Day 2


Symmetry (n): 1. A relationship of characteristic correspondence, equivalence, or identity among constituents of a system or between different systems. 2. Exact correspondence of form and constituent configuration on opposite sides of a dividing line, or plane, or about a center or axis. 3. Beauty as a result of balance or harmonious arrangement.

Why do we study symmetry? Re-read the previous definition carefully. Symmetry is a fascinating mineralogy topic, and descriptions of symmetry are fundamental to the descriptions of crystals. As with any topic requiring spatially complex and three-dimensional perspectives, symmetry can be confusing and difficult to understand. So..., why do we study symmetry? It is because the physical properties of all minerals (including crystal forms, cleavage, fracture, hardness, thermal conductivity, and interactions with light) tend to vary in directions that have the same symmetry as the atomic structure of the mineral. (The preceeding statement is paraphrased from a fundamental postulate of physics - known as Neumann's Principle). Therefore, mineral symmetry can be a tremendous aid in identifying, characterizing, and understanding minerals.

All two-dimensional patterns and three-dimensional objects, including minerals, may be grouped according to the symmetry they possess. Symmetry can be defined as "invariance to an operation". A symmetry element is the geometrical feature, such as a point, line or plane, that we can use to visualize order in an arrangement. A symmetry operation is the act of changing an object or arrangement of objects, by inversion, rotation, or reflection. Symmetry can be defined as "invariance to an operation,Ó which is to say that one or more points in a design, etc. will appear unchanged, or unmoved, by the symmetry operation. Objects that are invariant to rotation are said to contain a rotation axis. Objects invariant to a reflection are said to contain a mirror plane.

¥ If we ignore the possibility of translation (that's covered in lecture #6), symmetry operations also have the property that repeating the operation will eventually return a part of an object or pattern to its original position.

The collection of symmetry operations (rotation, reflection, inversion and their combinations) that characterize an object all intersect at its center, i.e., in a point. Therefore, the collection of symmetry operations that characterize a crystal are termed the point symmetry, point group, or crystal class of the object.

Objects invariant to the combined operation of rotation about an axis followed by reflection across a plane perpendicular to that axis contain a rotoreflection axis. If an object possesses 2-fold rotoreflection about one axis, it possesses 2-fold rotoreflection symmetry about all axes. For this reason, 2-fold rotoreflection symmetry is given a special name: inversion. Because the orientation of the axis does not matter, the symmetry element is termed an inversion center (the one point unmoved by this combined operation). Every rotoreflection operation may be represented instead by the combined operations of rotation followed by inversion (rotoinversion). For various reasons, our text (and most other texts) use rotoinversion axes rather than rotoreflection axes.

A crystal may possess only certain combinations of symmetry elements. Only 32 possibilities exist and these are the 32 crystal classes or crystallographic point groups (see pages 31-32). Every mineral belongs to one of these crystal classes. Details of the 32 point groups are given in K&H (p. 63-100).

Only certain combinations of symmetry elements are possible and, for the same reasons, some symmetry elements require the presence of others. It is useful to remember that the line of intersection of any two mirror planes must be a rotation axis. If the mirror planes intersect at 90°, the line of intersection will be a 2-fold axis; at 60° = 3-fold, at 45° = 4-fold, at 30° = 6-fold. Think about it....


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