The Many Symmetries

of Planar Objects

The composition of a temple is based on symmetry, whose princi-ples architects should take the greatest care to master. Symmetryderives from proportion [. . . ]. Proportion is the mutual calibra-tion of each element of the work and of the whole, from which theproportional system is achieved. No temple can have any compo-sitional system without symmetry and proportion . . .

Vitrubius

Figure 3.A Vitruvian Man by Leonardo da Vinci

Leaving aside minor details such as the lateral view of the left feet asopposed to the front view for the right ones and the derived differencein the legs torsions the figure exhibits what we will call bilateralsymmetry. If we trace an imaginary vertical line through the center ofthe image, the resulting halves are mirror images of one another. Aconsequence of this feature is the 1-to-1 ratio between left and rightparts of the body, a proportion responsible for the feeling of balancein the Vitruvian man.

Figure 3.B Groundplan of the Temple of Diana at Ephesos

This groundplan also displays bilateral symmetry (the imaginary linenow being horizontal). But it exhibits other regularities as well. Thecolumns of the temple are placed in parallel lines and equally spacedwithin these lines, thus inducing a sense of order in their repetition.

Both the uniformity of the columns placement and the bilateralsymmetry can be precisely described in terms of invariance under cer-tain isometries, a fact that naturally leads us to a general definitionof symmetry.

The Basic Symmetries

To describe the effect of the different isometries we consider subsetsS of the plane E and their images under these isometries. In whatfollows we will call any such subset of the plane a planar figure (orsimply a figure).

Bilateral symmetry: the straight-lined mirror

The most frequent use of the word symmetry in common speech refersto the presence of bilateral symmetry. Briefly described, this is theinvariance of a figure under a reflection.

We say that a figure S has bilateral symmetry when there exists aline such that refl(S) = S.

It is important to notice that S is not fixed by refl. The onlypossible fixed points (i.e., points p in S for which refl(p) = p) arethose lying in .

But S is invariant under refl; that is, the figure S is indistinguishablefrom its reflection refl(S) (even though most points in S changedtheir position). Thus, if we reflect the word bod with respect to avertical line passing through the centre of the o

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bod

we observe that the word has bilateral symmetry. In the reflectiondefining the symmetry the letter o is invariant but the letters b andd are not. In fact the reflection turns one into the other while makingthem to swap places.

Whereas bilateral symmetry is pervasive both in nature and in dif-ferent art forms, its ubiquitousness is maybe best seen in architecture.We have already mentioned it as a canon in Vitruvius and shown thegroundplan of a classical temple as an example.

Figure 3.C.1 Groundplans of the Cathedral of Amiens, the Church of Invalides

(Paris), the Pantheon (Paris), the Pisa Cathedral, St. Elizabeth Cathedral (Marburg),

and St. Paul Cathedral (London).

Figure 3.C.2 Elevations or sections of the Church of Santo Spirito (Florence), the

Girard College (Philadelphia), the Cathedral of Ulm, St. Elizabeth Cathedral (Marburg)

(two plates), and the Magasins Bon March (Paris).

Rotational symmetry

Invariance under an isometry naturally induces forms of symmetryother than bilateral by considering isometries other than reflections.In the case of rotations we obtain rotational symmetry.

We say that a figure S has rotational symmetry when there exist apoint O and an angle such that rotO,(S) = S.

Rotational symmetry is also common in architecture. One can seeit in domes, floors, fountains, and windows. The latter, because oftheir resemblance to roses, came to be known in the middle ages asrosettes.

Figure 3.D Design of rosettes in the Cathedrals of Strasbourg (left) and Lyon

(right).

Rotational symmetry is found in various contexts beyond architecture.Occasionally, inspiration from architectural design is evident, as in thefollowing Tabriz carpet.

Figure 3.E The Dome of the Sheikh Lotf Allah Mosque in Isfahan, Iran, (left) and a

Tabriz carpet with a dome design (right).

Other objects have no direct links with architectural design.

Figure 3.F Rosette 22 by Jerry Matchett.

Central symmetry: the one-point mirror

A particular case of rotation is that with an angle of 180 (half-turn).In this case the image P of a point P lies on the line determined byP and the center O of the rotation, and is opposed to P .

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O

P

P

We can think of such a rotation as a mirror consisting of a singlepoint. Unlike the line-shaped mirror:

orientation is preserved,

one-point mirrors do not partition the plane as line-shaped mirrorsdo. There is no division between this side and the other side of themirror.

The symmetry associated to half-turns is known as central symmetry.

A well-known centrally symmetric figure is the symbol representing theYing-yang principle of Taoism.

With its absence of boundaries, central symmetry provides a fittingrepresentation of the Ying-yang principle which postulates that forcesthat appear to be opposite are bound together, interwoven, and inter-dependent in the universe, giving rise to each other in turn.

