glide-reflection, i.e. reflection followed by translation.
All translations which occur are assumed to form a discrete subgroup of the group symmetries being described. Each group is denoted in orbifold notation by a finite string made up from the following symbols:
the symbol o, which is called a wonder and also a handle because it topologically represents a torus closed surface. Patterns repeat by two translation.
the symbol , which is called a miracle and represents a topological crosscap where a pattern repeats as a mirror image without crossing a mirror line.
A string written in boldface represents a group of symmetries of Euclidean 3-space. A string not written in boldface represents a group of symmetries of the Euclidean plane, which is assumed to contain two independent translations. Each symbol corresponds to a distinct transformation:
an integern to the left of an asterisk indicates a rotation of order n around a gyration point
an integer n to the right of an asterisk indicates a transformation of order 2n which rotates around a kaleidoscopic point and reflects through a line
the symbol indicates infinite rotational symmetry around a line; it can only occur for bold face groups. By abuse of language, we might say that such a group is a subgroup of symmetries of the Euclidean plane with only one independent translation. The frieze groups occur in this way.
the exceptional symbol o indicates that there are precisely two linearly independent translations.
Good orbifolds
An orbifold symbol is called good if it is not one of the following: p, pq, *p, *pq, for p,q≥2, and p≠q.
Chirality and achirality
An object is chiral if its symmetry group contains no reflections; otherwise it is called achiral. The corresponding orbifold is orientable in the chiral case and non-orientable otherwise.
Subtracting the sum of these values from 2 gives the Euler characteristic. If the sum of the feature values is 2, the order is infinite, i.e., the notation represents a wallpaper group or a frieze group. Indeed, Conway's "Magic Theorem" indicates that the 17 wallpaper groups are exactly those with the sum of the feature values equal to 2. Otherwise, the order is 2 divided by the Euler characteristic.
Equal groups
The following groups are isomorphic:
1* and *11
22 and 221
*22 and *221
2* and 2*1.
This is because 1-fold rotation is the "empty" rotation.
Two-dimensional groups
The symmetry of a 2D object without translational symmetry can be described by the 3D symmetry type by adding a third dimension to the object which does not add or spoil symmetry. For example, for a 2D image we can consider a piece of carton with that image displayed on one side; the shape of the carton should be such that it does not spoil the symmetry, or it can be imagined to be infinite. Thus we have n• and *n•. The bullet is added on one- and two-dimensional groups to imply the existence of a fixed point. Similarly, a 1D image can be drawn horizontally on a piece of carton, with a provision to avoid additional symmetry with respect to the line of the image, e.g. by drawing a horizontal bar under the image. Thus the discrete symmetry groups in one dimension are *•, *1•, ∞• and *∞•. Another way of constructing a 3D object from a 1D or 2D object for describing the symmetry is taking the Cartesian product of the object and an asymmetric 2D or 1D object, respectively.
Correspondence tables
Spherical
Euclidean plane
Frieze groups
Wallpaper groups
Orbifold Signature
Coxeter
Hermann– Mauguin
Speiser Niggli
Polya Guggenhein
Fejes Toth Cadwell
*632
p6m
C6v
D6
W16
632
+
p6
C6
C6
W6
*442
p4m
C4
D*4
W14
4*2
p4g
CII4v
Do4
W24
442
+
p4
C4
C4
W4
*333
]
p3m1
CII3v
D*3
W13
3*3
p31m
CI3v
Do3
W23
333
]+
p3
CI3
C3
W3
*2222
pmm
CI2v
D2kkkk
W22
2*22
cmm
CIV2v
D2kgkg
W12
22*
pmg
CIII2v
D2kkgg
W32
22×
pgg
CII2v
D2gggg
W42
2222
+
p2
C2
C2
W2
**
pm
CIs
D1kk
W21
*×
cm
CIIIs
D1kg
W11
××
pg
CII2
D1gg
W31
o
p1
C1
C1
W1
Hyperbolic plane
A first few hyperbolic groups, ordered by their Euler characteristic are: