Orbifold notation

In geometry, orbifold notation (or orbifold signature) is a system, invented by William Thurston and popularized by the mathematician John 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 point groups on the sphere (), the frieze groups and wallpaper groups of the Euclidean plane (), and their analogues on the hyperbolic plane ().

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 (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 after an asterisk counts as
  • 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

Bentley Snowflake13
A perfect snowflake would have *6• symmetry,
Pentagon symmetry as mirrors 2005-07-08
The pentagon has symmetry *5•, the whole image with arrows 5•.
Flag of Hong Kong
The Flag of Hong Kong has 5 fold rotation symmetry, 5•.

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.

Correspondence tables


Fundamental domains of reflective 3D point groups
(*11), C1v=Cs (*22), C2v (*33), C3v (*44), C4v (*55), C5v (*66), C6v
Spherical digonal hosohedron2
Order 2
Spherical square hosohedron2
Order 4
Spherical hexagonal hosohedron2
Order 6
Spherical octagonal hosohedron2
Order 8
Spherical decagonal hosohedron2
Order 10
Spherical dodecagonal hosohedron2
Order 12
(*221), D1h=C2v (*222), D2h (*223), D3h (*224), D4h (*225), D5h (*226), D6h
Spherical digonal bipyramid2
Order 4
Spherical square bipyramid2
Order 8
Spherical hexagonal bipyramid2
Order 12
Spherical octagonal bipyramid2
Order 16
Spherical decagonal bipyramid2
Order 20
Spherical dodecagonal bipyramid2
Order 24
(*332), Td (*432), Oh (*532), Ih
Tetrahedral reflection domains
Order 24
Octahedral reflection domains
Order 48
Icosahedral reflection domains
Order 120
Spherical Symmetry Groups[1]
Coxeter Schönflies Hermann–Mauguin Order
Polyhedral groups
*532 [3,5] Ih 53m 120
532 [3,5]+ I 532 60
*432 [3,4] Oh m3m 48
432 [3,4]+ O 432 24
*332 [3,3] Td 43m 24
3*2 [3+,4] Th m3 24
332 [3,3]+ T 23 12
Dihedral and cyclic groups: n=3,4,5...
*22n [2,n] Dnh n/mmm or 2nm2 4n
2*n [2+,2n] Dnd 2n2m or nm 4n
22n [2,n]+ Dn n2 2n
*nn [n] Cnv nm 2n
n* [n+,2] Cnh n/m or 2n 2n
[2+,2n+] S2n 2n or n 2n
nn [n]+ Cn n n
Special cases
*222 [2,2] D2h 2/mmm or 22m2 8
2*2 [2+,4] D2d 222m or 2m 8
222 [2,2]+ D2 22 4
*22 [2] C2v 2m 4
2* [2+,2] C2h 2/m or 22 4
[2+,4+] S4 22 or 2 4
22 [2]+ C2 2 2
*22 [1,2] D1h=C2v 1/mmm or 21m2 4
2* [2+,2] D1d=C2h 212m or 1m 4
22 [1,2]+ D1=C2 12 2
*1 [ ] C1v=Cs 1m 2
1* [2,1+] C1h=Cs 1/m or 21 2
[2+,2+] S2=Ci 21 or 1 2
1 [ ]+ C1 1 1

