# Hexagonal (6) Space Groups

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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To the right of each space group is a listing of coordinate points. These are the coordinates to which a general point (x,y,z) is transformed by the space group. Origins (called "equivalent points" in the International Tables), are additional points around which the points are transformed. For example, (0,0,0) refers to a 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 also transformed to (1/2+x,1/2+y,1/2+z) and so on.

3-fold and 6-fold coordinates are tabulated with respect to axes intersecting at 60 degrees. In this oblique coordinate system, coordinates tend to be simple

 168     P6(+x, +y, +z); (-y, +x-y, +z); (+y-x, -x, +z); (-x, -y, +z); (+y, +y-x, +z); (+x-y, +x, +z); 169     P61(+x, +y, +z); (-y, +x-y, 1/3+z); (+y-x, -x, 2/3+z); (-x, -y, 1/2+z); (+y, +y-x, 5/6+z); (+x-y, +x, 1/6+z); 170     P65(+x, +y, +z); (-y, +x-y, 2/3+z); (+y-x, -x, 1/3+z); (-x, -y, 1/2+z); (+y, +y-x, 1/6+z); (+x-y, +x, 5/6+z); 171     P62(+x, +y, +z); (-y, +x-y, 2/3+z); (+y-x, -x, 1/3+z); (-x, -y, +z); (+y, +y-x, 2/3+z); (+x-y, +x, 1/3+z); 172     P64(+x, +y, +z); (-y, +x-y, 1/3+z); (+y-x, -x, 2/3+z); (-x, -y, +z); (+y, +y-x, 1/3+z); (+x-y, +x, 2/3+z); 173     P63(+x, +y, +z); (-y, +x-y, +z); (+y-x, -x, +z); (-x, -y, 1/2+z); (+y, +y-x, 1/2+z); (+x-y, +x, 1/2+z);