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The symmetry of a single close-packed layer of spheres is 6*mm* . It has 2-
, 3- and 6- fold axes of rotation normal to its plane as shown in Fig. 3. In
addition it has three symmetry planes--one perpendicular to the *x* -axis,
one perpendicular to the *y* axis and the third equally inclined to
*x* and *y* . When two or more layers are stacked over each other in a
close-packing the resulting structure retains all the three symmetry planes and
has at least 3-fold axes parallel to [00.1] through the points 000,
0 and 0 as shown in Fig.
4. Such a structure belongs to the trigonal system and has a space group *P*3*m*1
or *R*3*m*1, according as the lattice is hexagonal or rhombohedral. This
represents the lowest symmetry of a close-packing of spheres comprised of a
completely arbitrary periodic stacking sequence of close-packed layers. If the
arbitrariness in stacking successive layers in the unit cell is limited then
higher symmetries can also result. It can be shown^{2,6} that it is possible
to have three additional symmetry elements, namely, a centre of symmetry
(, a mirror plane perpendicular to [00.1], and a screw axis
6_{3}. It was shown by Belov^{7} that consistent combinations of these
symmetry elements can give rise to only eight possible space groups:

*P*3*m*1, , , *P*6_{3}*mc*

*P*6_{3}/*mmc*, *R*3*m*, and

**Copyright © 1981, 1997 International Union of
Crystallography**