Proof That There Are 17 Wallpaper Groups

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Proof That There Are 17 Wallpaper Groups

A wallpaper is a two dimensional repeating pattern with two independent translations that fill the plane. A wallpaper lattice L is the set of all points that the origin gets mapped to under translation. We select two nonparallel, nonzero vectors a and b in L, where a is shortest and b is also as short as possible, then L is spanned by a and b. All the elements of L are of the form ma + nb, where n and m are integers. By examining all possible basic parallelograms determined by a and b, we can conclude that there are only five different types of lattices. Parallelogram: | a | < | b | < | a – b | < | a + b | Rectangular: | a | < | b | < | a – b | = | a + b | Rhombic: | a | < | b | = | a – b | < | a + b | Square: | a | = | b | < | a – b | = | a + b | Hexagonal: | a | = | b | = | a – b | < | a + b | Parallelogram _Rectangular Rhombic Square Hexagonal

An isometry of _ℝ2_{is a distance-preserving transformation of the plane. There are only four} types of isometries of the plane: translations, rotations, reflections and glide reflections. We define R to be the matrix that represents an anticlockwise rotation through an angle of  about the origin. R = _cos  –sin _{sin  cos  . For example, R}_ 90 = _0 –1_{1 0 , R}_ 180 = _–1 0_{0 –1 = –I, R}_ – 90 = __{–1 0 .}0 1_

We define M to be the matrix that represents a reflection whose mirror makes an angle of  with the positive horizontal axis. M = _cos 2 sin 2_{sin 2 –cos 2 . For example, M}_ 0 = _1 0_{0 –1}_ represents a reflection about the horizontal axis, and M90 = _–1 0_{0 1 represents a reflection}_ about the vertical axis. M45 = _0 1_{1 0 represents a reflection about the line y = x, and M}_ 135 =

 

 

0 –1

–1 0 represents a reflection about the line y = –x.

The symmetry group of a wallpaper pattern is said to be a wallpaper group. The point group of a wallpaper group G is the set of all orthogonal matrices corresponding to the isometries in G.

Lemma:

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point group of G is { I } or { I, –I }. Proof:

The only possible nontrivial transformation in G is 180 rotation. That is, if we rotate it 180, we get the same figure. There are no possible reflections. Hence the point group of G must be a subgroup of { I, –I }  Z2 . If we exclude 180 rotation, then the

point group can only be { I }. If we include 180 rotation, then the point group must be { I, –I }. Hence we have established all possible point groups of G.

Theorem:

A parallelogram lattice produces two distinct wallpaper groups: p1 and p2. Proof:

Let G be a wallpaper group corresponding to a pattern with a parallelogram lattice, and having a point group J. From above lemma, J is { I } or { I, –I }.

If J = { I }, then we define G to be the group p1. This group p1 only contains translations. If J = { I, –I }, then we define G to be the group p2, which has translations and 180rotations. Since the point groups of p1 and p2 are not isomorphic, so p1 and p2 are not isomorphic.

p1 p2 no rotation no reflection no glide reflection rotation of order 2 no reflection no glide reflection Lemma:

If G is a wallpaper group corresponding to a pattern with a rectangular lattice, then the possible point groups of G are { I }, { I, –I } { I, M0 }, { I, M90 } and { I, –I, M0 , M90 }.

Proof:

The point group J of G can have 180 rotation, vertical reflection and horizontal reflection. Thus J must be a subgroup of { I, –I, M0 , M90 }  K4 . The subgroups of K4 are K4 , Z2 and { I }. If J is isomorphic to K4 , then J = { I, –I, M0 , M90 }. If J is isomorphic

to Z2 , then we have 3 possibilities: { I, –I } { I, M0 }, { I, M90 }. These sets are closed and are groups. The only other possible group left is { I }. Hence we conclude that J must equal { I }, { I, –I } { I, M0 }, { I, M90 } or { I, –I, M0 , M90 }.

