Im3m, Ia3d

Steven Dutch, Professor Emeritus, Natural and Applied Sciences, Universityof Wisconsin - Green Bay

For the isometric space groups, we encounter a visualization problem. There is no longer a single principal symmetry direction we can look along. The diagonal 3-fold symmetry axes rotate the motif into planes perpendicular to the plane of the diagram so they are seen edgewise. So for isometric space groups,  three modes of visualization are employed. First is an oblique drawing of the cubic unit cell with the R motif on a smaller cube. Second is an oblique drawing with stereograms replacing the small cube. The stereograms are drawn in standard crystallographic style without any attempt to represent the projections in perspective. Finally there is a view of the unit cell and stereograms viewed perpendicular to a face. For cases like screw axes where only one octant of the cube or stereogram might be present, only that octant is portrayed.

229 Im3m

Hexoctahedral point groups in a body centered cubic lattice

(+x, +y, +z); (+z, +x, +y); (+y, +z, +x); (+x, +z, +y); (+y, +x, +z); (+z, +y, +x)
(+x, -y, -z); (+z, -x, -y); (+y, -z, -x); (+x, -z, -y); (+y, -x, -z); (+z, -y, -x)
(-x, +y, -z); (-z, +x, -y); (-y, +z, -x); (-x, +z, -y); (-y, +x, -z); (-z, +y, -x)
(-x, -y, +z); (-z, -x, +y); (-y, -z, +x); (-x, -z, +y); (-y, -x, +z); (-z, -y, +x)
(-x, -y, -z); (-z, -x, -y); (-y, -z, -x); (-x, -z, -y); (-y, -x, -z); (-z, -y, -x)
(-x, +y, +z); (-z, +x, +y); (-y, +z, +x); (-x, +z, +y); (-y, +x, +z); (-z, +y, +x)
(+x, -y, +z); (+z, -x, +y); (+y, -z, +x); (+x, -z, +y); (+y, -x, +z); (+z, -y, +x)
(+x, +y, -z); (+z, +x, -y); (+y, +z, -x); (+x, +z, -y); (+y, +x, -z); (+z, +y, -x)

230 Ia3d

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Created 23 July 2001, Last Update 11 June 2020