Symmetry of three-dimensional patterns space groups

1 Symmetry of three-dimensional patterns space groups 

There are parallels between the two-and three-dimensional cases. Naturally, mirror lines in two dimensions become mirror planes, and glide lines in two dimensions become glide planes. The glide translation vector, t, is constrained to be equal to half of the relevant lattice vector, T, for the same reason that the two-dimensional glide vector is half of a lattice translation (Chapter 3). 

In addition, the combination of three-dimensional symmetry elements gives rise to a completely new symmetry operator, the screw axis. Screw axes are rototranslational symmetry elements, constituted by a combination of rotation and translation. A screw axis of order n operates on an object by (a) a rotation of 2n/n counter clockwise and then a translation by a vector t parallel to the axis, in a positive direction. The value of n is the order of the screw axis. For example, a screw axis running parallel to the c-axis in an orthorhombic crystal would entail a counter-clockwise rotation in the a - b plane, (001), followed by a translation parallel to +c. This is a right-handed screw rotation. Now if the rotation component of the operator is applied n times, the total rotation is 

Crystals and Crystal Structures. Richard J. D. Tilley 2006 John Wiley Sons, Ltd 

CHS BUILDING CRYSTAL STRUCTURES FROM LATTICES AND SPACE GROUPS 

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