Theorem:

A rectangular lattice produces five distinct wallpaper groups that are different from those produced by a parallelogram lattice: pm, pg, pgg, pmg and pmm.

Proof:

Let G be a wallpaper group corresponding to a pattern with a rectangular lattice, and having a point group J. From above lemma, J is { I }, { I, –I } { I, M0 }, { I, M90 } or { I, –I, M0 , M90 }. If J = { I }, then G only has translations and thus is isomorphic to p1. This does not produce any new groups.

If J = { I, –I }, then G only has translations and 180 rotations, and thus is isomorphic to p2. This does not produce any new groups.

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contains translations and horizontal reflections. The group pm is not isomorphic to p1 or p2, because neither p1 nor p2 contains reflections.

Suppose that M0 is realised by a glide reflection. We define G to be the group pg, which contains translations and horizontal glide reflections. The group pg is not isomorphic to p1 or p2 (which do not contain glide reflections), or pm (which contains reflections).

If J = { I, M90 }, then M90 can be realised either by a vertical reflection or a vertical glide reflection. By changing perspective, G is isomorphic to either pm or pg.

If J = { I, –I, M0 , M90 }, then M0 and M90 can be realised either by reflections or glide reflections. Suppose that M0 and M90 are realised by glide reflections. We define G to be the group pgg, which contains translations, horizontal and vertical glide reflections. This is the only group of order 4 so far, and so is not isomorphic to any earlier groups.

Suppose that M0 is realised by a glide reflection, and M90 is realised by a reflection. We define G to be the group pmg, which contains translations, vertical reflections and horizontal glide reflections. This is the only group of order 4 so far to contain reflections, and so is not isomorphic to any earlier groups.

Suppose that M0 and M90 are realised by reflections. We define G to be the group pmm, which contains translations, horizontal and vertical reflections. This is the only group so far to contain reflections in 2 directions, and so is not isomorphic to any earlier groups.

pm pg pgg pmg pmm

no rotation reflection in 1

direction glide reflections along reflection axes

no rotation no reflection glide reflection rotation of order 2 no reflection glide reflection rotation of order 2 reflection in 1 direction rotation of order 2 reflection in 2 directions rotations along reflection axes Lemma:

If G is a wallpaper group corresponding to a pattern with a rhombic (centred-rectangular) lattice, then the possible point groups of G are { I }, { I, –I } { I, M0 }, { I, M90 } and { I, –I, M0 , M90 }.

Proof:

The point group J of G can have 180 rotation, vertical reflection and horizontal reflection.

Thus J must be a subgroup of { I, –I, M0 , M90 }  K4 . Hence, as above, we conclude that J must equal { I }, { I, –I } { I, M0 }, { I, M90 } or { I, –I, M0 , M90 }.

Theorem:

A rhombic lattice produces two distinct wallpaper groups that are different from those produced by a parallelogram or rectangular lattice: cm and cmm.

Proof:

Let G be a wallpaper group corresponding to a pattern with a rhombic lattice, and having a point group J. From above lemma, J is { I }, { I, –I } { I, M0 }, { I, M90 } or { I, –I, M0 , M90 }. { I } and { I, –I } have been previously explored and do not produce any new groups.

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is realised by a horizontal reflection and a horizontal glide reflection. We define G to be the group cm, which contains translations, horizontal reflections and horizontal glide reflections. The group cm is not isomorphic to p1, p2, pg or pgg (which do not contain reflections), or pmm or pmg (which contain rotations), or pm (since some glide reflections do not occur along reflection axes).

If J = { I, M90 }, then M90 can be realised by a vertical reflection and a vertical glide reflection. By changing perspective, G is isomorphic to cm.

