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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 perpendicularmirror 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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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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Symmetry, Groups and Crystal Structures

Symmetry, Groups and Crystal Structures

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