Euclidean plane

Frieze groups

Frieze groups
IUC Cox Schön*
and Conway nickname[2]
p1 [∞]+
CDel node h2.pngCDel infin.pngCDel node h2.png
Frieze group 11
Frieze example p1
Frieze hop
(T) Translations only:
This group is singly generated, by a translation by the smallest distance over which the pattern is periodic.
p11g [∞+,2+]
CDel node h2.pngCDel infin.pngCDel node h4.pngCDel 2x.pngCDel node h2.png
Frieze group 1g
F ᖶ F ᖶ F ᖶ F ᖶ
Frieze example p11g
Frieze step
(TG) Glide-reflections and Translations:
This group is singly generated, by a glide reflection, with translations being obtained by combining two glide reflections.
p1m1 [∞]
CDel node.pngCDel infin.pngCDel node.png
Frieze group m1
Frieze example p1m1
Frieze sidle
(TV) Vertical reflection lines and Translations:
The group is the same as the non-trivial group in the one-dimensional case; it is generated by a translation and a reflection in the vertical axis.
p2 [∞,2]+
CDel node h2.pngCDel infin.pngCDel node h2.pngCDel 2x.pngCDel node h2.png
Frieze group 12
Frieze example p2
Frieze spinning hop
spinning hop
(TR) Translations and 180° Rotations:
The group is generated by a translation and a 180° rotation.
p2mg [∞,2+]
CDel node.pngCDel infin.pngCDel node h2.pngCDel 2x.pngCDel node h2.png
Frieze group mg
Frieze example p2mg
Frieze spinning sidle
spinning sidle
(TRVG) Vertical reflection lines, Glide reflections, Translations and 180° Rotations:
The translations here arise from the glide reflections, so this group is generated by a glide reflection and either a rotation or a vertical reflection.
p11m [∞+,2]
CDel node h2.pngCDel infin.pngCDel node h2.pngCDel 2.pngCDel node.png
Frieze group 1m
Frieze example p11m
Frieze jump
(THG) Translations, Horizontal reflections, Glide reflections:
This group is generated by a translation and the reflection in the horizontal axis. The glide reflection here arises as the composition of translation and horizontal reflection
p2mm [∞,2]
CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.png
Frieze group mm
Frieze example p2mm
Frieze spinning jump
spinning jump
(TRHVG) Horizontal and Vertical reflection lines, Translations and 180° Rotations:
This group requires three generators, with one generating set consisting of a translation, the reflection in the horizontal axis and a reflection across a vertical axis.
*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.

Wallpaper groups

Fundamental domains of Euclidean reflective groups
(*442), p4m (4*2), p4g
Uniform tiling 44-t1 Tile V488 bicolor
(*333), p3m (632), p6
Tile 3,6 Tile V46b
17 wallpaper groups[3]
Coxeter Hermann–
Fejes Toth
*632 [6,3] p6m C(I)6v D6 W16
632 [6,3]+ p6 C(I)6 C6 W6
*442 [4,4] p4m C(I)4 D*4 W14
4*2 [4+,4] p4g CII4v Do4 W24
442 [4,4]+ p4 C(I)4 C4 W4
*333 [3[3]] p3m1 CII3v D*3 W13
3*3 [3+,6] p31m CI3v Do3 W23
333 [3[3]]+ p3 CI3 C3 W3
*2222 [∞,2,∞] pmm CI2v D2kkkk W22
2*22 [∞,2+,∞] cmm CIV2v D2kgkg W12
22* [(∞,2)+,∞] pmg CIII2v D2kkgg W32
22× [∞+,2+,∞+] pgg CII2v D2gggg W42
2222 [∞,2,∞]+ p2 C(I)2 C2 W2
** [∞+,2,∞] pm CIs D1kk W21
[∞+,2+,∞] cm CIIIs D1kg W11
×× [∞+,(2,∞)+] pg CII2 D1gg W31
o [∞+,2,∞+] p1 C(I)1 C1 W1

Hyperbolic plane

Poincaré disk model of fundamental domain triangles
Example right triangles (*2pq)
H2checkers 237
H2checkers 238
Hyperbolic domains 932 black
H2checkers 23i
H2checkers 245
H2checkers 246
H2checkers 247
H2checkers 248
H2checkers 24i
H2checkers 255
H2checkers 256
H2checkers 257
H2checkers 266
H2checkers 2ii
Example general triangles (*pqr)
H2checkers 334
H2checkers 335
H2checkers 336
H2checkers 337
H2checkers 33i
H2checkers 344
H2checkers 366
H2checkers 3ii
H2checkers 666
Infinite-order triangular tiling
Example higher polygons (*pqrs...)
Hyperbolic domains 3222
H2chess 246a
H2chess 248a
H2chess 246b
H2chess 248b
Uniform tiling 552-t1
Uniform tiling 66-t1
Uniform tiling 77-t1
Uniform tiling 88-t1
Hyperbolic domains i222
H2chess 24ia
H2chess 24ib
H2chess 24ic
H2chess iiic

