Layer group
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[1][2]
| 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 | p42m | 58 | p421m |
| 59 | p4m2 | 60 | p4b2 | 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 | p31m | 72 | p3m1 | ||||
| Hexagonal | |||||||||
| 73 | p6 | 74 | p6 | 75 | p6/m | 76 | p622 | 77 | p6mm |
| 78 | p6m2 | 79 | p62m | 80 | p6/mmm | ||||
Correspondence Between Layer Groups and Plane Groups
[edit | edit source]The surjective mapping from a layer group to a wallpaper group (plane group) can be obtained by disregarding symmetry elements along the stacking direction, typically denoted as the z-axis, and aligning the remaining elements with those of the plane groups.[3] The resulting surjective mapping provides a direct correspondence between layer groups and plane groups (wallpaper groups).
| # | Layer Group | # | Plane Group |
|---|---|---|---|
| 1 | p1 | 1 | p1 |
| 2 | p1 | 2 | p2 |
| 3 | p112 | 2 | p2 |
| 4 | p11m | 1 | p1 |
| 5 | p11a | 1 | p1 |
| 6 | p112/m | 2 | p2 |
| 7 | p112/a | 2 | p2 |
| 8 | p211 | 3 | pm |
| 9 | p2111 | 4 | pg |
| 10 | c211 | 5 | cm |
| 11 | pm11 | 3 | pm |
| 12 | pb11 | 4 | pg |
| 13 | cm11 | 5 | cm |
| 14 | p2/m11 | 6 | p2mm |
| 15 | p21/m11 | 7 | p2mg |
| 16 | p2/b11 | 7 | p2mg |
| 17 | p21/b11 | 8 | p2gg |
| 18 | c2/m11 | 9 | c2mm |
| 19 | p222 | 6 | p2mm |
| 20 | p2122 | 7 | p2mg |
| 21 | p21212 | 8 | p2gg |
| 22 | c222 | 9 | c2mm |
| 23 | pmm2 | 6 | p2mm |
| 24 | pma2 | 7 | p2mg |
| 25 | pba2 | 8 | p2gg |
| 26 | cmm2 | 9 | c2mm |
| 27 | pm2m | 3 | pm |
| 28 | pm21b | 3 | pm |
| 29 | pb21m | 4 | pg |
| 30 | pb2b | 3 | pm |
| 31 | pm2a | 3 | pm |
| 32 | pm21n | 4 | pg |
| 33 | pb21a | 4 | pg |
| 34 | pb2n | 5 | cm |
| 35 | cm2m | 5 | cm |
| 36 | cm2e | 3 | pm |
| 37 | pmmm | 6 | p2mm |
| 38 | pmaa | 6 | p2mm |
| 39 | pban | 10 | p4 |
| 40 | pmam | 7 | p2mg |
| 41 | pmma | 6 | p2mm |
| 42 | pman | 9 | c2mm |
| 43 | pbaa | 7 | p2mg |
| 44 | pbam | 8 | p2gg |
| 45 | pbma | 7 | p2mg |
| 46 | pmmn | 10 | p4 |
| 47 | cmmm | 9 | c2mm |
| 48 | cmme | 6 | p2mm |
| 49 | p4 | 10 | p4 |
| 50 | p4 | 10 | p4 |
| 51 | p4/m | 10 | p4 |
| 52 | p4/n | 12 | p4gm |
| 53 | p422 | 11 | p4mm |
| 54 | p4212 | 12 | p4gm |
| 55 | p4mm | 11 | p4mm |
| 56 | p4bm | 12 | p4gm |
| 57 | p42m | 11 | p4mm |
| 58 | p421m | 12 | p4gm |
| 59 | p4m2 | 11 | p4mm |
| 60 | p4b2 | 12 | p4gm |
| 61 | p4/mmm | 11 | p4mm |
| 62 | p4/nbm | 11 | p4mm |
| 63 | p4/mbm | 12 | p4gm |
| 64 | p4/nmm | 11 | p4mm |
| 65 | p3 | 13 | p3 |
| 66 | p3 | 16 | p6 |
| 67 | p312 | 14 | p3m1 |
| 68 | p321 | 15 | p31m |
| 69 | p3m1 | 14 | p3m1 |
| 70 | p31m | 15 | p31m |
| 71 | p31m | 17 | p6mm |
| 72 | p3m1 | 17 | p6mm |
| 73 | p6 | 16 | p6 |
| 74 | p6 | 13 | p3 |
| 75 | p6/m | 16 | p6 |
| 76 | p622 | 17 | p6mm |
| 77 | p6mm | 17 | p6mm |
| 78 | p6m2 | 14 | p3m1 |
| 79 | p62m | 15 | p31m |
| 80 | p6/mmm | 17 | p6mm |
See also
[edit | edit source]References
[edit | edit source]- ^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
- ^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
- ^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
External links
[edit | edit source]- 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
- CVM 1.1: Vibrating Wallpaper by Frank Farris. He constructs layer groups from wallpaper groups using negating isometries.