Translational symmetry: repeated mirrors

We say that a figure S has translational symmetry when there existsa vector v such that transv (S) = S.

How does a translational invariant figure look like? Assume that vis horizontal and points to the right, like . Assume as well thatthe figure contains a letter A. Since the whole figure is invariant bytranslation, the letter A will move to a position already containingan A. That is, the figure contains the pair

A A

Repeating this reasoning produces a third A, and a fourth, a fifth, . . . .

A A A A A A

Also, the space left empty by our first A needs to be filled by anotherA which had to be originally located at its left. Again, repeating thisreasoning shows that the figure must have (if we start with no morethan the A) the following aspect:

A A A A A A A A A A A A A A A A A A A A A

Strictly speaking, we would be at pains to find translationally symmet-ric figures in any form of art. Human creations have the limitation offiniteness. But infinity can be suggested in finite fragments of trans-lationally symmetric figures.

Figure 3.G Friezes at the Temple of Hathor, Denderah, Egypt.

As with rotational invariance, architecture provided a name to thesetranslationally symmetric patterns (frieze). Hence the use of the ex-pression frieze pattern to denote translationally symmetric designs.

Figure 3.H Pavement at Rhodes, Greece.

Glidal symmetry

The next definition is the natural close to our family of basic symme-tries.

We say that a figure S has glidal symmetry when there exists a line and a vector v parallel to such that glide,v (S) = S.

The footprint we encountered in the previous chapter, convenientlyrepeated, may give us a first idea of the visual appearance of a glidallysymetrical figure.

Glidal symmetry can be found, as well, in different forms of artwork.For example, it is observable in the following Maori design.

Figure 3.I Maori Kowhaiwhai pattern in the Ngaru style, New Zealand.

It is also present (with a vertical direction) in the Japanese patternshown below.

Figure 3.J Seigaiha (waves from the blue ocean) pattern, Japan.

Note: Every figure having glidal symmetry is a frieze.

A few differences between the three patterns above deserve emphasis:

The Maori pattern has bilateral symmetry (around a horizontal lineat its center) whereas the footpath has not.

The Maori pattern has translational symmetry for the translationwith vector v (the vector associated to the glide) whereas the set offootprints does not.

The Seigaiha has two possible translations with different directions.This fact allows one (actually forces one) to extend the design all overthe plane.

Figures having two translational symmetries with non-parallel trans-lation vectors are called wallpapers. Architectural glossary appearsagain, this time through the humble activity of wall decoration.

The Arithmetic of Isometries

Plane figures may posses invariance with respect to various isometries.The Strasbourg rosette has both rotational and bilateral symmetry.Figures 3.I and 3.J have glidal, translational and bilateral invariance,and the latter possesses translational invariance with respect to twonon-parallel vectors.

The family of isometries leaving a figure invariant has some structurethat makes it more than a loose collection of transformations. Tounderstand this structure a few definitions are necessary.

Let f, g : E E be two functions on the plane. The compositionf g : E E is the function associating to any x E the pointf(g(x)). That is:

x 7 g(x) 7 f(g(x)).

For instance, given O, and , the composition rotO, refl associatesto any point p on the plane the point p obtained by first reflectingp with respect to and then rotating the resulting point, say p, anangle around the center O.

p

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O

p

p............................................

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We may regard composition as an operation between isometries aswe regard addition as an operation between numbers.

The composition of two isometries produces a third one.

The number 0 has no effect when added to any other number.That is, one has 0 + x = x + 0 = x for all x. An isometry with asimilar behavior (viz composition) is the identity Id. Indeed, for anyisometry and any point p E

(Id )(p) = Id((p)) = (p)

and, similarly, ( Id)(p) = (p). That is, Id = Id = .

For any number x there exists another number y such that x+y =y + x = 0. Simply take y to be x. To prove that the correspondingstatement for composition of isometries holds true we can rely on theclassification of isometries. Indeed, it is easy to check that,

(i) for any line , refl refl = Id,

(ii) for any point O and any angle , rotO, rotO,360 = rotO,360 rotO, = Id,

(iii) for any vector v , transv transv = transv transv = Id,and

(iv) for any line and vector v parallel to , glide,v glide,v =glide,v glide,v = Id.

Here v denotes the vector whose endpoints are those of v butwith the opposite orientation.

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vv

We therefore see that for any isometry there exists an isometry such that = = Id. We call the inverse of and denotethis inverse by 1.

Another structural coincidence between addition of numbers andcomposition of isometries is the fact that for all isometries , , onehas

( ) = ( ) .

That is, it is irrelevant whether we first compute and thencompose with the result or, instead, we first compute andthen compose the result with .

