In mathematics, a layer group is a three-dimensional extension of a wallpaper group

Wallpaper group

A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art...

, with reflections in the third dimension. It is a space group

Space group

In mathematics and geometry, a space group is a symmetry group, usually for three dimensions, that divides space into discrete repeatable domains.In three dimensions, there are 219 unique types, or counted as 230 if chiral copies are considered distinct...

 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

Crystallographic point group

In crystallography, a crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a central point fixed while moving other directions and faces of the crystal to the positions of features of the same kind...

 with the main axis being perpendicular to the lattice plane.

Table of the 80 layer groups, organized by crystal system

Crystal system

In crystallography, the terms crystal system, crystal family, and lattice system each refer to one of several classes of space groups, lattices, point groups, or crystals...

 or lattice type, and by their point groups:

Triclinic
1	p1	2	p
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	p	51	p4/m	52	p4/n	53	p422
54	p4212	55	p4mm	56	p4bm	57	p2m	58	p21m
59	pm2	60	pb2	61	p4/mmm	62	p4/nbm	63	p4/mbm
64	p4/nmm
Trigonal
65	p3	66	p	67	p312	68	p321	69	p3m1
70	p31m	71	p1m	72	pm1
Hexagonal
73	p6	74	p	75	p6/m	76	p622	77	p6mm
78	pm2	79	p2m	80	p6/mmm

See also

• Point group
  Point group
  In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O...
  
  
• Crystallographic point group
  Crystallographic point group
  In crystallography, a crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a central point fixed while moving other directions and faces of the crystal to the positions of features of the same kind...
  
  
• Space group
  Space group
  In mathematics and geometry, a space group is a symmetry group, usually for three dimensions, that divides space into discrete repeatable domains.In three dimensions, there are 219 unique types, or counted as 230 if chiral copies are considered distinct...
  
  
• Rod group
  Rod group
  In mathematics, a rod group is a three-dimensional line group whose point group is one of the axial crystallographic point groups. This constraint means that the point group must be the symmetry of some three-dimensional lattice....
  
  
• Frieze group
  Frieze group
  A frieze group is a mathematical concept to classify designs on two-dimensional surfaces which are repetitive in one direction, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art...
  
  
• Wallpaper group
  Wallpaper group
  A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art...
  
  

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
• CVM 1.1: Vibrating Wallpaper by Frank Farris. He constructs layer groups from wallpaper groups using negating isometries.