SingSurf Home Wallpaper app Help Maths Orbifolds Groups Code

Orbifold and topology

An alternative way of specifying the groups is using Thurston's orbifold notation. It uses numbers to represent the order of rotation points, * for a reflection, x for a glide-reflection and o for the "wonder" with no additional symmetries other than translations. Symbols are listed in order, number before a * represent cyclic rotations and those after dihedral rotations.

Each type of symmetry has a cost:

The "Magic theorem" states that for Wallpaper groups these costs must sum to 2. The theorem is derived from the calculating the Euler characteristic of a tessellation of the plane from multiple copies of the fundamental domain. Each vertex and edge is given a weighting, for example a C3 vertex has a weight of $\frac13$ as three copies of the vertex contribute to a single vertex in the tessellations, likewise edge have a weight of $\frac12$ and D2 vertex has a weight of $\frac14$. Costs for each symmetry are found from these weights and a calculation show the Euler characteristic of the plane is $2-\text{total cost}=0$. A sketch proof of the result can be found in Geometry and the Imagination: The Euler characteristic of an orbifold

Some of the edges of the fundamental domain can be identified, so it forms a topological manifold, $M$. If are lines of reflection it will be a manifold with boundary. This manifold is either a sphere (p2, p3, p4, p6) a disk (pmm, p3m1, p4m, p6m, pmg, cmm, p31m, p4g) a cylinder (pm), Klein bottle (pg), Möbius band (cm), real-projective plane (pgg) or torus (p1).

Further structure is added to form an orbifold where cyclic rotations give conical points, reflections lines give boundary points and dihedral rotations form corners on the boundary. There is a group actions $\mathbb{R}^2/\Gamma$ associated with each symmetry in the orbifold.

Pure gyrations
These patterns are generated by a three or four cyclic rotation points. The two edges adjacent each rotation point are identified so the underlying manifold forms a topological sphere, with Euler characteristic, χ=2. Each n-fold rotation point gives a conical point in the orbifold with cost $1-\frac{1}{n}$.
p2: 2222 Four C2 rotations. Cost: $\frac12+\frac12+\frac12+\frac12=2$ Manifold: sphere Euler characteristic: χ=2 Chromatic number: 4
p3: 333 Three C3 rotations. Cost: $\frac23+\frac23+\frac23=2$ Manifold: sphere Euler characteristic: χ=2 Chromatic number: 4
p4: 442 Two C4 rotations and one C2 rotation. Cost: $\frac34+\frac34+\frac12=2$ Manifold: sphere Euler characteristic: χ=2 Chromatic number: 4
p6: 632 One C6, one C3 and one C2 rotation. Cost: $\frac5/6+\frac23+\frac12=2$ Manifold: sphere Euler characteristic: χ=2 Chromatic number: 4
Pure kaleidoscopes
These patterns are generated by three or four dihedral rotations. The edges of the fundamental domain are lines of reflection, giving rise to a boundary of the orbifold. Hence the underlying manifold is a disk with Euler characteristic, χ=1. The boundary, *, has a cost of 1 and each n-fold dihedral rotation has a cost of $\frac{n-1}{2n}$.
pmm: *2222 One reflection, four D2 rotations. Cost: $1+\frac14+\frac14+\frac14+\frac14=2$ Manifold: disk Euler characteristic: χ=1 Chromatic number: 4
p3m1: *333 One reflection, three D3 rotations. Cost: $1+\frac26+\frac26+\frac26=2$ Manifold: disk Euler characteristic: χ=1 Chromatic number: 4
p4m: *442 One reflection, two D4 rotations and one D2 rotation. Cost: $1+\frac38+\frac38+\frac14=2$ Manifold: disk Euler characteristic: χ=1 Chromatic number: 4
p6m: *632 One reflection, one D6, one D3 and one D2 rotation. Cost: $1+\frac{5}{12}+\frac26+\frac14=2$ Manifold: disk Euler characteristic: χ=1 Chromatic number: 4
Parallel reflections
This patterns is generated by two parallel reflections and a translation. Two edges of the fundamental domain are lines of reflection, giving rise to a boundary of the orbifold, the other two are identified. Hence the underlying manifold is a cylinder with Euler characteristic, χ=0.
pm: ** Two reflections. Cost: $1+1=2$ Manifold: cylinder Euler characteristic: χ=0 Chromatic number: 4
Hybrid types
These patterns have both cyclic points, reflections and sometimes dihedral rotations. The lines of reflection give a boundary of the orbifold, so the underlying manifold is a disk with Euler characteristic, χ=1. The cyclic rotations correspond to conical points and the dihedral rotations give corners.
pmg: 22* Two C2 rotations and one line of reflections. Cost: $\frac12+\frac12+1=2$ Manifold: disk Euler characteristic: χ=1 Chromatic number: 4
cmm: 2*22 One C2 rotation, one reflection and two D2 reflections. Cost: $\frac12+1+\frac14+\frac14=2$ Manifold: disk Euler characteristic: χ=1 Chromatic number: 4
p31m: 3*3 One C3 rotation, one reflection and one D3 rotation. Cost: $\frac13+1+\frac26=2$ Manifold: disk Euler characteristic: χ=1 Chromatic number: 4
p4g: 4*2 One C4 rotation, one reflections and one D2 rotation. Cost: $\frac34+1+\frac14=2$ Manifold: disk Euler characteristic: χ=1 Chromatic number: 4
Glide-reflections and cross-caps
pg: xx Two parallel glide-reflections. Cost: $1+1=2$ Manifold: Klein bottle Euler characteristic, χ=0. Chromatic number: 6
cm: *x A reflection and a glide-reflections. Cost: $1+1=2$ Manifold: Möbius band Euler characteristic: χ=0. Chromatic number: 6
pgg: 22x Two C2 rotations, and a glide-reflection. Cost: $\frac12+\frac12+1=2$ Manifold: real-projective plane RP² Euler characteristic: χ=1. Chromatic number: 6
The wonder: just translations
This pattern has no additional symmetries. Opposite edges are identified, so the underlying manifold is a torus with Euler characteristic, χ=0.
p1: o Cost: $2$ Manifold: torus Euler characteristic: χ=0. Chromatic number: 7

Chromatic number for orbifolds and tilings

Given a wallpaper group with a specified lattice, we can consider the set of tilings that are invariant under the that group. Similar to the four-colour theorem for planer graphs we can consider the Chromatic number for the wallpaper group:

The smallest number χ, such that any tiling invariant under the group can be coloured with χ colours such that:

• No two tiles with the same colour share a common edge
• The colouring is invariant under that tiling

Calculating the chromatic number is easy, it is the same as the chromatic number of the underlying topological manifold of the orbifold, and the chromatic numbers of the each type of manifold are well known - Torus: 7, Möbius band: 6, Klein bottle: 6, Projective Plane: 6, sphere: 4, disk: 4, open cylinder or annulus: 4. All that remains is to find a tiling that achieves the maximum bound. Again there are well known graphs on the various manifold that achieve the maximum, so we simply have unwrap these graphs to form a tiling of the plane. We give examples of each of these below and discuss how the underlying manifold is derived.

P1, manifold: torus, chromatic number: 7

Given a topological space X and a group G we can form the quotient space by factoring out the group action. Say with the P1 group

Using a motif of Beanish from Tales of the BeanWorld.

Factoring out the group action generated by the two translations gives a single cell with opposite edges are identified.

We can wrap the image around a cylinder, so the blue arrows match, and then join the top of the cylinder to the bottom to form a torus:

The torus has Euler characteristic χ=0, and following the Heawood conjecture any tiling with this symmetry can be coloured with 7 colours, so that no two faces of the same colour share an edge. One such tiling is based on hexagons.

Unit cell	Wallpaper pattern	Wrapped around a torus

CM, manifold: Möbius band, chromatic number: 6

Folding along the reflection lines gives a strip with glide-reflection. The fundamental domain has top and bottom identified in opposite directions, and forms a Möbius band.

A strip formed by folding along the reflection lines, has a glide-reflection.	The unit cell has top and bottom identified, but in opposite directions:	Joining edges forms a Möbius band.

The Euler characteristic of the Möbius band is 0, and in 1910 Heinrich Tietze found a subdivision into six mutually adjacent regions. This can be used as the basis of a tiling of the plane requiring 6 colours.

Unit cell	Wallpaper pattern	Wrapped around a Möbius band

PG: xx

Wallpaper pattern	Unit cell

This has Euler characteristic χ=0, and chromatic number 6.