If J = { I, –I, M0 , M90 }, then M0 and M90 can be realised by either a reflection or a glide reflection. We define G to be the group cmm, which contains horizontal and vertical reflections and glide reflections. The group cmm is not isomorphic to p1, p2, pg, pgg, pm, pmg or cm (which do not contain reflections in 2 directions), or pmm (since some rotations do not occur along reflection axes).

cm cmm

no rotation reflection in 1 direction

some glide reflections not along reflection axes

rotation of order 2 reflection in 2 directions

some rotations not along reflection axes

Lemma:

If G is a wallpaper group corresponding to a pattern with a square lattice, then the possible point groups of G are { I }, { I, –I }, { I, M0 }, { I, M45 }, { I, M90 }, { I, M135 }, { I, –I, M0 , M90 }, { I, –I, M45 , M135 }, { I, –I, R90 , R–90 } and { I, –I, R90 , R–90 , M0 , M45 , M90 , M135 }.

Proof:

The point group J of G can have rotation of order 4, vertical reflection, horizontal reflection and two diagonal reflections. Thus J must be a subgroup of { I, –I, R90 , R–90 , M0 , M45 , M90 , M135 }  D4 . The subgroups of D4 are D4 , K4 , Z4 , Z2 and { I }. If J is isomorphic to D4 , then J = { I, –I, R90 , R–90 , M0 , M45 , M90 , M135 }.

If J is isomorphic to K4 , then it can have 180 rotation and perpendicular reflections. If it contains 2 reflections such as M0 and M45 , whose mirrors are not perpendicular, it will not be isomorphic to K4 . Thus the only possible groups are { I, –I, M0 , M90 } and { I, –I, M45 , M135 }. If J is isomorphic to Z4 , then it must have rotations of order 4 and no reflections. The only possible point group is { I, –I, R90 , R–90 }.

If J is isomorphic to Z2 , then it contains the identity and a transformation of order 2. The only possible groups are { I, –I }, { I, M0 }, { I, M45 }, { I, M90 } and { I, M135 }.

Theorem:

A square lattice produces three distinct wallpaper groups that are different from those produced by a parallelogram, rectangular or rhombic lattice: p4, p4g and p4m.

Proof:

Let G be a wallpaper group corresponding to a pattern with a square lattice, and having a point group J. From above lemma, J is { I }, { I, –I }, { I, M0 }, { I, M45 }, { I, M90 }, { I, M135 }, { I, –I, M0 , M90 }, { I, –I, M45 , M135 }, { I, –I, R90 , R–90 } and { I, –I, R90 , R–90 , M0 , M45 , M90 , M135 }. { I }, { I, –I }, { I, M0 }, { I, M90 } and { I, –I, M0 , M90 } have been previously explored and do not produce any new groups.

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produce any new wallpaper groups.

The group { I, –I, M45 , M135 } is isomorphic to { I, –I, M0 , M90 } and so does not produce any new groups.

If J = { I, –I, R90 , R–90 }, we define G to be the group p4, which contains translations and 90 rotations. This is the only group so far to contain rotations of order 4, and so is not isomorphic to any earlier groups.

If J = { I, –I, R90 , R–90 , M0 , M45 , M90 , M135 }, M0 and M90 can be realised by reflections. If M45 and M135 can be realised by glide reflections, then we define G to be the group p4g, which has rotations of order 4 and reflections in 2 directions. This is the first group of order 8 so far, and so is not isomorphic to any earlier groups.

If M45 and M135 can be realised by reflections, then we define G to be the group p4m, which has rotations of order 4 and reflections in 4 directions. This is the only group so far to contain reflections in 4 directions, and so is not isomorphic to any earlier groups.

p4 p4g p4m rotation of order 4 no reflection rotation of order 4 reflection in 2 directions rotation of order 4 reflection in 4 directions Lemma:

If G is a wallpaper group corresponding to a pattern with a hexagonal lattice, then the possible point groups of G are { I }, { I, –I }, { I, M30k } for 0 ≤ k ≤ 5, { I, –I, M0 , M90 }, { I, –I, M30 , M120 }, { I, –I, M60 , M150 }, { I, R120 , R–120 }, { I, R120 , R–120 , M30 , M90 , M150 }, { I, R120 , R–120 , M0 , M60 , M120 }, { I, –I, R60 , R120 , R–120 , R–60 } and { I, –I, R60 , R120 , R–120 , R–60 , M0 , M30 , M60 , M90 , M120 , M150 }.