Important note: A feature breaking down the analogy betweenaddition of numbers and composition of isometries is the fact thatin the former the order of the operands is irrelevant whereas in thelatter it is not. For any numbers x and y we have x + y = y + x.In contrast, for the composition described three slides ago a briefobservation shows that (rotO, refl)(p) 6= (refl rotO,)(p).

For sets A,B we will denote by A B the set of all ordered pairs(a, b) with a A and b B.

A group is a set G endowed with a function : G G G and adesignated element e satisfying the following properties:

(associative) for all x, y, z G, x (y z) = (x y) z,

(neutral element) for all x G, x e = e x = x, and

(inverse) for all x G there exists x1 G such that x x1 =x1 x = e.

If, in addition, one has that x y = y x for all x, y G then we saythat G is commutative.

The set Isom(E) of plane isometries endowed with the composition is a group. It is not commutative.

We say that H is a subgroup of G when H G and H is a groupwhen endowed with . This means that h g H for all h, g Hand that the three group properties are satisfied.

Examples of subgroups of Isom(E), are the group Transl(E) of trans-lations on the plane and, for each point O, the group Rot(E, O) ofrotations with centre O.

Given a figure S on the plane we denote by Sym(S) the set of isome-tries leaving S invariant.

Proposition 1 Sym(S) is a subgroup of Isom(E).

We call Sym(S) the group of symmetries of S. In the barest case, it isthe trivial group consisting on the identity Id only; no other isometryleaves S invariant.

Otherwise, we say that S has (non-trivial) symmetries.

For instance, both patterns below have a group with four elements.

Figure 3.K Design with birds from India (left) and floor plan by F.L. Wright (right).

For the design on the left, these are the reflections reflv and reflhwith vertical and horizontal axes respectively, the rotation rot180, andthe identity Id. For the one in the right, they are the rotationsrot90, rot180, rot270 and the identity. The group structure in eachcase that is, the way these isometries relate to each other undercomposition can be succintly expressed by the following tables

Id reflv reflh rot180Id Id reflv reflh rot180

reflv reflv Id rot180 reflhreflh reflh rot180 Id reflvrot180 rot180 reflh reflv Id

Id rot90 rot180 rot270Id Id rot90 rot180 rot270

rot90 rot90 rot180 rot270 Id

rot180 rot180 rot270 Id rot90rot270 rot270 Id rot90 rot180

These two tables show different structures for the isometries of thetwo patterns above. This is apparent in the fact that every isometry inthe group for the pattern in the left gives the identity when composedwith itself (that is, refl2v = refl

2h = rot

2180 = Id) but such a property

fails to occur for the other group with rot90 and rot270.

Note: two figures may have the same group structure and yet havedifferent kinds of symmetry. The simplest example is given by thegroundplan in Figure 3.B and the Ying-yang symbol. Both have asymmetry group with two elements and a table

Id Id Id

Id

but in the first case is a reflection (yielding bilateral symmetry) andin the second a rotation of 180 (yielding central symmetry).

A number of isometries (all reflections and some rotations) have theproperty that, for some positive integer n, n = Id. For an elementx in a group G, we call the order of x the smallest positive integern such that xn = e. In case no such integer exists we say that xhas order infinity. Thus, for instance, a reflection has order 2, a 45

rotation has order 8, and a (non-trivial) translation has order infinity.

Rosettes and Whirls

The class of all possible groups is too large to give an idea of itsmany possible structures. Even the class of all finite groups is so.Finite groups of the form Sym(S) for some figure S on the plane are,nonetheless, amenable to such a description. As a first step towardsthis description consider the following three-legged stars.

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rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr

rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr

rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr

rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr

1

2 3 rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr

rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr

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Both are invariant under two-nontrivial rotations, with angles 120

and 240. The one on the left, in addition, is also invariant underthree different reflections (with axes 1, 2 and 3 in the figure).

These are all the isometries leaving these objects invariant. In addition,they behave under composition as displayed in the following tables

Id rot120 rot240 refl1 refl2 refl3Id Id rot120 rot240 refl1 refl2 refl3

rot120 rot120 rot240 Id refl3 refl1 refl2rot240 rot240 Id rot120 refl2 refl3 refl1refl1 refl1 refl2 refl3 Id rot120 rot240refl2 refl2 refl3 refl1 rot240 Id rot120refl3 refl3 refl1 refl2 rot120 rot240 Id

and

Id rot120 rot240Id Id rot120 rot240

rot120 rot120 rot240 Id

rot240 rot240 Id rot120

That is, the symmetry groups of the two stars are given by thesetables. Groups with either of these tables are usually denoted by D3and C3, respectively.