Unit cell	Wallpaper pattern	Wrapped around a Klein Bottle. (source)

PGG: 22x

Wallpaper pattern	Unit cell	Wrapped around a real projective plane.

This has Euler characteristic χ=1, and chromatic number 6.

Unit cell	Wallpaper pattern	Wrapped around a Klein Bottle.

The cell division here is adapted from mathcurve.com

PM, manifold: cylinder, chromatic number 4

Take a horizontal strip with top and bottom identified, fold along the vertical mirror lines, forming a concertina. Identify top and bottom edges to form a cylinder. The green edges correspond to the two lines of reflection, which always form a boundary on the orbifold.

A strip formed by folding along the reflection lines.	The unit cell has top and bottom identified, so it forms a cylinder.	Joining edges forms a cylinder.

The cylinder has Euler characteristic χ=0, and had chromatic number 4, so four colours suffice.

Fundamental domain	Wallpaper pattern	Orbifold

Groups generated by cyclic rotations

In terms of colouring the interesting groups are those with a chromatic number greater than 4 are shown above. The rest are either topological spheres or disks, and can be coloured with 4 colours. First we consider the groups generated by cyclic rotations P2 (2222), P3 (333), P4 (442) and P6 (632).

P2, manifold: sphere, chromatic number 4

The P2 is generated by four C2 rotations, around the mid-points of a rectangular fundamental domain. The two halves of each edge are identified, so the edges join up to form a sphere.

Unit cell	Top of sphere	Bottom of sphere

A tessellation requiring 4 colours is easy to find. Start by considering a tetrahedral polygonization of the sphere, with four mutually adjacent triangular faces, each of a different colour. Such a polygonization can be arranged so that each of the four rotation points lie inside one of the four faces. Next unwrap this to form the fundamental domain, and apply the symmetries to form the tiling.

Unit cell	Top of sphere	Bottom of sphere

P3, P4, P6

A similar process can be used to the other pure rotational groups, P3, P4 and P6.

P3: 333	P4: 442	P6: 632

In each case the pattern is generated by three rotations. The tiling is chosen so that each rotation point lies in the centre of a face, the fourth triangular face is chosen in the middle of the fundamental domain.

Groups generated by dihedral reflections

The groups PMM, P3M1, P4M and P6M, are all generated by dihedral rotations. Each edge of the fundamental domain is a reflection line which form an edge in the underlying manifold. Hence the manifold is a topological disk with chromatic number 4.

Finding tilings that require four colours is easy here. Take an domain with a small triangle in the centre and draw lines outward from the corners. This forms a square or triangle with four mutually adjacent regions.

PMM domain	PMM wallpaper	P3M1 domain	P3M1 wallpaper
P4M domain	P4M wallpaper	P6M domain	P6M wallpaper

For these groups the orbifold is a polygon corresponding to the fundamental domain, the reflection lines form the boundary of the orbifold, each dihedral rotation point giving a vertex. Topologically this is a disk.

Orbifolds for PMM, P3M1, P4M and P6M

Mixed types

The groups CMM, P31M and P4G all have both cyclic rotations and dihedral rotations. The lines of reflection gives an edge around the underlying manifold, a topological disk, with corners for each dihedral rotation. Cyclic rotation points lie in the middle of the disk. Each pattern has chromatic number 4. PMG is similar, it has a line of reflection and two cyclic rotation points, by no dihedral rotations.

PMG domain	PMG wallpaper	PMG underlying manifold	CMM domain	CMM wallpaper	CMM underlying manifold
P4G domain	P4G wallpaper	P4G orbifold	P31M domain	P31M wallpaper

Summery of colourings

References

• Wikipedia: Wallpaper groups.
• nLab: Riemannian orbifold
• Patrick J. Morandi The Classification of Wallpaper Patterns: From Group Cohomology to Escher's Tessellations 2003 - Describes how cohomology can be used to classify wallpaper patterns.
• João Guerreiro Orbifolds and Wallpaper Patterns 2009
• J. H. Conway, H. Burgiel, C. Goodman-Stauss, The Symmetries of Things, Taylor & Francis Group, New York, 2008. - A good illustrated example of wallpaper patterns and orbifolds.

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SingSurf Home Wallpaper app Help Maths Orbifolds Groups Code

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