Orbifold notation

Definition of the notation

The following types of Euclidean transformation can occur in a group described by orbifold notation:

reflection through a line (or plane)

translation by a vector

rotation of finite order around a point

infinite rotation around a line in 3-space

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:

positive integers 1 , 2 , 3 , …

the infinity symbol, ∞

the asterisk, *

the symbol o (a solid circle in older documents), which is called a wonder and also a handle because it topologically represents a torus (1-handle) closed surface. Patterns repeat by two translation.

the symbol × (an open circle in older documents), 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 integer n 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 (or plane)

an × indicates a glide reflection

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.

The Euler characteristic and the order

The Euler characteristic of an orbifold can be read from its Conway symbol, as follows. Each feature has a value:

n without or before an asterisk counts as n − 1 n

n after an asterisk counts as n − 1 2 n

asterisk and × count as 1

o counts as 2.

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. (In three dimensions these groups exist in an n-fold digonal orbifold and are represented as nn and *nn.)

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.

Frieze groups

^*Schönflies's point group notation is extended here as infinite cases of the equivalent dihedral points symmetries ^§The diagram shows one fundamental domain in yellow, with reflection lines in blue, glide reflection lines in dashed green, translation normals in red, and 2-fold gyration points as small green squares.

Hyperbolic plane

A first few hyperbolic groups, ordered by their Euler characteristic are: