Computers and Structures 94–95 (2012) 34–44

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Computers and Structures

journal homepage: www.elsevier .com/locate/compstruc

Symmetry recognition in group-theoretic computational schemesfor complex structural systems

Alphose Zingoni ⇑Department of Civil Engineering, University of Cape Town, Rondebosch 7701, Cape Town, South Africa

a r t i c l e i n f o

Article history:Received 6 October 2011Accepted 14 December 2011Available online 16 January 2012

Keywords:SymmetrySymmetric structureSymmetry operationSymmetry groupGroup theorySymmetry recognition

0045-7949/$ - see front matter � 2012 Elsevier Ltd. Adoi:10.1016/j.compstruc.2011.12.004

⇑ Tel.: +27 21 650 2601; fax: +27 21 650 5864.E-mail address: alphose.zingoni@uct.ac.za

a b s t r a c t

Group theory provides a formal means for exploiting symmetry in the analysis of physical systems. Incurrent group-theoretic formulations, it is assumed that the symmetry properties of the system areself-evident, and the symmetry group of the problem is deduced by the analyst and assigned as an inputparameter. However, for complex systems with a large number of nodes or elements, the symmetry prop-erties may not be obvious. The present paper proposes a procedure for the systematic search and iden-tification of the symmetries of 2D and 3D structural configurations, and hence for the automaticrecognition of the symmetry group to be used in a group-theoretic analysis of the system.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The exploitation of symmetry in structural analysis is a well-established concept dating back many years [1]. A system is saidto exhibit symmetry if it can be turned into one or more new con-figurations physically indistinguishable from the initial configura-tion, by the application of one or more symmetry operations suchas reflections, rotations or combinations of these.

Group theory (and associated representation theory) has pro-vided the mathematical tool for the study of physical systems pos-sessing symmetry. Applications to problems in physics andchemistry are well-known [2–5]. Within the discipline of engineer-ing mechanics, specifically the mechanics of solids and structures,group theory has been applied to simplify problems of the vibrationof a variety of structures [6–13], the bifurcation of rods, thin shellsand skeletal space structures [14–17], the stability analysis of skel-etal structures [18], the statics and kinematics of trusses and frames[7,19–23], and the numerical computation of finite-elementmatrices [24–27].

The common feature of all group-theoretic methods is thedecomposition of the vector space of the problem into a numberof smaller subspaces spanned by symmetry-adapted variables asbasis vectors. If the formulation involves a stiffness matrix, this isrendered into block-diagonal form by the procedure, each blockbeing associated with a particular subspace of the problem. The

ll rights reserved.

subspace problems (each of dimension a fraction of that of the ori-ginal problem) are solved-for independently of each other, thusachieving considerable simplification of the problem and overallreductions in computational effort. Not only does the group-theoretic approach give computational benefits; it also enablesvaluable insights to be gained into the structural mechanics of aproblem [10,11,23,28,29]. Applications of group theory to prob-lems in solid and structural mechanics are growing all the time,as may be seen in a recent review of the subject [30]. The essentialattribute in all these problems is symmetry. It is the prerequisitefor the application of group theory. It should be pointed out thatapart from group-theoretic methods, other approaches have alsobeen developed for exploiting symmetry in structures, such asthose based on concepts of graphs [31–33], or combinations ofgraphs and group theory [34,35].

In all group-theoretic procedures developed to date, it is as-sumed that the symmetry properties of the system are self-evi-dent, and that there is instant recognition of the symmetry groupof the problem by the analyst. Thus the symmetry must first berecognized by a human as a condition for the use of group-theo-retic simplifications. However, for complex systems with a largenumber of structural members, nodes or joints, or any other phys-ical system with a large number of points arranged in a seeminglyrandom manner, the symmetry properties may not at all be obvi-ous to the human eye. Even if one succeeds in identifying somesymmetry properties of the system, these might not be the onlyones, and others will possibly remain hidden. It is the view ofthe author that a fully automated group-theoretic procedureshould have the capability of identifying the symmetry of the

A. Zingoni / Computers and Structures 94–95 (2012) 34–44 35

problem on the basis of the given geometric data alone (withouthuman intervention), yet to date no comprehensive procedurefor the automatic recognition of symmetry in problems of solidand structural mechanics has been documented. This paper is amore complete account of the work that was presented at a recentconference [36], and contains a substantial amount of additionalnew findings.

