Symmetry Elements
The concept of symmetry explains how similar objects (known as motifs) are repeated systematically in space to produce ordered structures where all objects have specific and predictable positions. Symmetry operations "act" on a given object to produce sets of identical objects in prescribed positions. In mineralogy, the interaction of symmetry elements on atoms determines crystal structures, and the systematic repetition of atoms in space is the mechanism that allows unit cells grow into beautiful crystal forms exhibited in hand samples.
The illustrations used below are illustrative of the types of symmetry that are demonstrated in the dance forms used in this module. This is just a start. For a more systematic and comprehensive coverage of symmetry and cyrstallography, follow these links to Teaching Activities and other Web-Based Resources for Teaching Crystallography.
Types of Symmetry
Point Group SymmetrySymmetry elements that all pass through a single point (at the center of a crystal) include:
- Symmetry with respect to a plane: these are mirror planes; objects on one side of the plane are exactly reproduced on the opposite side of the plane; note that the "handedness" of an object changes--a right hand on one side of the mirror becomes a left hand on the other side.
Figure 11.08 from Dyar, Gunter, and Tasa (2008), Mineralogy and Optical Mineralogy. Used by permission of the Mineralogical Society of America. Mirror plane--symmetry with respect to a plane.
- Symmetry with respect to a line: these are rotation axes; possible axes are 2-, 3-, 4-, and 6-fold axes; in these operations, an object is reproduced after a 180o, 120o, 90o, or 60o degree rotation; you can rotate either clockwise or counter-clockwise and the "handedness" of the object remains the same.
Figure 11.04 from Dyar, Gunter, and Tasa (2008), Mineralogy and Optical Mineralogy. Used by permission of the Mineralogical Society of America. The "proper" rotation axes: 2-, 3-, 4-, and 6-fold axes.
Figure 12.13 from Dyar, Gunter, and Tasa (2008), Mineralogy and Optical Mineralogy. Used by permission of the Mineralogical Society of America. Another view of the "proper" rotation axes.
Figure 4.4a from Dyar, Gunter, and Tasa (2008), Mineralogy and Optical Mineralogy. Used by permission of the Mineralogical Society of America. Cube rotating around three mutually perpendicular 4-fold axes.
- Symmetry with respect to a point--this is a "center of symmetry" or "inversion point"; objects are perfectly reproduced equidistant on the opposite side of the point, but the object is inverted and "handedness" is also reversed.
Figure 11.09 from Dyar, Gunter, and Tasa (2008), Mineralogy and Optical Mineralogy. Used by permission of the Mineralogical Society of America. Symmetry with respect to a point--a center of symmetry or inversion point.
- Symmetry that combines rotation and inversion--these are "rotoinversion axes;" handedness reverses in these operations.
Figure 12.15a-e from Dyar, Gunter, and Tasa (2008), Mineralogy and Optical Mineralogy. Used by permission of the Mineralogical Society of America. (a) "Bar 1," center of symmetry or inversion, (b) "Bar 2," mirror plane, (c) "Bar 3," (d) "Bar 4," (e) "Bar 6," 3-fold axis perpendicular to a mirror plane.
Combinations of Symmetry Elements
"Symmetry begets more symmetry." But, the symmetry elements can only come together in specific combinations and at specific angles. For example, a 4-fold axis (a line) that resides in a mirror plane must necessarily generate a second mirror plane located 90o to the first. Or, two mirror planes intersect in a line, and that line actually defines a new rotation axis, but the type of axis is determined by the fixed angle between the planes: two mirrors that intersect at 90o create a 2-fold axis; 45o creates a 4-fold axis; 60o creates a 3-fold axis; 30o creates a 6-fold axis; and all other angles of intersection are not permitted because a multiplicity of mirror planes (i.e. no symmetry at all) will be generated.
Figure 11.12g from Dyar, Gunter, and Tasa (2008), Mineralogy and Optical Mineralogy. Used by permission of the Mineralogical Society of America. A combination of 2-fold axis and mirror planes in the 2-D point group 2mm.
Figure 11.13a from Dyar, Gunter, and Tasa (2008), Mineralogy and Optical Mineralogy. Used by permission of the Mineralogical Society of America. A combination of 4-fold axis and mirror planes in the 2-D point group 4mm.
Translational Symmetry
Translation repeats objects by movement along a line at specific distances and angles. Translational symmetry operations include:
Figure 11.3 from Dyar, Gunter, and Tasa (2008), Mineralogy and Optical Mineralogy. Used by permission of the Mineralogical Society of America.
Figure 11.14a from Dyar, Gunter, and Tasa (2008), Mineralogy and Optical Mineralogy. Used by permission of the Mineralogical Society of America. Translational symmetry in one dimension.
- Glide Planes: a combination of mirror operations and translation.
Figure 11.10 from Dyar, Gunter, and Tasa (2008), Mineralogy and Optical Mineralogy. Used by permission of the Mineralogical Society of America. Glide plane--a combination of mirror and translation symmetries.
- Screw Axes: a combination of rotation axes and translation.
Figure 11.23 from Dyar, Gunter, and Tasa (2008), Mineralogy and Optical Mineralogy. Used by permission of the Mineralogical Society of America. Screw axis, a combination of rotation and translation.
Figure 11.69b from Dyar, Gunter, and Tasa (2008), Mineralogy and Optical Mineralogy. Used by permission of the Mineralogical Society of America. Four fold screw axis, 41, showing 1/4 translation for every 90o turn of the motif.
Unit Cells
Unit cells are defined by translation vectors, with specific lengths and at required angles, to define 2- and 3- dimensional lattices. The symmetry elements identified above must occur at specific locations in the unit cell, and the symmetry elements that are permitted are dictated by the intrinsic symmetry of the unit cell itself.
Figure 11.14b from Dyar, Gunter, and Tasa (2008), Mineralogy and Optical Mineralogy. Used by permission of the Mineralogical Society of America.
Using Art Prints to Visualize Symmetry: The periodic drawings (tesselations) of M.C. Escher and Francois Brisse are often used to demonstrate 2-D lattices and the permitted locations of symmetry elements. Click on these links to find illustrative and fun teaching activities.
Now, lt's see how these symmetry operations can be demonstrated in dance forms.