E. Hitzer

**Interactive Visualization of
Plane Space Groups with the Space Group Visualizer**

*preprint*,
36 pp. (2010), with 32 color figures.

**Introduction:**
This set of instructions shows how to successfully display the 17 two-dimensional
(2D) space groups in the interactive crystal symmetry software Space Group Visualizer
(SGV) [6]. The SGV is described in [4]. It is based on a new type of powerful
geometric algebra visualization platform [5].

The principle is to select in the SGV a three-dimensional super space group and by
orthogonal projection produce a view of the desired plane 2D space group. The choice
of 3D super space group is summarized in the lookup table Table 1. The direction of
view for the orthographic projection needs to be adapted only for displaying the plane
2D space groups Nos. 3, 4 and 5. In all other cases space group selection followed by
orthographic projection immediately displays one cell of the desired plane 2D space
group.

The full symmetry selection, interactivity and animation features for 3D space
groups offered by the SGV software become thus also available for plane 2D space
groups. A special advantage of this visualization method is, that by canceling the orthographic
projection (remove the tick mark of Orthographic View in drop down menu
Visualization), every plane 2D space group is seen to be a *subgroup* of a corresponding
3D super space group.

The convenient lookup table Table 1 lists column by column: 2D crystal system,
the 2D space group number [1], the Hermann-Maugin name of the 2D space group [1],
(and the Coxeter-Hestenes-Holt symbols [2, 3] in brackets) the crystal system of 3D
super space group, the 3D point group number of the 3D super space group [1], the
3D space super group number [1], and the the Hermann-Maugin name of the 3D space
super group (and the Coxeter-Hestenes-Holt symbols [2, 3] in brackets).

The colors become much more vivid with a choice of a black background, which
is strongly recommended for presentations: Drop down menu Visualization, Color
Scheme, Black Background. For printability this set of instructions uses white background
pictures created with the SGV.

The SGV can be purchased from [7]. The current instructions apply to the Space
Group Visualizer, version 2.2.5.

**References: **

[1] T. Hahn, Int. Tables of Crystallography A, Springer, Dordrecht, 2005; Online:
it.iucr.org

[2] D. Hestenes, J. Holt, The Crystallographic Space Groups in Geometric Algebra,
Journal of Mathematical Physics, Vol. 48, No. 2 (2007).

[3] H.S.M. Coxeter, Discrete groups generated by reflections, Ann. of Math. 35, pp.
588?621 (1934); H.S.M. Coxeter, W.O.J. Moser, Generators and Relations for
Discrete Groups, Springer, 4th ed., 1980.

[4] E.Hitzer, C. Perwass, Interactive 3D Space Group Visualization with CLUCalc
and the Clifford Geometric Algebra Description of Space Groups, accepted for
Proc. 8th Int. Conf. on Clifford Algebras (ICCA8) and their Appl. in Math. Phys.,
26-30 May 2008, Las Campinas, Brazil (2008).

[5] C. Perwass, CLUCalc - a visual calculator, www.clucalc.info

[6] C. Perwass, E. Hitzer, Space Group Visualizer, www.spacegroup.info (2005).

[7] The Space Group Visualizer can be purchased from Raytrix GmbH,
www.raytrix.de.
Free download for non-commercial users:
www.spacegroup.info .

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