# f432, f4132, i432

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.

 209      f432 Origins: (0,0,0);  (0,1/2,1/2); (1/2,0,1/2); (1/2,1/2,0) Simple space group. 432 clusters in an F lattice. First 12 points are a 23 cluster. Second 12 generated from first by interchanging + and -, and interchanging y and z. (+x,+y,+z); (+z,+x,+y); (+y,+z,+x)(+x,-y,-z); (+z,-x,-y); (+y,-z,-x) (-x,+y,-z); (-z,+x,-y); (-y,+z,-x)(-x,-y,+z); (-z,-x,+y); (-y,-z,+x) (-x,-z,-y); (-y,-x,-z); (-z,-y,-x)(-x,+z,+y); (-y,+x,+z); (-z,+y,+x) (+x,-z,+y); (+y,-x,+z); (+z,-y,+x(+x,+z,-y); (+y,+x,-z); (+z,+y,-x) 210      f4132Looks superficially like 203 (Fd3) except the interior point clusters are rotated rather than reflected. First 12 points are a 23 cluster. Second set derived as for f432 and shifting 1/4 in each direction. (+x,+y,+z); (+z,+x,+y); (+y,+z,+x)(+x,-y,-z); (+z,-x,-y); (+y,-z,-x) (-x,+y,-z); (-z,+x,-y); (-y,+z,-x)(-x,-y,+z); (-z,-x,+y); (-y,-z,+x) (1/4-x,1/4-z,1/4-y); (1/4-y,1/4-x,1/4-z); (1/4-z,1/4-y,1/4-x) (1/4-x,1/4+z,1/4+y); (1/4-y,1/4+x,1/4+z); (1/4-z,1/4+y,1/4+x) (1/4+x,1/4-z,1/4+y); (1/4+y,1/4-x,1/4+z); (1/4+z,1/4-y,1/4+x (1/4+x,1/4+z,1/4-y); (1/4+y,1/4+x,1/4-z); (1/4+z,1/4+y,1/4-x) 211      i432 Origins: (0,0,0); (1/2,1/2,1/2) Simple space group. 432 clusters in an I lattice. First 12 points are a 23 cluster. Second 12 generated from first by interchanging + and -, and interchanging y and z. (+x,+y,+z); (+z,+x,+y); (+y,+z,+x)(+x,-y,-z); (+z,-x,-y); (+y,-z,-x) (-x,+y,-z); (-z,+x,-y); (-y,+z,-x)(-x,-y,+z); (-z,-x,+y); (-y,-z,+x) (-x,-z,-y); (-y,-x,-z); (-z,-y,-x)(-x,+z,+y); (-y,+x,+z); (-z,+y,+x) (+x,-z,+y); (+y,-x,+z); (+z,-y,+x(+x,+z,-y); (+y,+x,-z); (+z,+y,-x)