Crystallographic Groups, Groupoids, and Orbifolds

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In this note, We first discuss the relationship among crystallographic lattice groups, space groups, and point groups by using a short exact sequence, then in footnotes indicate the classification of those groups. We then introduce screw and glide groupoids as an extension of point groups in a new exact sequence, and list the one-translational-dimension screw and glide groupoids, which require torus and truncated cylinder projection representations in addition to the spherical projection used for point groups. We then briefly discuss the two and three translational dimension groupoids associated with the remaining point groups. Examples of space groups and their groupoid ... continued below

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7 pages

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Johnson, C.K. September 11, 2000.

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Description

In this note, We first discuss the relationship among crystallographic lattice groups, space groups, and point groups by using a short exact sequence, then in footnotes indicate the classification of those groups. We then introduce screw and glide groupoids as an extension of point groups in a new exact sequence, and list the one-translational-dimension screw and glide groupoids, which require torus and truncated cylinder projection representations in addition to the spherical projection used for point groups. We then briefly discuss the two and three translational dimension groupoids associated with the remaining point groups. Examples of space groups and their groupoid based nomenclature, which is mainly the extended Hermana-Mauguin international crystallographic nomenclature system plus a specific type of coset decomposition, are then given. Next the crystallographic orbifolds are defined and some application problems associated with orbifolds discussed. Finally, the derivation of might be called orbifoldoids is suggested as future research.

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7 pages

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OSTI as DE00771400

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  • Workshop on Orbifolds, Groupoids, and Their Application, Bangor (GB), 09/11/2000--09/15/2000

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  • Report No.: P00-108740
  • Grant Number: AC05-00OR22725
  • Office of Scientific & Technical Information Report Number: 771400
  • Archival Resource Key: ark:/67531/metadc716267

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  • September 11, 2000

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  • Sept. 29, 2015, 5:31 a.m.

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  • Jan. 21, 2016, 7:08 p.m.

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Johnson, C.K. Crystallographic Groups, Groupoids, and Orbifolds, article, September 11, 2000; Tennessee. (digital.library.unt.edu/ark:/67531/metadc716267/: accessed September 24, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.