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

Hyperbolic Symmetry Groups[4]
-1/χ Orbifolds Coxeter
84 *237 [7,3]
48 *238 [8,3]
42 237 [7,3]+
40 *245 [5,4]
36 - 26.4 *239, *2 3 10 [9,3], [10,3]
26.4 *2 3 11 [11,3]
24 *2 3 12, *246, *334, 3*4, 238 [12,3], [6,4], [(4,3,3)], [3+,8], [8,3]+
22.3 - 21 *2 3 13, *2 3 14 [13,3], [14,3]
20 *2 3 15, *255, 5*2, 245 [15,3], [5,5], [5+,4], [5,4]+
19.2 *2 3 16 [16,3]
18+2/3 *247 [7,4]
18 *2 3 18, 239 [18,3], [9,3]+
17.5 - 16.2 *2 3 19, *2 3 20, *2 3 21, *2 3 22, *2 3 23 [19,3], [20,3], [20,3], [21,3], [22,3], [23,3]
16 *2 3 24, *248 [24,3], [8,4]
15 *2 3 30, *256, *335, 3*5, 2 3 10 [30,3], [6,5], [(5,3,3)], [3+,10], [10,3]+
14+2/5 - 13+1/3 *2 3 36 ... *2 3 70, *249, *2 4 10 [36,3] ... [60,3], [9,4], [10,4]
13+1/5 *2 3 66, 2 3 11 [66,3], [11,3]+
12+8/11 *2 3 105, *257 [105,3], [7,5]
12+4/7 *2 3 132, *2 4 11 ... [132,3], [11,4], ...
12 *23∞, *2 4 12, *266, 6*2, *336, 3*6, *344, 4*3, *2223, 2*23, 2 3 12, 246, 334 [∞,3] [12,4], [6,6], [6+,4], [(6,3,3)], [3+,12], [(4,4,3)], [4+,6], [∞,3,∞], [12,3]+, [6,4]+ [(4,3,3)]+

See also


  1. ^ Symmetries of Things, Appendix A, page 416
  2. ^ Frieze Patterns Mathematician John Conway created names that relate to footsteps for each of the frieze groups.
  3. ^ Symmetries of Things, Appendix A, page 416
  4. ^ Symmetries of Things, Chapter 18, More on Hyperbolic groups, Enumerating hyperbolic groups, p239
  • John H. Conway, Olaf Delgado Friedrichs, Daniel H. Huson, and William P. Thurston. On Three-dimensional Orbifolds and Space Groups. Contributions to Algebra and Geometry, 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-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5

External links

Circle Limit III

Circle Limit III is a woodcut made in 1959 by Dutch artist M. C. Escher, in which "strings of fish shoot up like rockets from infinitely far away" and then "fall back again whence they came".It is one of a series of four woodcuts by Escher depicting ideas from hyperbolic geometry. Dutch physicist and mathematician Bruno Ernst called it "the best of the four".

Cyclic symmetry in three dimensions

In three dimensional geometry, there are four infinite series of point groups in three dimensions (n≥1) with n-fold rotational or reflectional symmetry about one axis (by an angle of 360°/n) that does not change the object.

They are the finite symmetry groups on a cone. For n = ∞ they correspond to four frieze groups. Schönflies notation is used. The terms horizontal (h) and vertical (v) imply the existence and direction of reflections with respect to a vertical axis of symmetry. Also shown are Coxeter notation in brackets, and, in parentheses, orbifold notation.

Dihedral symmetry in three dimensions

In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as abstract group is a dihedral group Dihn ( n ≥ 2 ).

Glide reflection

In 2-dimensional geometry, a glide reflection (or transflection) is a type of opposite isometry of the Euclidean plane: the composition of a reflection in a line and a translation along that line.

A single glide is represented as frieze group p11g. A glide reflection can be seen as a limiting rotoreflection, where the rotation becomes a translation. It can also be given a Schoenflies notation as S2∞, Coxeter notation as [∞+,2+], and orbifold notation as ∞×.

List of finite spherical symmetry groups

Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry.

This article lists the groups by Schoenflies notation, Coxeter notation, orbifold notation, and order. John Conway uses a variation of the Schoenflies notation, based on the groups' quaternion algebraic structure, labeled by one or two upper case letters, and whole number subscripts. The group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, "±", prefix, which implies a central inversion.Hermann–Mauguin notation (International notation) is also given. The crystallography groups, 32 in total, are a subset with element orders 2, 3, 4 and 6.

List of planar symmetry groups

This article summarizes the classes of discrete symmetry groups of the Euclidean plane. The symmetry groups are named here by three naming schemes: International notation, orbifold notation, and Coxeter notation.

There are three kinds of symmetry groups of the plane:

2 families of rosette groups – 2D point groups

7 frieze groups – 2D line groups

17 wallpaper groups – 2D space groups.