For each integer n > 3 we can consider regular stars (both plain-legged and crooked-legged) with n legs and we obtain as their groupof symmetries the group Dn (with n rotations and n reflections) andCn (with n rotations only). The groups D2 and C2 are already knownto us, the first being the group of symmetries for the Indian pattern inFigure 3.K and the second that for the Yin-yang symbol. Finally, wecan extend these definitions by taking C1 = {Id} (the trivial group)and D1 = {Id, refl} (the group of an object having only one bilateralsymmetry).

We call Cn the cyclic group of order n and Dn the dyhedral groupof order n. Figures S such that Sym(S) is Cn for some n 2 areoften called whirls (or vortices If, instead, Sym(S) is a Dn with n 2then we use the term rosette.

The collection of all groups Dn and Cn exhausts all possible finitesymmetry groups.

Theorem 2 Let S be a plane figure such that Sym(S) is finite.Then there exists n such that either Sym(S) = Cn or Sym(S) = Dn.

Let S be a rosette or whirl with group Sym(S) of symmetries Dn orCn, respectively. We call motif any subset M of S such that any twoimages in M , rot 360

n(M), rot 2360

n(M), . . . , rot (n1)360

n

(M) have no

common points (except, if it is in S, the center of rotation), and theunion of all of them yields S.

For instance, the two figures below are motifs for the crooked-leggedstar at the beginning of this section and for the whirl at the right ofFigure 3.D, respectively.

We next look at figure with infinite symmetry group.

Friezes

We mentioned that figures exhibiting translational symmetry are calledfriezes. We now want to be more precise. For, on the one hand, wewant to exclude wallpapers such as the one in Figure 3.J and, on theother hand, we also want to exclude objects such as the followingbar-code:

In both cases we have too many translations leaving the figure in-variant. In a wallpaper, this is because the object is invariant undertranslations with non-parallel vectors. And in the bar-code, becausethis invariance occurs for all translations with horizontal vector, nomatter the length of the vector. In the former the motif spreads inall directions, in the latter, being thin in the extreme, it fails to beperceived.

We will call frieze (or frieze pattern) a subset S of the plane possessingtranslational invariance for a vector v and such that for any vectorw satisfying transw (S) = S we have

w = kv for some k Z. Wecall v the basic vector of the frieze.

We remark that v could also be taken as basic vector. We remarkas well that the definition above amounts to saying that the group oftranslations leaving S invariant is transv .

Take any line perpendicular to the basic vector v and let =transv (). The part of S lying between and

is called the unit cellof the frieze. For instance, for the Maori pattern:

Remark: Different choices of position for the original line yielddifferent unit cells but these are all equivalent in that they producethe same frieze. For instance, the following two are possible unit cellsfor the Maori pattern.

Although not required in the definition, we usually deal with friezeswhose unit cell is bounded. That is, it can be included in a finite rect-angle. Unbounded unit cells, while logically possible, are not feasiblein practice.

Other possible symmetries of friezes besides the translational arisefrom possible symmetries of the unit cell. We will see, they give riseto seven possible frieze groups. To symplify the exposition of thesegroups we will call horizontal the direction of the basic vector vand, consequently, we will call vertical the direction perpendicular tothat of v .

p111 We create an instance of this simplest case by repeating a p.

We obtain the following:

p p p p p p p p p p p p p p p p p p p p

All translations of the form transkv

(with k Z) leave this friezeinvariant. There is no other apparent symmetry.

To obtain additional symmetries one needs to impose them on theunit cell.

p1m1 Replacing p bypbone obtains

pb

pb

pb

pb

pb

pb

pb

pb

pb

pb

pb

pb

pb

pb

pb

pb

pb

pb

pb

pb

a pattern with a visible bilateral symmetry with respect to a horizontalaxis (sometimes called the backbone at the middle of the unit cell. Abonus symmetry in these friezes is the glidal one. Indeed, if reflmdenotes the reflection around the backbone then

reflm transv = transv reflm = glidev

and S is invariant under glidev since it is so under both reflm andtransv .

The elements of Sym(S) generated by the translations and reflm aretherefore of the form transkv or glidekv , for some k Z, togetherwith reflm.

Group structure is given by the following equalities, for k, q Z,

transkv transqv = trans(k+q)vreflm transkv = glidekvtranskv reflm = glidekvreflm glidekv = transkvglidekv reflm = transkv

glidekv transqv = glide(k+q)vtranskv glideqv = glide(k+q)vglidekv glideqv = trans(k+q)v .

pm11 Friezes of type pm11 are those obtained with a bilaterally

symmetrical unit cell with an axis perpendicular to the basic vector.For instance, the cell p q which yields

p q p q p q p q p q p q p q p q p q p q

As in the p1m1 case we get an extra symmetry but this time of thesame type namely, bilateral invariance with respect to a vertical axis.Consider a line r1 (such that reflr1(p) = q) and r2 (parallel to r1 andsuch that the length of v is twice the distance between r1 and r2):

q

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p

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pr1 r2

Then,reflr2 = transv reflr1

and it follows that reflr2 leaves S invariant.

