GY 302 Crystallography and Mineralogy Supplementary Lecture Notes 2 Crystal Systems and Symmetry Symmetry explanations below are from http://www.cartage.org.lb Plagioclase Triclinic 6 Gypsum, Orthoclase Monoclinic 5 Aragonite, Staurolite Orthorhombic 4 Quartz, Calcite Trigonal 3b Apatite Hexagonal 3a Zircon Tetragonal 2 Halite, Galena, Pyrite Cubic 1 Mineral examplesAngles between axesAxes Crystal System 2-fold Rotation Axis - If an object appears identical after a rotation of 180o, that is twice in a 360o rotation, then it is said to have a 2-fold rotation axis (360/180 = 2). Note that in these examples the axes we are referring to are imaginary lines that extend toward you perpendicular to the page or blackboard. A filled oval shape represents the point where the 2-fold rotation axis intersects the page.http://www.cartage.org.lb 3-fold Rotation Axis - Objects that repeat themselves upon rotation of 120o are said to have a 3-fold axis of rotational symmetry (360/120 =3), and they will repeat 3 times in a 360o rotation. A filled triangle is used to symbolize the location of 3-fold rotation axis. 4-fold Rotation Axis - If an object repeats itself after 90o of rotation, it will repeat 4 times in a 360o rotation, as illustrated previously. A filled square is used to symbolize the location of 4-fold axis of rotational symmetry. 6-fold Rotation Axis -If rotation of 60o about an axis causes the object to repeat itself, then it has 6-fold axis of rotational symmetry (360/60=6). A filled hexagon is used as the symbol for a 6-fold rotation axis. 2-fold Rotoinversion - The operation of 2-fold rotoinversion involves first rotating the object by 180 degrees then inverting it through an inversion center. This operation is equivalent to having a mirror plane perpendicular to the 2-fold rotoinversion axis. 3-fold Rotoinversion - This involves rotating the object by 120o (360/3 = 120), and inverting through a center. A cube is good example of an object that possesses 3-fold rotoinversion axes.Diagrams below are from Ness, W. D., 2000. Introduction to Mineralogy. Oxford University Press, New York 442pp 4-fold Rotoinversion - This involves rotation of the object by 90o then inverting through a center. Note that an object possessing a 4- fold rotoinversion axis will have two faces on top and two identical faces upside down on the bottom, if the axis is held in the vertical position. 6-fold Rotoinversion - involves rotating the object by 60o and inverting through a center. Note that this operation is identical to having the combination of a 2-fold rotation axis perpendicular to a mirror
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