# p432, p4232

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 or two octants of the cube or stereogram might contain motifs, only those octants are portrayed.

 207      p432 Simple space group. 432 clusters in a P 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) 208       p4232 (+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) (1/2-x,1/2-z,1/2-y); (1/2-y,1/2-x,1/2-z); (1/2-z,1/2-y,1/2-x)(1/2-x,1/2+z,1/2+y); (1/2-y,1/2+x,1/2+z); (1/2-z,1/2+y,1/2+x)(1/2+x,1/2-z,1/2+y); (1/2+y,1/2-x,1/2+z); (1/2+z,1/2-y,1/2+x)(1/2+x,1/2+z,1/2-y); (1/2+y,1/2+x,1/2-z); (1/2+z,1/2+y,1/2-x) First 12 points are a 23 cluster. Second 12 are the second 12 of a 432 cluster translated to (1/2,1/2,1/2) Looks like I23 and Pn3 except central cluster is rotated 90 degrees. Not an I lattice because the central set of 12 points is not the same as the corner sets.

Created 23 July 2001, Last Update 11 June 2020