In a similar manner we obtain bilateral invariance with respect to alllines bisecting pairs p q or pairs q p. To describe all these verticalaxes of reflection, fix a reflection axis 0 (e.g, as r1 above) and, for allk Z, let k = transk

2v (0). We then have

transkv transqv = trans(k+q)vreflk transqv = refl(kq)transkv reflq = refl(k+q)

reflk reflq = trans(kq)v .

p1a1 This kind of frieze has glidal symmetry without bilateral in-variance:

p b p b p b p b p b p b p b p b p b p b

Visible elements in its group are of the form transkv or glide(k+ 12)v ,

for k Z, and compose as follows:

transkv transqv = trans(k+q)vglide(k+ 1

2)v transqv = glide(k+q+ 1

2)v

transkv glide(q+ 12)v = glide(k+q+ 1

2)v

glide(k+ 12)v glide(q+ 1

2)v = trans(k+q+1)v .

p112 This type of frieze features central symmetries:

p d p d p d p d p d p d p d p d p d p dAgain, we add central symmetries with centers in between ps andds and we find that additional central symmetries, with centers inbetween ds and ps are generated in Sym(S). Indicating these centersof symmetry with a small circle we have the following situation:

d pp p

As we did for pm11, fix a center O0 of central symmetry and letOk = transk

2v (O0), for k Z. The elements of the form transkv or

rotOk,180, for k Z, are all in Sym(S) and compose as follows:

transkv transqv = trans(k+q)vrotOk,180 transqv = rotOk+q,180

transkv rotOq,180 = rotOqk,180

rotOk,180 rotOq,180 = transOkq ,180.

pma2 It is also possible to have both central and bilateral symme-

tries as in the pattern

p q b d p q b d p q b d p q b d p q b d

We also have glidal symmetry (with vector 12v ) but not bilateral

symmetry around the backbone. The situation around the unit cell isthe following:

p q b d

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....

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..........

....

.. .... .... .... .... ..... .... .... .... .... .

We now describe Sym(S). We fix a center of half-turn O0 and a

vertical reflection axis 0. We take 0 at a distancelength(v )

4of O0 and

to its right. Then, isometries of the form transkv , reflk , rotOk,180,or glide(k+ 1

2)v , for k Z, are all in Sym(S). We have the following

(known) equalities:

transkv transqv = trans(k+q)vreflk transqv = refl(kq)transkv reflq = refl(k+q)

reflk reflq = trans(kq)vrotOk,180 transqv = rotOk+q,180

transkv rotOq,180 = rotOqk,180

rotOk,180 rotOq,180 = transOkq ,180

glide(k+ 12)v transqv = glide(k+q+ 1

2)v

transkv glide(q+ 12)v = glide(k+q+ 1

2)v

glide(k+ 12)v glide(q+ 1

2)v = trans(k+q+1)v

plus the following new ones:

rotOk,180 reflq = glide(kq+ 12 )v

reflk rotOq ,180 = glide(kq 12 )v

glide(k+ 12)v reflq = rotOk+q ,180

reflk glide(q+ 12 )v = rotOkq ,180

rotOk,180 glide(q+ 12 )v = reflk+q

glide(k+ 12)v rotOq ,180 = reflk+q .

pmm2 The last type of frieze is the richest in terms of symmetries.

It has central, bilateral (both around the backbone and around verticalaxes) and glidal symmetries:

pb

qd

pb

qd

pb

qd

pb

qd

pb

qd

pb

qd

pb

qd

pb

qd

pb

qd

pb

qd

pb

qd

pb

qd

Using the conventions set above we can represent the symmetriesabout the unit cell as follows

pb

qd

pb

qd

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....

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....

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....

.... .... .... ..... .... .... .... .... ...

Using previous notations we see that isometries of the form transkv ,reflk , rotOk,180, or glidekv , for k Z, as well as reflm, are all inSym(S). We have already described how some of these isometriescompose with each other. The compositions we havent describedobey (agreeing that 0 passes through O0) the following identities

reflk glideqv = rotOkq ,180

glidekv reflq = rotOk+q,180

rotOk,180 glideqv = reflkqglidekv rotOq,180 = reflk+q .

The seven types of frieze described above occur in various forms of artacross time and cultures. An extensive collection of these occurrencescan be found in The Grammar of Ornament published in 1856 by OwenJones.

Our description of the seven types of frieze leaves two open issues.

