# Tetragonal (4mm); 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.

 99      P4mm (+x, +y, +z); (-x, -y, +z); (-x, +y, +z); (+x, -y, +z); (+y, +x, +z); (-y, -x, +z); (-y, +x, +z); (+y, -x, +z); 100      P4bm (+x, +y, +z); (-x, -y, +z); (1/2-x, 1/2+y, +z); (1/2+x, 1/2-y, +z); (+y, +x, +z); (-y, -x, +z); (1/2-y, 1/2+x, +z); (1/2+y, 1/2-x, +z); 101     P42cm (+x, +y, +z); (-x, -y, +z); (-x, +y, 1/2+z); (+x, -y, 1/2+z); (+y, +x, +z); (-y, -x, +z); (-y, +x, 1/2+z); (+y, -x, 1/2+z); 102     P42nm (+x, +y, +z); (-x, -y, +z); (1/2-x, 1/2+y, 1/2+z); (1/2+x, 1/2-y, 1/2+z); (+y, +x, +z); (-y, -x, +z); (1/2-y, 1/2+x, 1/2+z); (1/2+y, 1/2-x, 1/2+z); 103     P4cc (+x, +y, +z); (-x, -y, +z); (-x, +y, 1/2+z); (+x, -y, 1/2+z); (-y, +x, +z); (+y, -x, +z); (+y, +x, 1/2+z); (-y, -x, 1/2+z) 104     P4nc (+x, +y, +z); (-x, -y, +z); (1/2-x, 1/2 +y, 1/2+z); (1/2+x, 1/2-y, 1/2+z); (-y, +x, +z); (+y, -x, +z); (1/2+y, 1/2 +x, 1/2+z); (1/2-y, 1/2 -x, 1/2+z) 105     P42mc (+x, +y, +z); (-x, -y, +z); (-x, +y, +z); (+x, -y, +z);  (+y, +x, 1/2+z); (-y, -x, 1/2+z); (-y, +x, 1/2+z); (+y, -x, 1/2+z); 106     P42bc (+x, +y, +z); (-y, +x, 1/2+z); (1/2-x, 1/2+y, +z); (1/2+y, 1/2+x, 1/2+z) (-y, -x, +z); (+y, -x, 1/2+z); (1/2+x, 1/2-y, +z); (1/2-x, 1/2-y, 1/2+z); 107     I4mmOrigins: (0, 0, 0 (1/2.1/2, 1/2); (+x, +y, +z); (-x, -y, +z); (-x, +y, +z); (+x, -y, +z); (+y, +x, +z); (-y, -x, +z); (-y, +x, +z); (+y, -x, +z) 108     I4cm Origins: (0, 0, 0 (1/2.1/2, 1/2); (+x, +y, +z); (-x, -y, +z); (1/2-x, 1/2+y, +z); (1/2+x, 1/2-y, +z); (+y, +x, +z); (-y, -x, +z); (1/2-y, 1/2+x, +z); (1/2+y, 1/2-x, +z) 109     I41mdOrigins: (0, 0, 0 (1/2.1/2, 1/2); (+x, +y, +z); (-x, -y, +z); (+y, 1/2+x, 1/4+z); (-y, 1/2-x, 1/4+z) (-x, +y, +z); (+x, -y, +z); (-y, 1/2+x, 1/4+z); (+y, 1/2-x, 1/4+z) 110     I41cd Origins: (0, 0, 0);  (1/2.1/2, 1/2); (+x, +y, +z); (-x, -y, 1/2+z); (+y, 1/2+x, 3/4+z); (-y, 1/2-x, 3/4+z) (-x, +y, +z); (+x, -y, 1/2+z); (-y, 1/2+x, 1/4+z); (+y, 1/2-x, 1/4+z)

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Created 30 March 1999, Last Update 15 December 2011

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