2. A brief review of symmetry recognition

The need for systematic procedures for the detection and iden-tification of symmetries of images and patterns has been longrecognised within the fields of electronic and manufacturing engi-neering. Configurations in 2-dimensional space have received themost attention. Highnam [37] considered a planar point set, andproposed an algorithm for locating all axes of mirror symmetry,by reducing the 2-dimensional symmetry problem to linear pat-tern matching. Similarly, Atallah [38] presented an algorithm foridentifying axes of symmetry of a planar figure comprising seg-ments, circles and points, while Marola [39] proposed an algorithmfor finding axes of symmetry of planar images, based on the iden-tification of the centroids of the image and other related sets ofpoints.

An interesting method for the detection of symmetry in planarpolygons, based on signal analysis of the distribution of peaks inthe transformed space, has also been advanced [40]. The idea isthat if an object has rotational symmetry, its signal space will haveperiodic peaks, and by analysing the distribution of these, one candeduce the order of rotation symmetry of the object. In this ap-proach, a predetermination of the centroid of the object is notrequired.

In proposing an algorithm for classifying cyclic and dihedralsymmetries in finite 2-dimensional patterns, Lucchese [41]exploited the fact that the Fourier transforms of symmetric pat-terns bear a characteristic relationship to the patterns themselves.Cyclic patterns possess rotation symmetries only, whereas dihe-dral patterns possess both rotation and reflection symmetries.The procedure put forward by Lucchese involved identifying npairs of orthogonal lines in the plots of the zero loci of the norma-lised Fourier transform, and distinguishing between cyclic anddihedral symmetry from the properties of the histogram describ-ing the distribution of points of the pattern with respect to therotation angle.

If a pattern of points can be represented by a plane algebraiccurve, or if the object itself is a plane algebraic curve, the symmetryof the curve can be deduced from the algebraic structure of thecoefficients of the polynomial describing the curve. Lebmeir andRichter-Gebert [42] adopted a complex representation of the poly-nomial for such a curve, enabling the detection of the symmetryproperties of the curve.

Another area in which recognition of symmetry has been of vi-tal importance is design and assembly of machine components.With computer-aided design in mind, Tate and Jared [43] pro-posed a method for symmetry detection based on the identifica-tion of matching loops. They defined a loop as a closed path alongthe edges marking a boundary of a face, a particularly useful con-cept for solids with planar faces (e.g. the platonic solids). Theareas enclosed by all such loops are calculated; for each loop,the centroid of the enclosed area, and the normal to the loopplane at the centroid, are determined. For each group of matchingloops, axes and planes of symmetry are constructed. A method forthe identification of the primary symmetry axes was alsoindicated.

The problem of 3-dimensional polyhedral objects whose 2-dimensional projections have skewed rotational symmetry was

tackled by Zou and Lee [44], who presented an algorithm fordetecting this type of symmetry. Such skewed rotational symmetryin 2 dimensions arises from viewing a 3-dimensional object (withtrue symmetry) from an arbitrary direction. The basic idea is that ifthe vertices of the 2D drawing have skew rotational symmetry, itmust be possible to fit an ellipse to the vertices. If this is so, the ver-tices on the ellipse can then be mapped onto the circle whose par-allel projection is the ellipse. The order of rotational symmetry ofthat face of the solid is then given by the cyclic periodicity of thevertices on the circle.