Non-Euclidean crystallographic group

In mathematics, a non-Euclidean crystallographic group, NEC group or N.E.C. group is a discrete group of isometries of the hyperbolic plane. These symmetry groups correspond to the wallpaper groups in euclidean geometry. A NEC group which contains only orientation-preserving elements is called a Fuchsian group, and any non-Fuchsian NEC group has an index 2 Fuchsian subgroup of orientation-preserving elements.

The hyperbolic triangle groups are notable NEC groups. Others are listed in Orbifold notation.

Order-4 heptagonal tiling

In geometry, the order-4 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {7,4}.

Order-6 hexagonal tiling

In geometry, the order-6 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,6} and is self-dual.

Order-6 octagonal tiling

In geometry, the order-6 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,6}.

Order-6 pentagonal tiling

In geometry, the order-6 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,6}.

Order-8 octagonal tiling

In geometry, the order-8 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,8} and is self-dual.

Order-8 square tiling

In geometry, the order-8 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,8}.

Point groups in three dimensions

In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries.

Symmetry groups of objects are isometry groups. Accordingly, analysis of isometry groups is analysis of possible symmetries. All isometries of a bounded 3D object have one or more common fixed points. We choose the origin as one of them.

The symmetry group of an object is sometimes also called full symmetry group, as opposed to its rotation group or proper symmetry group, the intersection of its full symmetry group and the rotation group SO(3) of the 3D space itself. The rotation group of an object is equal to its full symmetry group if and only if the object is chiral.

The point groups in three dimensions are heavily used in chemistry, especially to describe the symmetries of a molecule and of molecular orbitals forming covalent bonds, and in this context they are also called molecular point groups.

Finite Coxeter groups are a special set of point groups generated purely by a set of reflectional mirrors passing through the same point. A rank n Coxeter group has n mirrors and is represented by a Coxeter–Dynkin diagram. Coxeter notation offers a bracketed notation equivalent to the Coxeter diagram, with markup symbols for rotational and other subsymmetry point groups.

Quarter order-6 square tiling

In geometry, the quarter order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of q{4,6}. It is constructed from *3232 orbifold notation, and can be seen as a half symmetry of *443 and *662, and quarter symmetry of *642.

Rhombitriapeirogonal tiling

In geometry, the rhombtriapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of rr{∞,3}.

Rhombitrioctagonal tiling

In geometry, the rhombitrioctagonal tiling is a semiregular tiling of the

hyperbolic plane. At each vertex of the tiling there is one triangle and one octagon, alternating between two squares. The tiling has Schläfli symbol rr{8,3}. It can be seen as constructed as a rectified trioctagonal tiling, r{8,3}, as well as an expanded octagonal tiling or expanded order-8 triangular tiling.

Square lattice

In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice, denoted as Z2. It is one of the five types of two-dimensional lattices as classified by their symmetry groups; its symmetry group in IUC notation as p4m, Coxeter notation as [4,4], and orbifold notation as *442.Two orientations of an image of the lattice are by far the most common. They can conveniently be referred to as the upright square lattice and diagonal square lattice; the latter is also called the centered square lattice. They differ by an angle of 45°. This is related to the fact that a square lattice can be partitioned into two square sub-lattices, as is evident in the colouring of a checkerboard.

Wallpaper group

A wallpaper group (or plane symmetry group or plane crystallographic 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, especially in textiles and tiles as well as wallpaper.

A proof that there were only 17 distinct groups of possible patterns was first carried out by Evgraf Fedorov in 1891 and then derived independently by George Pólya in 1924. The proof that the list of wallpaper groups was complete only came after the much harder case of space groups had been done. The seventeen possible wallpaper groups are listed below in § The seventeen groups.

Wallpaper groups are two-dimensional symmetry groups, intermediate in complexity between the simpler frieze groups and the three-dimensional space groups. Wallpaper groups categorize patterns by their symmetries. Subtle differences may place similar patterns in different groups, while patterns that are very different in style, color, scale or orientation may belong to the same group.

Consider the following examples:

Examples A and B have the same wallpaper group; it is called p4m in the IUC notation and *442 in the orbifold notation. Example C has a different wallpaper group, called p4g or 4*2 . The fact that A and B have the same wallpaper group means that they have the same symmetries, regardless of details of the designs, whereas C has a different set of symmetries despite any superficial similarities.

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