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 ().
The following types of Euclidean transformation can occur in a group described by orbifold notation:
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:
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 orbifold symbol is called good if it is not one of the following: p, pq, *p, *pq, for p,q>=2, and p≠q.
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 of an orbifold can be read from its Conway symbol, as follows. Each feature has a value:
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.
The following groups are isomorphic:
This is because 1-fold rotation is the "empty" rotation.
A perfect snowflake would have *6• symmetry, |
The pentagon has symmetry *5•, the whole image with arrows 5•. |
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.
(*11), C_{1v}=C_{s} | (*22), C_{2v} | (*33), C_{3v} | (*44), C_{4v} | (*55), C_{5v} | (*66), C_{6v} |
---|---|---|---|---|---|
Order 2 |
Order 4 |
Order 6 |
Order 8 |
Order 10 |
Order 12 |
(*221), D_{1h}=C_{2v} | (*222), D_{2h} | (*223), D_{3h} | (*224), D_{4h} | (*225), D_{5h} | (*226), D_{6h} |
Order 4 |
Order 8 |
Order 12 |
Order 16 |
Order 20 |
Order 24 |
(*332), T_{d} | (*432), O_{h} | (*532), I_{h} | |||
Order 24 |
Order 48 |
Order 120 |
Orbifold Signature |
Coxeter | Schönflies | Hermann–Mauguin | Order |
---|---|---|---|---|
Polyhedral groups | ||||
*532 | [3,5] | I_{h} | 53m | 120 |
532 | [3,5]^{+} | I | 532 | 60 |
*432 | [3,4] | O_{h} | m3m | 48 |
432 | [3,4]^{+} | O | 432 | 24 |
*332 | [3,3] | T_{d} | 43m | 24 |
3*2 | [3^{+},4] | T_{h} | m3 | 24 |
332 | [3,3]^{+} | T | 23 | 12 |
Dihedral and cyclic groups: n=3,4,5... | ||||
*22n | [2,n] | D_{nh} | n/mmm or 2nm2 | 4n |
2*n | [2^{+},2n] | D_{nd} | 2n2m or nm | 4n |
22n | [2,n]^{+} | D_{n} | n2 | 2n |
*nn | [n] | C_{nv} | nm | 2n |
n* | [n^{+},2] | C_{nh} | n/m or 2n | 2n |
n× | [2^{+},2n^{+}] | S_{2n} | 2n or n | 2n |
nn | [n]^{+} | C_{n} | n | n |
Special cases | ||||
*222 | [2,2] | D_{2h} | 2/mmm or 22m2 | 8 |
2*2 | [2^{+},4] | D_{2d} | 222m or 2m | 8 |
222 | [2,2]^{+} | D_{2} | 22 | 4 |
*22 | [2] | C_{2v} | 2m | 4 |
2* | [2^{+},2] | C_{2h} | 2/m or 22 | 4 |
2× | [2^{+},4^{+}] | S_{4} | 22 or 2 | 4 |
22 | [2]^{+} | C_{2} | 2 | 2 |
*22 | [1,2] | D_{1h}=C_{2v} | 1/mmm or 21m2 | 4 |
2* | [2^{+},2] | D_{1d}=C_{2h} | 212m or 1m | 4 |
22 | [1,2]^{+} | D_{1}=C_{2} | 12 | 2 |
*1 | [ ] | C_{1v}=C_{s} | 1m | 2 |
1* | [2,1^{+}] | C_{1h}=C_{s} | 1/m or 21 | 2 |
1× | [2^{+},2^{+}] | S_{2}=C_{i} | 21 or 1 | 2 |
1 | [ ]^{+} | C_{1} | 1 | 1 |
IUC | Cox | Schön^{*} Struct. |
Diagram^{§} Orbifold |
Examples and Conway nickname^{[2]} |
Description |
---|---|---|---|---|---|
p1 | [∞]^{+} |
C_{∞} Z_{∞} |
∞∞ |
F F F F F F F F hop |
(T) Translations only: This group is singly generated, by a translation by the smallest distance over which the pattern is periodic. |
p11g | [∞^{+},2^{+}] |
S_{∞} Z_{∞} |
∞× |
F ᖶ F ᖶ F ᖶ F ᖶ 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 | [∞] |
C_{∞v} Dih_{∞} |
*∞∞ |
Λ Λ Λ Λ Λ Λ Λ Λ 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]^{+} |
D_{∞} Dih_{∞} |
22∞ |
S S S S S S S S spinning hop |
(TR) Translations and 180° Rotations: The group is generated by a translation and a 180° rotation. |
p2mg | [∞,2^{+}] |
D_{∞d} Dih_{∞} |
2*∞ |
V Λ V Λ V Λ V Λ 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] |
C_{∞h} Z_{∞}×Dih_{1} |
∞* |