Suresh and Sirpotdar [45] proposed an automated procedure forexploiting symmetry in practical computer-aided design. Theirprocedure made use of concepts of group theory, and involvedsymmetry detection, construction of the basic repeating unit (the‘‘symmetry cell’’), and formulation of the symmetry-reduced prob-lem. This paper is one of the very few which have attempted toautomate and integrate the steps of the group-theoretic procedure.

A review of other symmetry-detection strategies may be seenin Ref. [43]. These methods are mostly based on pattern match-ing. We also note that most of the developed algorithms aregeared towards 2-dimensional problems. One of the most gen-eral approaches involves fitting a template of known symmetryonto the object, and exploring all possible locations, orientationsand scales. Ten dimensions are involved: 9 combinations of thethree translations {x,y,z} and the three rotations {hx,hy,hz}, andthe scale s.

In summary, it appears that many of these methods require theuser to specify the type of symmetry to be searched for. Also, thealgorithms are suited for the detection of symmetry based onthe shape of objects, or the positions of boundary points, and can-not detect the symmetries of configurations with interior points(e.g. a solid finite element with not only face, edge and cornernodes, but also interior nodes; a crystal structure with interiorpositions of atoms).

As will be seen, the procedure that will be proposed in thispaper will comprise a systematic search of the symmetries of agroup of points without any assumptions as to the type of sym-metry the configuration might have, and taking into account allpoints of a system without the need for first defining the bound-aries of the system. Before the procedure is described in detail,we will begin with a description of the various symmetry opera-tions and how these combine to form symmetry groups, whichare of fundamental importance in all group-theoretic exploita-tions of symmetry.

3. Symmetry and symmetry groups

For finite objects and most physical systems encountered inengineering applications, symmetry operations are of the followingtypes:

(i) reflections in planes of symmetry, which we will denote byrl, where l is the plane of symmetry,

(ii) rotations about an axis of symmetry, which we will denoteby Cn, if the angle of rotation is 2p/n,

(iii) rotation-reflections, which we will denote by Sn; these rep-resent a rotation through an angle 2p/n, combined with areflection in the plane perpendicular to the axis of rotation,

(iv) an inversion through the centre of symmetry (that is, the onepoint of a finite object which remains unmoved by all sym-metry operations), which we will denote by i; an inversion isa special case of Sn with n = 2.

Now, a set of elements {a,b,c, . . . ,g, . . .} comprises a group G ifthe following axioms are satisfied:

36 A. Zingoni / Computers and Structures 94–95 (2012) 34–44

(i) The product c of any two elements a and b of the group,denoted by c = ab, must be a unique element which alsobelongs to the group.

(ii) Among the elements of G, there must exist an identity ele-ment e which, when multiplied with any element a of thegroup, leaves the element unchanged: ea = ae = a.

(iii) For every element a of G, there must exist another element dalso belonging to the group G, such that ad = da = e; d isreferred to as the inverse of a, and denoted by a�1.

(iv) The order of the multiplication of three or more elements ofG does not affect the result (that is, multiplication is associa-tive): (ab)c = a(bc).

When all elements of G are symmetry operations, then thegroup G is called a symmetry group. Classification of symmetrygroups is usually based on the types of symmetry elements makingthem up. For example, groups denoted by Cn and Cnv all possess asingle n-fold axis of rotational symmetry (giving n rotation ele-ments, one of which is the identity element e), with the Cnv groupspossessing an additional n reflection elements. Groups Cn and Cnvare of order n and 2n respectively, the order of a group being sim-ply the total number of elements comprising it. Groups Cn belongto the family of cyclic groups. Assuming the n-fold axis is vertical,if we combine rotation-reflections and vertical reflections with ele-ments of groups Cn, we obtain new symmetry groups containingrotation-reflections, such as groups Sn (also cyclic) and Cnh (Abeliangroups of order 2n).

Table 1Symmetry-group database. The number in brackets denotes the order of the group.

