice point in the cell.
• C = centered on the two faces perpendicular to the c axis of the unit cell.  It has 2 complete lattice points within the cell, that is, 1 complete point corresponding to the 8/8 of vertices and 2/2 that correspond to the faces perpendicular to the c axis. In total: 8/8 + 2/2 = 2.
• I = centered in the body of the cell. It has 2 complete reticular points within the cell, 1 complete in the center of the body, and the usual 8/8.
• F = centered in all faces of the cell.  It has 4 complete lattice points within the cell, 6/2 corresponding to the centers of the faces and the usual 8/8.
• R = primitive, identical cell axes and cell angles, or hexagonal two times body centered containing 3 complete lattice points, that is, 2 on a diagonal of the cell body and the usual 8/8.

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