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Published byClifton Scott Modified over 2 years ago

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Three-Dimensional Symmetry How can we put dots on a sphere?

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The Seven Strip Space Groups

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Simplest Pattern: motifs around a symmetry axis (5) Equivalent to wrapping a strip around a cylinder

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Symmetry axis plus parallel mirror planes (5m)

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Symmetry axis plus perpendicular mirror plane (5/m)

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Symmetry axis plus both sets of mirror planes (5m/m)

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Symmetry axis plus perpendicular 2-fold axes (52)

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Symmetry axis plus mirror planes and perpendicular 2-fold axes (5m2)

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The three- dimensional version of glide is called inversion

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Axial Symmetry (1,2,3,4,6 – fold symmetry) x 7 types = 35 Only rotation and inversion possible for 1- fold symmetry ( = 30) 3 other possibilities are duplicates 27 remaining types

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Isometric Symmetry Cubic unit cells Unifying feature is surprising: four diagonal 3-fold symmetry axes 5 isometric types + 27 axial symmetries = 32 crystallographic point groups Two of the five are very common, one is less common, two others very rare

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The Isometric Classes

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Non-Crystallographic Symmetries There are an infinite number of axial point groups: 5-fold, 7-fold, 8-fold, etc, with mirror planes, 2-fold axes, inversion, etc. In addition, there are two very special 5-fold isometric symmetries with and without mirror planes. Clusters of atoms, molecules, viruses, and biological structures contain these symmetries Some crystals approximate these forms but do not have true 5-fold symmetry, of course.

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Icosahedral Symmetry

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Icosahedral Symmetry Without Mirror Planes

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Why Are Crystals Symmetrical? Electrostatic attraction and repulsion are symmetrical Ionic bonding attracts ions equally in all directions Covalent bonding involves orbitals that are symmetrically oriented because of electrostatic repulsion

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Malformed Crystals

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Why Might Crystals Not Be Symmetrical? Chemical gradient Temperature gradient Competition for ions by other minerals Stress Anisotropic surroundings

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Regardless of Crystal Shape, Face Orientations and Interfacial Angles are Always the Same

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We Can Project Face Orientation Data to Reveal the Symmetry

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Projections in Three Dimensions are Vital for Revealing and Illustrating Crystal Symmetry

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