Group Elements

C1 (1) eCi (2) e, iC2 (2) e, C2

Cs (2) e, rC2v (4) e, C2, rx, ry

C2h (4) e, C2, rh, iC3 (3) e;C3;C

�13

C3v (6) e;C3;C�13 ;r1;r2;r3

C3h (6) e;C3;C�13 ;rh ; S3; S

�13

C4 (4) e;C4;C�14 ;C2

S4 (4) e; S4; S�14 ;C2

C4v (8) e;C4;C�14 ;C2;rx;ry;r1;r2

C4h (8) e;C4;C�14 ;C2;rh; S4; S

�14 ; i

C6 (6) e;C6;C�16 ;C3;C

�13 ;C2

S6 (6) e; S6; S�16 ; S3; S

�13 ; i

C6v (12) e;C6;C�16 ;C3;C

�13 ;C2;ra;rb;rc ;r1;r2;r3

C6h (12) e;C6;C�16 ;C3;C

�13 ;C2;rh; S6; S

�16 ; S3; S

�13 ; i

D2 (4) e;Cx2;C

y2;C

z2

D2h (8) e;Cx2;C

y2;C

z2; i;rxy;rxz;ryz

D2d (8) e; S4; S�14 ;C2;C

a2;C

b2;ra;rb

D3 (6) e;C3;C�13 ;Ca

2;Cb2;C

c2

D3h (12) e;C3;C�13 ; S3; S

�13 ;rh;C

12;C

22;C

32;r1;r2;r3

D3d (12) e;C3;C�13 ; S6; S

�16 ; i;Ca

2;Cb2;C

c2;ra;rb;rc

D4 (8) e;C4;C�14 ;C2;C

a2;C

b2;C

12;C

22

D4h (16) e;C4;C�14 ;C2; S4; S

�14 ; i;rh;C

x2;C

y2;C

12;C

22;rx;ry;r1;r2

D6 (12) e;C6;C�16 ;C3;C

�13 ;C2;C

a2;C

b2;C

c2;C

12;C

22;C

32

D6h (24) e;C6;C�16 ;C3;C

�13 ;C2; S6; S

�16 ; S3; S

�13 ; i;rh ;C

a2;C

b2;C

c2;

C12;C

22;C

32;ra;rb;rc ;r1;r2;r3

O (24) e;8C3;3C2;6C4;6Ci2

Oh (48) e;8C3;6C2;6C4;6S4;8S6; i;3rh;3Ci2;6ri

T (12) e, 8C3, 3C2

Th (24) e, 8C3, 3C2, 8S6, i, 3ri

Td (24) e, 8C3, 3C2, 6S4, 6ri

I (60) e, 24C5, 20C3, 15C2

Ih (120) e, 24C5, 20C3, 15C2, 20S6, 24S10, i, 15ri

In 3-dimensional configurations, there are groups with one n-fold principal axis as well as a system of 2-fold secondary axes atright angles to the principle axis. These are the dihedral groupsDn of order 2n. If we add a horizontal plane of symmetry to these,we obtain groups Dnh with 4n symmetry elements. If we add a ver-tical reflection plane which bisects the angle between two adjacent2-fold axes, we obtain the group Dnd, also with 4n elements.

Regular polyhedra are associated with more than one n-foldaxis of symmetry where n > 2. The tetrahedral, octahedral and ico-sahedral groups, denoted by T, O and I respectively, contain onlypure rotations. The tetrahedral group T has four 3-fold axes andthree 2-fold axes, giving a total of 12 elements (including the iden-tity element e). The octahedral group O has four 3-fold axes, three4-fold axes and six 2-fold axes, giving it a total of 24 elements. Theicosahedral group I has six 5-fold axes, ten 3-fold axes and fifteen2-fold axes, giving 60 elements in total. Adding planes of reflectionto the groups T, O and I gives new groups Td, Th, Oh and Ih which areof higher order.

