# Pm3,  Pn3

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.  200    Pm3 Simple space group. 2m3 clusters in a P lattice. First 12 points are a 23 cluster. Second 12 are a reflected 23, or a 23 cluster with one sign changed. (+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,-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)   201    Pn3 First 12 points are a 23 cluster. Second 12 are translated to (1/2,1/2,1/2) and reflected. Looks very similar to I23 except the central cluster is reflected. (+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-y,1/2-z); (1/2-z,1/2-x,1/2-y); (1/2-y,1/2-z,1/2-x);( 1/2-x,1/2+y,1/2+z); (1/2-z,1/2+x,1/2+y); (1/2-y,1/2+z,1/2+x);1/2+x,1/2-y,1/2+z); (1/2+z,1/2-x,1/2+y); (1/2+y,1/2-z,1/2+x) 1/2+x,1/2+y,1/2-z); (1/2+z,1/2+x,1/2-y); (1/2+y,1/2+z,1/2-x) 