Proof:

The point group J of G can have rotation of order 6, vertical and horizontal reflections, and reflections whose mirrors make 30, 60, 120 and 150 with the positive horizontal axis. Thus J must be a subgroup of { I, –I, R60 , R120 , R– 120 , R–60 , M0 , M30 , M60 , M90 , M120 , M150 }  D6 . The subgroups of D6 are D6 , Z6 , K4 , D3 , Z3 , Z2 and { I }.

If J is isomorphic to D6 , then J = { I, –I, R60 , R120 , R–120 , R–60 , M0 , M30 , M60 , M90 , M120 , M150 }.

If J is isomorphic to Z6 , then it must have rotations of order 6. The only possible point group is { I, –I, R60 , R120 , R–120 , R–60 }.

If J is isomorphic to K4 , then it can have 180 rotation and perpendicular reflections. If it contains 2 reflections such as M0 and M45 , whose mirrors are not perpendicular, it will not be isomorphic to K4 . Thus the only possible groups are { I, –I, M0 , M90 }, { I, –I, M30 , M120 } and { I, –I, M60 , M150 }.

If J is isomorphic to D3 , then it can have 120 rotation, horizontal reflection, and reflections whose mirrors make 60 and 120 with the positive horizontal axis. This is the case when J = { I, R120 , R–120 , M0 , M60 , M120 }. J can also have 120 rotation, vertical reflection, and reflections whose mirrors make 30 and 150 with the positive horizontal axis. This is the case when J = { I, R120 , R–120 , M30 , M90 , M150 }.

If J is isomorphic to Z3 , then it must have rotations of order 3. The only possible point group is { I, R120 , R–120 }.

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for 0 ≤ k ≤ 5. Theorem:

A hexagonal lattice produces five distinct wallpaper groups that are different from those produced by a parallelogram, rectangular, rhombic or square lattice: p3, p3m1, p31m, p6 and p6m.

Proof:

Let G be a wallpaper group corresponding to a pattern with a hexagonal lattice, and having a point group J. From above lemma, J is { I }, { I, –I }, { I, M30k } for 0 ≤ k ≤ 5, { I, –I, M0 , M90 }, { I, –I, M30 , M120 }, { I, –I, M60 , M150 }, { I, R120 , R–120 }, { I, R120 , R–120 , M30 , M90 , M150 }, { I, R120 , R–120 , M0 , M60 , M120 }, { I, –I, R60 , R120 , R–120 , R–60 } or { I, –I, R60 , R120 , R–120 , R–60 , M0 , M30 , M60 , M90 , M120 , M150 }.

{ I }, { I, –I }, { I, M30k } for 0 ≤ k ≤ 5, and { I, –I, M0 , M90 } have been previously explored and do not produce any new groups.

The groups { I, –I, M30 , M120 } and { I, –I, M60 , M150 } are isomorphic to { I, –I, M0 , M90 } and so do not produce any new groups.

If J = { I, R120 , R–120 }, we define G to be the group p3, which contains translations and 120 rotations. This is the only group of order 3 so far, and so is not isomorphic to any earlier groups. If J = { I, R120 , R–120 , M0 , M60 , M120 }, M0 , M60 and M120 can be realised by reflections. We define G to be the group p3m1. This is the only group of order 6 so far, and so is not isomorphic to any earlier groups.

If J = { I, R120 , R–120 , M30 , M90 , M150 }, M30 , M90 , M150 can be realised by reflections. We define G to be the group p31m. This is the second group of order 6 so far, and so is not isomorphic to any earlier groups, and not isomorphic to p3m1, where all rotations occur along reflection axes.

If J = { I, –I, R60 , R120 , R–120 , R–60 }, we define G to be the group p6, which contains translations and 60 rotations. This is the only group so far to contain rotation of order 6, and so is not isomorphic to any earlier groups.