Table 1 summarises the symmetry elements of the so-calledpoint groups (associated with crystals but also very common instructural configurations), as well as the icosahedral groups (whichare not possible for crystals but perfectly feasible for structuralconfigurations). Reference axes and reflection planes are labelledas x, y, z (or xy, xz, yz) where these can be related to the Cartesiancoordinate directions, or otherwise as 1,2,3, . . . or a,b,c, . . .,depending on whether they pass through corners, mid-sides ormid-faces of configurations. The element rh refers to the reflectionin the horizontal plane, where the principal axis (for Cnh and Dnh

groups) is assumed to be vertical. A comprehensive table of thistype serves as a database from which the applicable symmetrygroup can be picked, once all the symmetry elements associatedwith a structural or other physical configuration have been identi-fied. The range of possible engineering configurations is wider thanthe symmetries encountered in crystallography, but all we need todo is to expand the listing of symmetry groups to include all pos-sible configurations. A database of symmetry groups is an essentialcomponent of the proposed scheme.

The present paper proposes a procedure for the systematicsearch and identification of the symmetries of a group of pointsin real space, which will be referred to as a ‘‘constellation’’ (byanalogy to a group of stars) for the purposes of this paper. Such agroup of points may represent the nodes of a complex finite ele-ment (in finite element analysis), the connection points of a spacetruss with a large number of members, the atoms making up acomplex molecule, the stars in a cosmic structure held togetherby a gravitational field, etc. The points may occur as just concen-trated masses or neutral positions, or they may be associated withforce and/or displacement vectors. Algorithms for the search andidentification of symmetry will be presented for both 2-dimen-sional (plane) and 3-dimensional (space) configurations.

To give a feel for the end objectives of the proposed search,Fig. 1 shows constellations of points in the xy plane. We assumethe search for symmetry has already been completed for each con-stellation, and the results are the reflection planes marked in eachdiagram. The six constellations are seen to have {1,3,4,5,6,8}reflection planes respectively, and a similar number of rotationoperations about the point at which the axes intersect (exceptfor set (a) which has only one axis). From this, we deducethat the configurations belong to the symmetry groups{C1v,C3v,C4v,C5v,C6v,C8v} respectively.

4. Symmetry-adapted coordinate system

For a symmetric physical system, the centre of symmetry is de-fined as the point at which all axes of symmetry and reflection

A. Zingoni / Computers and Structures 94–95 (2012) 34–44 37

planes intersect. Not all symmetric physical systems will have adefinite centre of symmetry, but the vast majority will. If a constel-lation with a total of m points has symmetry, and a centre of sym-metry exists, we proceed to find this as follows.

We begin by choosing an arbitrary origin and Cartesian-coordi-nate reference system x0y0z0, and then proceed to determine thecoordinates of each point i (i = 1,2, . . . ,m) on the basis of this coor-dinate system. The coordinates f�x; �y;�zg of the centroid of the con-stellation are calculated as follows:

�x ¼ 1m

Xm

i¼1

x0i; �y ¼ 1m

Xm

i¼1

y0i; �z ¼ 1m

Xm

i¼1

z0i ð1Þ

If the constellation has a centre of symmetry, then this must coincidewith the centroid of the system. Since we are searching for the sym-

Fig. 1. Examples of Cnv-symmetric constellations in the xy plan

metries of the constellation (if any), let us assume that a centre ofsymmetry O exists, and is therefore defined by

Oðx0; y0; z0Þ : f�x; �y;�zg ð2Þ

(Should we end up not finding any symmetries, then clearly such acentre of symmetry does not exist.) We then translate (withoutrotation) the coordinate system so that the origin coincides withthe centre of symmetry O. In the shifted reference system, whichwe will now denote by xyz, the location of O therefore becomes

Oðx; y; zÞ : f0;0;0g ð3Þ

The coordinates of all points i (i = 1,2, . . . ,m) of the constellation arethen transformed to the symmetry-adapted coordinate system asfollows:

x ¼ x0 � �x; y ¼ y0 � �y; z ¼ z0 � �z ð4Þ

e: (a) n = 1; (b) n = 3; (c) n = 4; (d) n = 5; (e) n = 6; (f) n = 8.