B B B B B B B B 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] |
D_{∞h} Dih_{∞}×Dih_{1} |
*22∞ |
H H H H H H H H 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. |
(*442), p4m | (4*2), p4g |
---|---|
(*333), p3m | (632), p6 |
Orbifold Signature |
Coxeter | Hermann– Mauguin |
Speiser Niggli |
Polya Guggenhein |
Fejes Toth Cadwell |
---|---|---|---|---|---|
*632 | [6,3] | p6m | C^{(I)}_{6v} | D_{6} | W^{1}_{6} |
632 | [6,3]^{+} | p6 | C^{(I)}_{6} | C_{6} | W_{6} |
*442 | [4,4] | p4m | C^{(I)}_{4} | D^{*}_{4} | W^{1}_{4} |
4*2 | [4^{+},4] | p4g | C^{II}_{4v} | D^{o}_{4} | W^{2}_{4} |
442 | [4,4]^{+} | p4 | C^{(I)}_{4} | C_{4} | W_{4} |
*333 | [3^{[3]}] | p3m1 | C^{II}_{3v} | D^{*}_{3} | W^{1}_{3} |
3*3 | [3^{+},6] | p31m | C^{I}_{3v} | D^{o}_{3} | W^{2}_{3} |
333 | [3^{[3]}]^{+} | p3 | C^{I}_{3} | C_{3} | W_{3} |
*2222 | [∞,2,∞] | pmm | C^{I}_{2v} | D_{2}kkkk | W^{2}_{2} |
2*22 | [∞,2^{+},∞] | cmm | C^{IV}_{2v} | D_{2}kgkg | W^{1}_{2} |
22* | [(∞,2)^{+},∞] | pmg | C^{III}_{2v} | D_{2}kkgg | W^{3}_{2} |
22× | [∞^{+},2^{+},∞^{+}] | pgg | C^{II}_{2v} | D_{2}gggg | W^{4}_{2} |
2222 | [∞,2,∞]^{+} | p2 | C^{(I)}_{2} | C_{2} | W_{2} |
** | [∞^{+},2,∞] | pm | C^{I}_{s} | D_{1}kk | W^{2}_{1} |
*× | [∞^{+},2^{+},∞] | cm | C^{III}_{s} | D_{1}kg | W^{1}_{1} |
×× | [∞^{+},(2,∞)^{+}] | pg | C^{II}_{2} | D_{1}gg | W^{3}_{1} |
o | [∞^{+},2,∞^{+}] | p1 | C^{(I)}_{1} | C_{1} | W_{1} |
Example right triangles (*2pq) | ||||
---|---|---|---|---|
*237 |
*238 |
*239 |
*23∞ | |
*245 |
*246 |
*247 |
*248 |
*∞42 |
*255 |
*256 |
*257 |
*266 |
*2∞∞ |
Example general triangles (*pqr) | ||||
*334 |
*335 |
*336 |
*337 |
*33∞ |
*344 |
*366 |
*3∞∞ |
*6^{3} |
*∞^{3} |
Example higher polygons (*pqrs...) | ||||
*2223 |
*(23)^{2} |
*(24)^{2} |
*3^{4} |
*4^{4} |
*2^{5} |
*2^{6} |
*2^{7} |
*2^{8} | |
*222∞ |
*(2∞)^{2} |
*∞^{4} |
*2^{∞} |
*∞^{∞} |
A first few hyperbolic groups, ordered by their Euler characteristic are:
-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)]^{+} |
... |
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 dimensionsIn 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 dimensionsIn 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 reflectionIn 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 groupsFinite 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 groupsThis 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 groupIn 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 tilingIn geometry, the order-4 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {7,4}.
Order-6 hexagonal tilingIn 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 tilingIn geometry, the order-6 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,6}.
Order-6 pentagonal tilingIn geometry, the order-6 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,6}.
Order-8 octagonal tilingIn 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 tilingIn 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 dimensionsIn 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 tilingIn 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 tilingIn geometry, the rhombtriapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of rr{∞,3}.
Rhombitrioctagonal tilingIn 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 latticeIn 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 groupA 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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