We gave, for each type, a subgroup of Sym(S) but the fact thatthis subgroup is the whole of Sym(S) (i.e., that there are no isometriesleaving S invariant and not listed by us) was not proved. Admittedly,these additional elements in Sym(S) are nowhere to be seen, but aproof of their non-existence remains a more solid argument.

We did not prove that an eighth type of frieze does not exist.

Theorem 3 Any frieze is of one of the following seven types: p111,p1m1, pm11, p1a1, p112, pma2, and pmm2. The elements ofSym(S), in each case, are exactly those in our description of thesegroups.

Theorem 3 shows that there are only seven kinds of frieze. Its proofalso suggests a way to classify any given one by checking the exis-tence of at most three symmetries (translations aside). The methodis succintly depicted by the following flowchart.

p111

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yes no

yes noyes no

yes no yes no yes no

Is there a vertical

reflection?

Is there a horizontal

reflection?

Is there a horizontal

reflection or glide?

Is there a

half-turn?

Is there ahorizontalreflection?

Is there a

half-turn?pmm2

pma2 pm11 p1m1 p1a1 p112

As an example, we see that when applied to the Egyptian friezes inFigure 3.G we obtain the type pm11 for these friezes lead by theanswers yes, no, no.

Similarly, for the pavement in Figure 3.H we get the answers no, no,yes and therefore the type p112.

Finally, for the Maori pattern in Figure 3.I the answers are no, yes,yes and hence its type is p1m1.

Wallpapers

We have classified all the possible finite groups of symmetries (thedihedral groups Dn and the cyclic groups Cn) as well as all possiblegroups of friezes. We finally turn our attention to wallpapers.

We call wallpaper (or wallpaper pattern) a subset S of the plane pos-sessing translational invariance for two non-parallel vectors v1 and

v2and such that for any vector w satisfying transw (S) = S there existtwo integers n1, n2 such that

w = n1v1 + n2

v2 .

Note: any vector of the form n1v1 + n2

v2 leaves S invariant sincev1 and

v2 do so. The contents of the definition is that these are allvectors leaving S invariant.

The next figure shows a vector w obtained as w = 3v1 + 4v2 .

w

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v1

v2

3v1

4v2

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As with friezes, take any point P on the plane and consider the quad-rangle with vertices P , transv1(P ), transv2(P ), and transv1+v2(P ).

P........................

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transv1(P )

v1

transv2(P )

v2

transv1+v2(P )

We call this quadrangle the unit cell of the wallpaper.

Remark: We already observed that the unit cell in a frieze is notuniquely determined. The same occurs with a wallpaper since neitherthe basic vectors v1 and

v2 are uniquely determined. For instance,one may have v1 and

v2 perpendicular and of the same length whichyields a square unit cell. But the pair w1 =

v1 andw2 =

v1 +v2 is

also a pair of basic vectors and yields a unit cell which is not square.

w2

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. . . . . . . . . . . . . ...............

. . . . . . . . . . . . . ....................

v1w1

v2....................

For all possible choices of basic vectors, by construction, the unitcell is a parallelogram. But, as we shall see, the presence of othersymmetries, besides the translational, may allow for choices of basicvectors which make the unit cell to be a rhombus or a rectangle, oreven a square. In such cases, we will take these choices.

A property of the unit cell is that the whole wallpaper S can be ob-tained by taking the union (that is, the putting together) of copies ofthe unit cell. These copies are the images of the unit cell under allthe elements in Transl(S). In addition, any two such copies do notoverlap.

motif a subset of S satisfying these properties (i.e., it spans thewallpaper with translated copies which do not overlap) which is notnecesarilly a parallelogram. The subset

has this property for M.C. Eschers wallpaper Eight heads.

Just as the only possible rotations leaving a frieze invariant are theidentity and half-turns, rotations leaving a wallpaper invariant musthave an angle equal to 60, 90, 120, or 180 (or, obviously, a multipleof these angles). This is known as the crystallographic restriction.

It is using this property (along with a number of other restrictions onthe possible isometries leaving a wallpaper invariant) that a classifi-cation theorem is proved showing the existence of exactly seventeendifferent wallpaper groups.

p1 This is the simplest wallpaper group. It consists only of trans-

lations. There are neither reflections, glides, nor rotations. The twobasic vectors may be inclined at any angle to each other.

pg This group contains glides. The axes of these glides are parallel

to one basic vector and perpendicular to the other (these axes appearas dotted lines in the picture below). There are neither rotations norreflections. The red rectangles on the right show (a choice of) unitcells for this pattern.

pgg This group contains no reflections, but it has half-turns and

two families of glides. The axes for the glides in these families areperpendicular, and the rotation centers which in the figures thatfollow we represent with ovals 0, 0, 0, 0 do not lie on these axes.The shaded rectangle with red boundary displays a unit cell.

pm This group contains reflections. The axes of these reflections

are parallel (and, actually, parallel to one basic vector and perpendic-ular to the other). There are neither rotations nor non-induced glides.

pmg This group contains reflections in only one direction and glides

with axes perpendicular to those of the reflections. It has half-turnswith centers on the glide axes, halfway between the reflection axes.

pmm This symmetry group contains reflections in two perpendicu-

lar directions and half-turns centered where the axes of these reflec-tions intersect. The unit cell is rectangular.

p2 This group differs from p1 only in that it contains half-turns.