38 A. Zingoni / Computers and Structures 94–95 (2012) 34–44

5. Searching for symmetry in 2 dimensions

For 2-dimensional problems, all points of the constellation lie ina single plane. For convenience, we assume this plane is the xyplane of the Cartesian coordinate system, but the considerationsthat follow apply to any other plane in the xyz space. Furthermore,and purely for ease of reference, we will take the xy plane as hor-izontal, and the z axis as vertical. With the centre of symmetry Obeing located at the origin {0,0,0} of the xyz coordinate system,the configuration therefore has one axis of symmetry coincidingwith the z axis (unless we do not find any symmetries after thesearch). It therefore remains to find all the symmetries of the con-figuration with respect to this one axis of symmetry.

As a strategy for searching for the reflection planes of the con-figuration in the xy plane, we begin by placing a rotor vector Yalong the positive y direction, and checking if the vertical planecontaining Y (and perpendicular to the xy plane) is a reflectionplane (i.e. plane of symmetry) of the constellation. We then rotateY about point O in the clockwise direction and through p successiveangular increments Dw of equal magnitude p/p, until the total an-gle of rotation attained is p. For example, if 180 increments areadopted (p = 180), Dw will be one degree.

Let Yj denote the rotated position of Y corresponding to a rota-tion of j(Dw), where j is an integer (j = 1,2, . . . ,p). We will refer tothe vertical plane containing Yj as simply the J plane (Fig. 2). Foreach j (j = 1,2, . . . ,p), we want to check if the J plane is a plane ofsymmetry for the constellation. If it is, then every point of the con-stellation must, without exception, reflect in the plane such that itsmirror image coincides with another point of the constellation. (If a

Fig. 2. The J plane (the vertical plane containing Yj).

(a)

Fig. 3. Points {1,2,3,4} reflecting in the J plane: (a) The images f10;20;30;40g do not coincidthe system; (b) The image 10 coincides with point 3, 20 coincides with 4, 30 coincides with

point happens to lie in the J plane itself, it simply reflects ontoitself.)

For a given Yj, we begin by calculating the mirror-image posi-tion of Point 1 of the constellation with respect to the J plane. (Thisis achieved by applying a reflection matrix operation on the coor-dinates of Point 1, in the usual manner.) If this position coincides(within pre-defined tolerances, e.g. within 0.01 of a coordinateunit) with the position of another point of the constellation, thenthe J plane is a reflection plane for at least two points of the con-stellation (Point 1 and its counterpart), so we go to Point 2 and testthis in the same manner. We carry on testing all the points until weencounter a point whose mirror image in the J plane does not coin-cide with any other point of the constellation, nor with itself. Assoon as that becomes the case, it means the J plane is not a reflec-tion plane of the system.

If we find that all m points of the constellation pass the test (i.e.every point has a mirror-image counterpart), then the J plane is in-deed a reflection plane of the constellation. Fig. 3 illustrates twopositions of the J plane for a 4-point constellation. In Fig. 3(a),the J plane is not a plane of symmetry, whereas in Fig. 3(b) it is.

Having tested all m points of the constellation for the position Yj

of the vector Y, we then increment j so that the position Yj becomesYj+1, and repeat the procedure to establish whether or not the new Jplane is a reflection plane. The search procedure is repeated for all j(j = 1,2, . . . ,p), and if the total number of reflection planes found isn, then the constellation belongs to the symmetry group Cnv. Algo-rithm 1 (shown in Fig. 4) summarises the procedure for searchingsymmetry planes for 2-dimensional configurations.

