“Layer group”的意思、由来-开放百科全书

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Layer group

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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	1}}
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	4}}	51	p4/m	52	p4/n	53	p422
54	p42₁2	55	p4mm	56	p4bm	57	4}}2m	58	4}}2₁m
59	4}}m2	60	4}}b2	61	p4/mmm	62	p4/nbm	63	p4/mbm
64	p4/nmm
Trigonal
65	p3	66	3}}	67	p312	68	p321	69	p3m1
70	p31m	71	3}}1m	72	3}}m1
Hexagonal
73	p6	74	6}}	75	p6/m	76	p622	77	p6mm
78	6}}m2	79	6}}2m	80	p6/mmm

References

{{Citation | last1=Hitzer | first1=E.S.M. | last2=Ichikawa | first2=D. | title=Representation of crystallographic subperiodic groups by geometric algebra | arxiv=1306.1280| journal=Electronic Proc. of AGACSE | issue=3, 17-19 Aug. 2008 | location=Leipzig, Germany | year=2008| bibcode=2013arXiv1306.1280H }}
{{Citation | editor1-last=Kopsky | editor1-first=V. | editor2-last=Litvin | editor2-first=D.B. | title=International Tables for Crystallography, Volume E: Subperiodic groups | url=http://it.iucr.org/E/ | publisher=Springer-Verlag | location=Berlin, New York | edition=5th | isbn=978-1-4020-0715-6 |doi= 10.1107/97809553602060000105 | year=2002 | volume=E}}

External links

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
[https://web.archive.org/web/20130702063256/http://www.maa.org/cvm/1998/01/vw/welcome.html CVM 1.1: Vibrating Wallpaper] by Frank Farris. He constructs layer groups from wallpaper groups using negating isometries.

2 : Euclidean symmetries|Discrete groups

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