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The 10 two-dimensional crystallographic point groups Interactive exercise Eugen Libowitzky Institute of Mineralogy and Crystallography 2012

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For teachers… (1) "Copy and paste" can be used to extend the exercise pages or to change the graphics objects. (2) Symmetry elements of a group and most graphics objects have been aligned at the page center. (3) The interactive fields and the symmetry elements of each point group are grouped. (4) The group m is available with vertical and horizontal m.

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Basics In two-dimensional lattices* and patterns the following local symmetries (i.e. without repetition by shift / translation; centered around one point) may occur: (1) Rotation about a 1-, 2-, 3-, 4-, 6-fold rotation axis (RA) (2) Reflection(s) through one (or more) mirror plane(s)** m * Remarks: Lattices have nothing to do with point groups! However, they constrain the possible rotation axes to those five given above! ** In two dimensions the term "mirror line" also makes sense.

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Sym. elementSym. operation Printed symbol Graphical symbol 1-fold RA360° rotation1none 2-fold RA180° rotation2 3-fold RA120° rotation3 4-fold RA90° rotation4 6-fold RA60° rotation6 mirror planereflectionm |

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Point groups… In two dimensions these 5 rotation axes alone or in combination with the mirror plane(s) result in exactly 10 possible combinations or arrangements of symmetry elements that intersect in one point: the 10 two-dimensional crystallographic* point groups * Remark: 5-, 7- and higher-fold rotation axes would result in an infinite number of two-dimensional point groups. However, these are not compatible with lattices. – See note (*) before!

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Symmetry An object is "symmetric" in a geometric sense, if it appears identical (in shape and orientation) after a symmetry operation (except "1" - equal to identity). Example: 2-fold rotation

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The 10 two-dim. cryst. point groups 2mm 3m3m4mm 6mm m (1m) 2 34 6 1

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The point group symbol (according to Hermann-Mauguin) In cases where only one symmetry element is present the symbol is just 1, 2, 3, 4, 6, m (the latter also 1m). In all other cases the rotation axis is written first, each set of symmetrically equivalent mirror planes behind. 2mm2mm 3m3m4mm4mm6mm6mm

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! Caution - Pitfall (1) ! The absolute orientation of symmetry elements in space has no impact on the point group assignment! Example: All three objects belong to point group m! )

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! Caution - Pitfall (2) ! "Two-dimensional" means that an object extends only in the very two dimensions of a plane. Even if we observe the object from the third dimension in space, there is no way for a symmetry operation between an apparent "front-" and "backside". Example: The two-dim. "Ampelmännchen" CANNOT be rotated about a 2-fold RA within the plane to the "backside"!

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! Caution - Pitfall (3) ! Symmetry is always complete, i.e. the operations must be repeatable until the starting point is reached again, and all parts of the object and even the symmetry elements must be included! A few examples: ) One detail "too much" prevents m! One "missing" detail prevents the complete 6-fold rotation and alle but one vertical m!

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! Caution - Pitfall (4) ! Quite frequently an object is assigned a certain symmetry that is well present but which underestimates the true symmetry and thus results in the wrong point group. An extreme example: graph of a snowflake (6mm) Symmetry well present, but underestimated / incomplete!

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! Caution - Pitfall (5) ! Symmetry elements or operations do not necessarily result in reproduction / repetition of object details. This is only true for general sites (remote from the symmetry elements). Special sites (exactly on a symmetry element) are repeated in themselves and therefore appear unchanged! A few examples: VZH

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Interactive exercise: (1) Assign the symmetry to the object in the page center. (2) Mouse click (left) on one of the 10 point group labels. (3) The symmetry elements appear in the object. RIGHT: Bright chimes sound. Green check symbol. Disappears after repeated mouse click. WRONG: Dull "error" sound. Disappears after three seconds or after repeated mouse click. CONTINUE: Return key or mouse click to the background.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 1 Homework, Page 124 Find the formulas for f + g, f – g, and fg.

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