Point Symmetry

Point symmetry deals with how a motif can be repeated about a fixed central point. In minerals, the motifs being repeated are either crystal faces on the outside of a hand specimen or particular arrangements of atoms in the internal structure. ^[1] The central point is the centre of the crystal or the origin of the unit cell. There are three fundamental point symmetry operations: reflection, rotation, and inversion. Every symmetry element in a crystal’s external form can be described using combinations of these three.

Reflection

A reflection is produced by a mirror plane - indicated by the letter “m” - that passes through the crystal so that the pattern on one side is the mirror image of the pattern on the other. ^[1] Only planes in specific orientations can be mirror planes in a given mineral. A monoclinic mineral has only one possible mirror plane. A triclinic mineral has none. An isometric mineral may have as many as nine. ^[1] The number and orientation of mirror planes is determined by the crystal’s internal structure, not by the observer’s choice.

Rotation

Rotational symmetry involves repeating a motif by uniform rotations about an axis. The notation is a capital “A” with a subscript indicating the number of repeats in 360° of rotation - so A4 means four identical wedge-shaped segments, each produced by a 90° rotation. ^[1] The possible rotation axes in crystals are 1-fold (A1), 2-fold (A2), 3-fold (A3), 4-fold (A4), and 6-fold (A6), corresponding to rotations of 360°, 180°, 120°, 90°, and 60° respectively. ^[1] Five-fold and seven-fold axes do not occur in crystals because no arrangement of atoms with those rotational symmetries can tile three-dimensional space without gaps - this is a geometric constraint, not an accident of nature.

A 1-fold rotation is essentially no symmetry at all: rotating an object 360° returns it to its original position, so every object trivially has A1. ^[1]

Inversion

A crystal with inversion symmetry - indicated by the letter “i” - has the property that any line drawn through the origin finds identical features equidistant on both sides. ^[1] In practice, this means that for every face, edge, or atom at position (x, y, z) in the crystal, there is an identical feature at position (-x, -y, -z). A crystal with inversion looks the same from opposite ends of any axis through its centre. Many common minerals have inversion symmetry; triclinic minerals with only A1 do not necessarily.

Combining the Operations

These three operations - reflection, rotation, and inversion - can be combined, but not arbitrarily. The combinations are limited because the symmetry elements must be compatible with each other and with the translational symmetry of the crystal lattice. The complete set of valid combinations defines the 32 point groups, also called the 32 crystal classes, which describe every possible combination of symmetry that a crystal can have. ^[1]

References

1. Nesse, W. D. (2017). Introduction to Mineralogy, 3rd ed. Oxford University Press.

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