Orbifold notation

In geometry, orbifold notation (or orbifold signature) is a system popularized by the mathematician John Horton Conway for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it describes the orbifold obtained by taking the quotient of Euclidean space by the group under consideration.

Groups representable in this notation include the wallpaper groups and frieze groups on the Euclidean plane (E²), the groups on the sphere (S²), and their analogues on the hyperbolic plane (H²).

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
• the infinity symbol,
• the asterisk, *
• the symbol o, which is called a wonder
• the symbol x, which is called a miracle

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 point
• an integer n to the right of an asterisk indicates a transformation of order 2n which rotates around a point and reflects through a line (or plane)
• an x 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.

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 after an asterisk counts as

• asterisk and x 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.

Other objects

The pentagon has symmetry *55, the whole image with arrows 55.

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 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 11, *11, 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

Euclidean plane

The fundamental domains in a *236 symmetry

The 17 plane wallpaper symmetry groups:^[1]

Frieze groups

7 Frieze groups

1. ∞∞ (hop): Translations only. This group is singly-generated, with a generator being a translation by the smallest distance over which the pattern is periodic. Consequently the group is isomorphic to Z, the group of integers.
2. ∞x (step): Glide-reflections and translations. This group is generated by a single glide reflection, with translations being obtained by combining two glide reflections. Consequently, this group is also isomorphic to Z.
3. ∞* (jump): Translations, the reflection in the horizontal axis and glide reflections. This group is isomorphic to the direct product Z × C2, and is generated by a translation and the reflection in the horizontal axis.
4. *∞∞ (sidle): Translations and reflections across certain vertical lines. The elements in this group correspond to isometries (or equivalently, bijective affine transformations) of the set of integers, and so it is isomorphic to a semidirect product of the integers with C2, and isomorphic to the infinite dihedral group. The group is generated by a translation and a reflection in a vertical axis. It is the same as the non-trivial group in the one-dimensional case
5. 22∞ (spinning hop): Translations and 180° rotations. Again, the transformations in this group correspond to isometries of the set of integers, and so the group is isomorphic to a semidirect product of Z and C2. The group is generated by a translation and a 180° rotation.
6. 2*∞ (spinning sidle): Reflections across certain vertical lines, glide-reflections, translations and rotations. The translations here arise from the glide reflections, so this group is generated by a glide reflection and a rotation. It is isomorphic to a semi-direct product of Z and C2.
7. *22∞ (spinning jump): Translations, glide reflections, reflections in both axes and 180° rotations. This group is the "largest" frieze group and requires three generators, with one generating set consisting of a translation, the reflection in the horizontal axis and a reflection across a vertical axis. It is isomorphic to C2 × (a semidirect product of Z and C2).

Spherical

The fundamental domains in a *532 symmetry

The spherical symmetry groups: (N=1,2,3,..)^[2]

Hyperbolic plane

The fundamental domains in the hyperbolic *237 symmetry

The hyperbolic symmetry groups: ^[3]

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

(-1/char)

• (84): *237
• (48): *238
• (42): 237
• (40): *245
• (24): *2.3.12, *246, *334, 3*4, 238
• (20): *2.3.15, *255, 5*2, 245
• (18+2/3): *247
• (18): *2.3.18, 239
• (16): *2.3.24, *248
• (15): *2.3.30, *256, *335, 3*5, 2.3.10
• (14+2/5): *2.3.36, *249
• (13+1/3): *2.3.60, *2.4.10
• (13+1/5): *2.3.66, 2.3.11
• (12+8/11): *2.3.105, *257
• (12+4/7): *2.3.132, *2.4.11 ... *23∞, *2.4.12, *266, 6*2
• (12): *336, 3*6, *344, 4*3, *2223, 2*23, 2.3.12, 246, 334

...

See also

• List of spherical symmetry groups
• List of planar symmetry groups
• Fibrifold notation - an extension of orbifold notation for 3d space groups

References

• J.H. Conway, O. Delgado Friedrichs, D.H. Huson, and W.P. Thurston. Three-dimensional orbifolds and space groups. Contributations to Geometry and Algebra, 42(2):475-507, 2001.
• J. H. Conway, D. H. Huson. The Orbifold Notation for Two-Dimensional Groups. Structural Chemistry, 13 (3-4): 247-257, August 2002.
• J. H. Conway (1992). "The Orbifold Notation for Surface Groups". In: M. W. Liebeck and J. Saxl (eds.), Groups, Combinatorics and Geometry, Proceedings of the L.M.S. Durham Symposium, July 5–15, Durham, UK, 1990; London Math. Soc. Lecture Notes Series 165. Cambridge University Press, Cambridge. pp. 438–447
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5

External links

• A field guide to the orbifolds (Notes from class on "Geometry and the Imagination" in Minneapolis, with John Conway, Peter Doyle, Jane Gilman and Bill Thurston, on June 17-28, 1991. See also PDF, 2006)
• 2DTiler Software for visualizing two-dimensional tilings of the plane and editing their symmetry groups in orbifold notation