There are neither reflections nor glides. The two basic vectors maybe inclined at any angle to each other. Its unit cell is therefore aparallelogram.

p3 This is the simplest group that contains a 120-rotation. It has

no reflections or glides. Centers of 120 rotations are indicated witha triangle H. The unit cell on the upper part of the figure in the rightis a rhombus made up with two equilateral triangles.

One can choose, however, for patterns in this group a hexagonal unitcell as shown in the lower part of that figure1.

1Strictly speaking, a hexagon is not a unit cell in the sense defined above since it is not a

quadrangle. Yet, we may equally obtain the whole wallpaper by glueing translational copies

of this hexagon and, for most p3 patterns, it displays the motif better than a quadrangular

unit cell.

p31m This group contains reflections (whose axes are inclined at

60 to one another) and rotations of order 3 (i.e., of 120). Some ofthe centers of rotation lie on the reflection axes, and some do not.There are some glides. Patterns in this group also admit hexagonalunit cells.

p3m1 This group is similar to p31m in that it contains reflections

and order-3 rotations. The axes of the reflections are again inclinedat 60 to one another, but for this group all of the centers of rotationlie on the reflection axes. There are some glides and the unit cell canbe taken hexagonal.

p4 This group has 90 rotations, that is, rotations of order 4 (whose

centers we indicate with a lozenge ). It also has half-turns. Thecenters of the latter are midway between the centers of the order-4rotations. There are no reflections. The unit cell can be chosen to bea square.

p4g Like p4, this group contains rotations of orders 2 and 4. It also

contains reflections. There are two perpendicular reflections passingthrough the centers of the half-turns. In contrast, the centers of theorder-4 rotations do not lie on any reflection axis. There are glides infour different directions. The unit cell can be chosen to be a square.

p4m This group also has both order-2 and order-4 rotations. In

addition, it has reflections whose axes are inclined to each other by45 so that four such axes pass through each order-4 rotation center.Every rotation center lies on some reflection axis. There are also twoglides passing through the center of each half-turn, with axes at 45

to the reflection axes. It has square unit cells.

p6 This group contains 60 rotations, that is, rotations of order

6. It also contains rotations of orders 2 and 3, but no reflections orglides. Centers of 60 rotations are indicated with a hexagon NHNHNH. Ithas hexagonal unit cells.

p6m This group has rotations of order 2, 3, and 6 as well as re-

flections. The axes of reflection meet at the centers of rotation. Atthe centers of the order-6 rotations, six reflection axes meet and areinclined at 30 to one another. There are also some glides and theunit cell can be chosen to be hexagonal.

cmm This group has perpendicular reflection axes, as does grouppmm, but it also has rotations of order 2. The centers of the rotationsdo not lie on the reflection axes. Unit cells, as for patterns of typecm, can be chosen to be rhombic.

cm This group contains reflections and glides with parallel axes.There are no rotations. The basic vectors may be inclined at anyangle to each other, but the axes of the reflections bisect the angleformed by the basic vectors.

There is at least one glide whose axis is not a reflection axis; itis halfway between two adjacent parallel reflection axes. This groupdisplays symmetrically staggered rows (i.e. there is a shift per rowof half the translation distance inside the rows) of identical objects,which have a symmetry axis perpendicular to the rows. Unit cells canbe chosen to be rhombic.

A brief sample

As in the case of friezes, the seventeen types of wallpaper describedabove occur in various forms of art across time and cultures. We nextgive a (necessarily brief) sample, taken mostly from The Grammar ofOrnament.

Figure 3.L.1 Mediaeval wall diapering (p1); Ceiling of Egyptian tomb (p2);

Egyptian tomb at Thebes (pm).

Figure 3.L.2 Mat on which Egyptian king stood (pg); Bronze vessel in Nimroud,

Assyria (cm); Mummy case (presently at the Louvre Museum) (pmm).

Figure 3.L.3 Cloth from Sandwich Islands (pmg); Bronze vessel in Nimroud,

Assyria (pgg); Egyptian (cmm).