Once the first two reflection planes have been identified, weneed not continue the search on the basis of the very small incre-ments Dw, but can proceed much more rapidly as follows. Let Y1

and Y2 be the first two reflection planes to be identified, and letb be the angle of separation between them (Fig. 5). We take advan-tage of the fact that the axes of symmetry of symmetric configura-tions belonging to the symmetry groups Cnv (n P 2) are allregularly separated by equal angles b = p/n. Thus, having foundY1 and Y2 (the first two reflection planes), the remaining reflectionplanes Yi of the constellation follow by successive clockwise rota-tions of Y2 through multiples of b, as follows:

Yi ¼ Y2 � ði� 2Þb ði ¼ 3;4; . . . ;n for group CnvÞ ð5Þ

Here the notation A � B denotes a clockwise rotation of vector Athrough an angle B. Thus in practice we only need to locate the firsttwo reflection planes, and all the other reflection planes will follow(without a need for searching) from Eq. (5).

With the angular separation b of reflection planes now estab-lished, the basic rotation operation (which turns the configuration

(b)

e with any of the points {1,2,3,4}, therefore the J plane is not a plane of symmetry of1, and 40 coincides with 2, therefore the J plane is a plane of symmetry of the system.

START

Set the required number of increments ,pand hence the angular increment ψΔ

Begin the search for symmetry planes by setting 0=j so that jY

coincides with the y axis, and the rotation ( ) 0=Δψj

For the J plane (rotation ( )ψΔj ), check if every point of the constellation has a mirror-image counterpart in the constellation. Begin by setting 1=i

Apply a reflection operation on the coordinates of Point i ),( ii yx

with respect to the J plane: ii PP ′→

Does iP′ (the reflection of iP ) coincide

with any point )...,1, 2,( mlPl = of the No

constellation, including iP itself ?

Yes The J plane is not a reflect- ion plane of the constellation

Point i has a mirror-image counterpart in the constellation, with respect to the J plane

Check the next point: 1+= ii (increment i )

No Is mi > ?

Yes

All m points of the constellation have mirror-image counterparts with respect to the J plane. Therefore the J plane is a reflection plane of the constellation

Rotate the J plane to the next position: 1+= jj (increment j )

No Is pj = ? Yes END

Fig. 4. Algorithm 1: Searching for symmetry in 2-dimensional space.

A. Zingoni / Computers and Structures 94–95 (2012) 34–44 39

onto itself) follows as Cn, which we define as a clockwise rotation of2b (=2p/n) about the centre of symmetry O. For example, the rota-

Fig. 5. Successive reflection planes separated by angle b.

tion operations (basic and full set) associated with the symmetrygroups C3v and C4v are as follows:

Group C3v: Basic rotation operation: C3 (clockwise rotationthrough 2p/3).Full set of rotation operations: C3; C2

3 ¼ C�13

� �; C3

3 ¼ eð Þ.Group C4v: Basic rotation operation: C4 (clockwise rotationthrough 2p/4).Full set of rotation operations: C4; C2

4 ð¼ C2Þ; C34ð¼ C�1

4 Þ;C4

4 ð¼ eÞ.

The notation C23 denotes two successive C3 operations (which is

the same result as an anticlockwise C3 operation (denoted by C�13 ÞÞ;

C33 denotes three successive C3 operations, which of course has the

same effect as the identity operation e. Rotations of the group C4vare interpreted in a similar way. Note that C2 denotes a rotationof p (the result is the same whether clockwise or anticlockwise).

If the points of the constellation are associated with directedvectors (such vectors representing forces, displacements, velocitiesor accelerations at the points), it is possible that the configurationof these vectors does not have the full symmetry of the points ofthe constellation, and this reduced symmetry may very well dic-tate the symmetry group governing the behaviour of the system.

(a)

40 A. Zingoni / Computers and Structures 94–95 (2012) 34–44

As an example, our algorithm will show that the constellation inFig. 6(a) (which comprises the points 1, 2 and 3) has three symme-try planes {Y1,Y2,Y3}, and hence belongs to the symmetry groupC3v, but its vectors {u1,u2,u3} are permuted not by the reflectionsin {Y1,Y2,Y3}, but by the rotations fC3;C

23;C

33g. Therefore the config-

uration of vectors {u1,u2,u3} belongs to the cyclic symmetry groupC3 (a subgroup of the group C3v). Similarly, with reference to the 4-point constellation of Fig. 6(b), the configuration of the pointsthemselves belongs to the symmetry group C4v, but the configura-tion of the associated vectors {u1,u2,u3,u4} belongs to the symme-try group C4 (a subgroup of the group C4v). Thus the proposedprocedure identifies both Cnv groups (comprising reflection androtation elements) and Cn groups (with rotations only).

