Space Groups

Last update 06. September 2010


Example visualisations

Group 15

Group 194

Group 225 (Gold)



There exist 230 symmetry space groups in 3D space, which are shown as 2D projections in the ITC Tables Volume A. The SpaceGroupVisualizer is a visualisation software that is the perfect addition to the ITC Tables Volume A, as it visualizes all 230 space groups in 3D, lets you freele rotate and move the crystal cells, offers real 3D impressions with Green/Magenta or Red/Cyan glasses, lets you select subsets of symmetries and animates the effect of symmetries on the unique cell elements. The SpaceGroupVisualizer therefore allows you to get good grasp on the symmetry structure of a space group much more easily.



  Download the new full version 2.3.1 (Windows)
This is now the full version of the SpaceGroupVisualizer, where all space groups are available.
  Here is a screenshot, and two screen movies (DivX): Movie 1, Movie 2.

The latest:

  Release Version of SpaceGroupVisualizer v2.3.1.

Recently added features:

  Interact even better with the 3D-Visualization by using a 3D-Mouse. The Space Group Visualizer supports 3D-Mice by 3dconnexion.
  If you have access to the online edition of ITC Volume A, the Space Group Visualizer always shows you the ITC group information for the currently visualized group in a separate window. You can browse through the groups either with the Space Group Visualizer or with the website and always get a synchronized view. You can enable this feature through the ‘Window’ Menu in the SGV Window.
  You can easily animate arbitrary rotations of the visualization: select the option ‘Animate Scene Rotation’ in the visualization menu and give the visualization a push with the mouse.

The initial version of this software was developed by C. Perwass and E. Hitzer during a visit of E. Hitzer at the university of Kiel in late summer 2005. The Geometric Algebra description of the symmetry operators is based on work by Jeremy Holt and David Hestenes.

The idea behind this software is to visualise the different space groups by drawing asymmetric elements at the general positions that are generated by the symmetries of a group. Note that this program is currently under development but it already generates a usable visualisation. Once you have downloaded the program ‘SpaceGroupVizSetup.exe’, execute it in order to install the software. You will then find an icon on your desktop that allows you to start the program.

Please send us your comments and suggestions for this script so we can develop a visualisation that is well suited for a large audience.



The following is a list of publications that describe the theoretical background of space groups.

  Point Groups and Space Groups in Geometric Algebra, by David Hestenes.
  Crystal Cell and Space Lattice Symmetries in Clifford Geometric Algebra, by E.M.S. Hitzer, C. Perwass .
  Full Geometric Description of all Symmetry Elements of Crystal Space Groups by the Suitable Choice of Only Three Vectors for Each Bravais Cell or Crystal Family, by E.M.S. Hitzer, C. Perwass.
  Interactive Visualization of Full Geometric Description of Crystal Space Groups, by C. Perwass, E.M.S. Hitzer.
  Three Vector Generation of Crystal Space Groups in Geometric Algebra, by E.M.S. Hitzer, C. Perwass.
  Space Group Visualizer for Monoclinic Space Groups, by C. Perwass, E.M.S. Hitzer.
  The Space Group Visualizer, by E.M.S. Hitzer, C. Perwass.
  The Hidden Beauty of Gold, by E.M.S. Hitzer, C. Perwass.
  Symmetry of Orthorhombic Materials and Interactive 3D-Visualization in Geometric Algebra, by D. Ichikawa, E.M.S. Hitzer.


COLOR="#3300CC">Symmetry of Orthorhombic Materials and Interactive 3D-Visualization in Geometric Algebra, by D. Ichikawa, E.M.S. Hitzer.