Steven Dutch, Professor Emeritus, Natural and Applied Sciences,
Universityof Wisconsin  Green Bay
To the right of each space group is a listing of coordinate points. These are thecoordinates to which a general point (x,y,z) is transformed by the space group. Origins(called "equivalent points" in the International Tables), areadditional points around which the points are transformed. For example, (0,0,0) refers toa corner of the unit cell, (1/2,1/2,1/2) to the center. All space groups have origin(0,0,0). For a space group with an additional origin (1/2,1/2,1/2), point (x,y,z) is alsotransformed to (1/2+x,1/2+y,1/2+z) and so on.
3fold and 6fold coordinates are tabulated with respect to axes intersecting at 60 degrees. In this oblique coordinate system, coordinates tend to be simple
149 P312 (+x,+y,+z); (y,+xy,+z);
(+yx,x,+z);


150 P321 (+x,+y,+z); (y,+xy,+z);
(+yx,x,+z);


151 P3_{1} 12 (+x,+y,+z); (y,+xy,1/3+z);
(+yx,x,2/3+z);


152 P3_{1} 21 (+x,+y,+z); (y,+xy,1/3+z);
(+yx,x,2/3+z);


153 P3_{2} 12 (+x,+y,+z); (y,+xy,2/3+z);
(+yx,x,1/3+z);


154 P3_{2} 21 (+x,+y,+z); (y,+xy,2/3+z);
(+yx,x,1/3+z);

Rhombohedra are cubes distorted along one of the three fold axes. Coordinates for rhombohedral symmetries can be described in terms of the rhombohedral cell edges or with respect to the threefold symmetry axis system. Coordinates here are given with respect to the threefold symmetry axis system.
155
R32 Origins: (0,0,0); (1/3,2/3,2/3); (2/3,1/3,1/3) (+x,+y,+z); (y,+xy,+z);
(+yx,x,+z);

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Created 13 October 1999, Last Update 11 June 2020