If J = { I, –I, R60 , R120 , R–120 , R–60 , M0 , M30 , M60 , M90 , M120 , M150 }, we define G to be the group p6m, which contains translations, reflections and 60 rotations. This is the only group of order 12, and so is not isomorphic to any other wallpaper groups.

p3 p3m1 p31m

rotation of order 3 no reflection

rotation of order 3 reflection in 3 directions all rotations along reflection

axes

rotation of order 3 reflection in 3 directions some rotations not along

reflection axes p6 p6m rotation of order 6 no reflection rotation of order 6 reflection in 6 directions

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17 Wallpaper Groups No Rotation

no reflection reflection in 1 direction

no glide reflection with glide reflection occur along reflection all glide reflections axes

some glide reflections do not occur along

reflection axes

p1 pg pm cm

Rotation of Order 2

no reflection reflection in 2 directions

no glide reflection with glide reflection reflection in 1 direction all rotations occur along reflection axes

some rotations do not occur along

reflection axes

p2 pgg pmg pmm cmm

reflection in 3 directions

no reflection all rotations occur along _{reflection axes} some rotations do not occur _{along reflection axes}

p3 p3m1 p31m

no reflection reflection in 2 directions reflection in 4 directions

p4 p4g p4m

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17 Wallpaper Tessellations No Rotation

no glide reflection with glide reflection occur along reflection all glide reflections axes

some glide reflections do not occur along

reflection axes

p1 pg pm cm

no glide reflection with glide reflection reflection in 1 direction all rotations occur along reflection axes

some rotations do not occur along

reflection axes

p2 pgg pmg pmm cmm

reflection in 3 directions

no reflection all rotations occur along _{reflection axes} some rotations do not occur _{along reflection axes}

p3 p3m1 p31m

p4 p4g p4m

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17 Wallpaper Designs No Rotation

no glide reflection with glide reflection all glide reflections along _{reflection axes} _{not along reflection axes}some glide reflections

p1 pg pm cm

no glide reflection with glide reflection reflection in 1 direction all rotations along _{reflection axes} _{along reflection axes}some rotations not

p2 pgg pmg pmm cmm Rotation of Order 3

all rotations occur along reflection axes

some rotations do not occur along reflection axes

p3 p3m1 p31m

p4 p4g p4m

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7 Frieze Patterns

No Reflection & No Rotation p1

none

Reflection Along Main Axis pm1

H

Reflection Perpendicular to Main Axis p1m V Rotation p2 R Glide Reflection pg G

Reflection & Glide Reflection pmg

VRG

Reflection in 2 Directions pmm

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Proof That There Are 7 Frieze Patterns

A frieze pattern is a design on a two-dimensional surface that is repetitive in one direction. A frieze pattern necessarily contains translations. Besides translations, it may also contain one or more of the following symmetries:

H = reflection in a horizontal mirror, V = reflection in a vertical mirror, R = 180° rotation, and

G = glide reflection along a horizontal axis.

For each frieze pattern, we can give it an HVRG symbol, depending on which of these four symmetries it possesses. For example, a frieze pattern with symbol VRG would have a symmetry group which contains a reflection in a vertical mirror, a 180° rotation and glide reflection along a horizontal axis, but not a reflection in a horizontal mirror.

Theoretically, there are sixteen possible HVRG symbols:

none, H, V, R, G, HV, HR, HG, VR, VG, RG, HVR, HVG, HRG, VRG, HVRG. However, the following observations allow us to exclude some of these 16 combinations:

If we have H, then we also have G:  G

If we have two out of { V, R, G }, then we have all three:

i.e. if we have V and R, we also have G: + R =  G if we have V and G, we also have R: + G =  R if we have R and G, we also have V: + G =  V

If we eliminate some of these possibilities using the above observations, we find that there are only seven remaining: none, H, V, R, G, VRG, HVRG.

Reference: . Symmetry In Geometry