In mathematics, a layer group is a three-dimensional extension of a wallpaper group, with reflections in the third dimension. It is a space group with a two-dimensional lattice, meaning that it is symmetric over repeats in the two lattice directions. The symmetry group at each lattice point is an axial crystallographic point group with the main axis being perpendicular to the lattice plane.
Table of the 80 layer groups, organized by crystal system or lattice type, and by their point groups:
Hitzer, E.S.M.; Ichikawa, D. (2008), "Representation of crystallographic subperiodic groups by geometric algebra", Electronic Proc. Of AGACSE (3, 17-19 Aug. 2008), Leipzig, Germany, arXiv:1306.1280, Bibcode:2013arXiv1306.1280H
Kopsky, V.; Litvin, D.B., eds. (2002), International Tables for Crystallography, Volume E: Subperiodic groups, International Tables for Crystallography, vol. E (5th ed.), Berlin, New York: Springer-Verlag, doi:10.1107/97809553602060000105, ISBN978-1-4020-0715-6
External linksedit
Bilbao Crystallographic Server, under "Subperiodic Groups: Layer, Rod and Frieze Groups"
Nomenclature, Symbols and Classification of the Subperiodic Groups, V. Kopsky and D. B. Litvin
by Frank Farris. He constructs layer groups from wallpaper groups using negating isometries.
May 02, 2024
layer, group, mathematics, layer, group, three, dimensional, extension, wallpaper, group, with, reflections, third, dimension, space, group, with, dimensional, lattice, meaning, that, symmetric, over, repeats, lattice, directions, symmetry, group, each, lattic. In mathematics a layer group is a three dimensional extension of a wallpaper group with reflections in the third dimension It is a space group with a two dimensional lattice meaning that it is symmetric over repeats in the two lattice directions The symmetry group at each lattice point is an axial crystallographic point group with the main axis being perpendicular to the lattice plane Table of the 80 layer groups organized by crystal system or lattice type and by their point groups Triclinic 1 p1 2 p1 Monoclinic inclined 3 p112 4 p11m 5 p11a 6 p112 m 7 p112 a Monoclinic orthogonal 8 p211 9 p2111 10 c211 11 pm11 12 pb11 13 cm11 14 p2 m11 15 p21 m11 16 p2 b11 17 p21 b11 18 c2 m11 Orthorhombic 19 p222 20 p2122 21 p21212 22 c222 23 pmm2 24 pma2 25 pba2 26 cmm2 27 pm2m 28 pm21b 29 pb21m 30 pb2b 31 pm2a 32 pm21n 33 pb21a 34 pb2n 35 cm2m 36 cm2e 37 pmmm 38 pmaa 39 pban 40 pmam 41 pmma 42 pman 43 pbaa 44 pbam 45 pbma 46 pmmn 47 cmmm 48 cmme Tetragonal 49 p4 50 p4 51 p4 m 52 p4 n 53 p422 54 p4212 55 p4mm 56 p4bm 57 p4 2m 58 p4 21m 59 p4 m2 60 p4 b2 61 p4 mmm 62 p4 nbm 63 p4 mbm 64 p4 nmm Trigonal 65 p3 66 p3 67 p312 68 p321 69 p3m1 70 p31m 71 p3 1m 72 p3 m1 Hexagonal 73 p6 74 p6 75 p6 m 76 p622 77 p6mm 78 p6 m2 79 p6 2m 80 p6 mmmSee also editPoint group Crystallographic point group Space group Rod group Frieze group Wallpaper groupReferences editHitzer E S M Ichikawa D 2008 Representation of crystallographic subperiodic groups by geometric algebra Electronic Proc Of AGACSE 3 17 19 Aug 2008 Leipzig Germany arXiv 1306 1280 Bibcode 2013arXiv1306 1280H Kopsky V Litvin D B eds 2002 International Tables for Crystallography Volume E Subperiodic groups International Tables for Crystallography vol E 5th ed Berlin New York Springer Verlag doi 10 1107 97809553602060000105 ISBN 978 1 4020 0715 6External links editBilbao Crystallographic Server under Subperiodic Groups Layer Rod and Frieze Groups Nomenclature Symbols and Classification of the Subperiodic Groups V Kopsky and D B Litvin CVM 1 1 Vibrating Wallpaper by Frank Farris He constructs layer groups from wallpaper groups using negating isometries Retrieved from https en wikipedia org w index php title Layer group amp oldid 1114254721, wikipedia, wiki, book, books, library,