Figure 3.M.1 Renaissance earthenware (p4); Saint-tienne Cathedral, Bourges,

France (p4m); Painted porcelain, China (p4g).

Figure 3.M.2 Wall tiling in the Alhambra, Spain (p3); Persian glazed tile (p31m);

Persian ornament (p3m1).

Figure 3.M.3 Persian ornament (p6); Persian glazed tile (p6m).

Flowcharts and Tables

Flowcharts allow for the classification of a given wallpaper, in thesame spirit as the flowchart for classifying friezes. The next figuregives one such possible flowchart classifying a wallpaper with at mostfour questions.

One may also summarize the different basic features of the seventeengroups in tabular form.

Rotation order denotes the highest possible order of a rotation in thegroup. [That is, the highest possible number n for which there is a rotation in the group such that n = Id but m 6= Id for all 1 m n 1.]

Glides are non-induced in the sense that their component reflectionand translation are not themselves elements of the group.

We use the word hexagonal to denote the possibility of chosing ahexagonal unit cell. [Recall, this happens when a possible choice for unit cellis a rhombus obtained by glueing together two equilateral triangles.]

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Type Unit cell Rotation Order Reflections Glides Other featuresp1 parallelogram 1 No Nop2 parallelogram 2 No Nopm rectangle 1 Yes Nopg rectangle 1 No Yescm rhombus 1 Yes Yes

pmm rectangle 2 Yes Nopmg rectangle 2 Yes Yes parallel reflection axespgg rectangle 2 No Yescmm rhombus 2 Yes Yes perpendicular

reflection axesp4 square 4 No No

p4m square 4 Yes Yes 4-fold centers onreflection axes

p4g square 4 Yes Yes 4-fold centers not onreflection axes

p3 hexagon 3 No Nop3m1 hexagon 3 Yes Yes all 3-fold centers on

reflection axesp31m hexagon 3 Yes Yes not all 3-fold centers

on reflection axesp6 hexagon 6 No No

p6m hexagon 6 Yes Yes

Symmetry and Repetition

There is an obvious feeling of repetition in figures having non-trivialsymmetries. Translational symmetry amounts to repeating the unitcell once and again, the whirl

repeats three times the figure

and the birds in

are all copies of the same bird.

A subset R S is said to be a fundamental region of S when

(i) the whole of S can be obtained by taking the union of copies ofR under different elements in Sym(S); that is

S =

Sym(S)

(R)

(ii) no proper subset R of R satisfies condition (i) above.

For example, the following subset

is a fundamental region for the Maori pattern

Figures S with a finite group of symmetries are obtained by glueingexactly n copies of the fundamental region (when Sym(S) = Cn) or2n such copies (when Sym(S) = Dn).

For friezes or wallpapers it is the unit cell which is obtained by glueinga finite number of copies of the fundamental region. The whole figurecan then be obtained by putting together copies of the unit cell.

There is repetition in symmetry. There is also order. The followingarrangement of rectangles exhibits six copies of a rectangle.

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But there is no symmetry in it, no law dictating how to obtain fromone of them the remaining five.

The Catalogue-makers

Madamina, il catalogo questo

Delle belle che am il padron mio;

Un catalogo egli che ho fattio;

Osservate, leggete con me.

W.A. Mozart Don Giovanni

[My lady, this is a list of the beauties that my master has loved; a list which I

have compiled. Observe, read along with me.]

A symmetric pattern is determined by two constituents: its symmetrygroup and its fundamental region. Once both are given, the entirepattern can be generated. A number of differences between these twoingredients stand out.

The first relates to meaning. Essentially, whichever meaning thepattern might carry will reside on its fundamental region; symmetrygroups are not meaning bearers.

The second relates to possibilities. Whereas there are no limits forthe possible fundamental regions of a pattern, its symmetry groups arerestricted to a few choices: the groups Cn and Dn, both for n 1,the seven groups of friezes, and the seventeen groups of wallpapers.

This catalogue of symmetry groups sets a frame within which thecreation of patterns will have to take place. In the words of a distin-guished art theorist (E. Gombrich), there is a solid core of fact whichrestricts the possibilities open to any pattern-maker I am referingto the laws of geometry.

I am not saying that the pattern-maker needs to be aware of this frame.The opposite is doubtless true and men have produced patterns withall symmetry structures (at least for moderate values of n in the caseof Cn and Dn) from sheer ingenuity.

The role of mathematics in this context reduces to proving that thesesymmetry structures are indeed the only possible and, maybe, to of-fer a list with a description of them. Mathematicians thus becomecatalogue-makers and in their dialogue with the artist they may say,as Leporello while unrolling Don Giovannis deeds to Doa Elvira, thisis a list [. . . ] a list which I have compiled. Observe, read along withme.