(b)

(c)

Fig. 7. Search field for 3-dimensional constellations: (a) spherical surface of radiusR and centred about O, with axis �r passing through point P {/,h} on the surface; (b)�rð/Þ as seen in the zh plane: 0 6 / 6 p/2; (c) h(h) (the projection of �r on the xyplane): 0 6 h < 2p.

6. Searching for symmetry in 3 dimensions

Symmetry groups associated with 3-dimensional configurationsinclude the dihedral groups Dnh of order 4n (i.e. with 4n symmetryelements, for a configuration with n-fold symmetry about the mainaxis of symmetry), the tetrahedral group Th of order 24 (describingthe symmetry of a regular tetrahedron), the octahedral group Oh oforder 48 (describing the symmetry of a cube), the icosahedralgroup Ih of order 120 (describing the symmetry of a regular icosa-hedron), among others.

Whereas for 2-dimensional configurations there is only one axisof symmetry, for 3-dimensional configurations more than one axisof symmetry may exist, and for each axis of symmetry, reflectionplanes and rotation operations may occur. We therefore need tosearch through space for all axes of symmetry, and for each iden-tified axis of symmetry, we need to search the associated (perpen-dicular) plane for reflection and rotation symmetries.

Taking the origin O of the xyz coordinate system at the centroidof the constellation (assumed to be the centre of symmetry), weadopt a spherical surface of radius R and centred about O, as thesearch field for axes of symmetry of the constellation (Fig. 7(a)).We define a spherical coordinate system {/,h} for points P lyingon the spherical surface, where / is the angle measured from thepositive z axis to the axis �r passing through O and P, and h is theangle measured from the positive y direction to the axis h (the pro-jection of �r on the xy plane).

We begin by letting / = 0 so that �r coincides with the z axis. Wecheck if �r (�z axis) is an axis of symmetry, by searching for reflec-tion planes and rotation symmetries as per Algorithm 1 (for planesystems). We then increment / in steps of D/ (ending when / = p/2), and for each value of /, we allow h to vary from zero to 2p inincrements of Dh (refer to Fig. 7(b) and (c) for the ranges of /and h respectively). Each �rð/; hÞ is checked to see if it is an axis

(a) (b)

Fig. 6. Examples of directed vectors at constellation points: (a) {u1,u2,u3} (C3

symmetry); (b) {u1,u2,u3,u4} (C4 symmetry).

of symmetry by searching for reflection planes and rotation sym-metries using Algorithm 1. The search for axes of symmetry there-fore covers the entire envelope of spherical space surroundingpoint O, to any desired level of accuracy or resolution.

The full procedure for searching for axes of symmetry and asso-ciated symmetry operations in 3-dimensional space is summarisedin Fig. 8 (Algorithm 2).

7. Illustrative examples

Consider the 8-point constellation with coordinates fx0; y0; z0g asfollows:

Point 1 : f2;0;2g Point 2 : f0;0;2gPoint 3 : f0;2;2g Point 4 : f2;2;2gPoint 5 : f2;0;0g Point 6 : f0;0;0gPoint 7 : f0;2;0g Point 8 : f2;2;0g

From Eq. (1), we find the position of the centroid of the eightpoints as {1, 1, 1}, which we will assume to be the centre of sym-metry O (Eq. (2)), if such a centre exists. Shifting the origin to thecentre of symmetry, and denoting Point i by Pi, the transformedcoordinates {x, y, z} become:

P1 : f1;�1;1g P2 : f�1;�1;1gP3 : f�1;1;1g P4 : f1;1;1gP5 : f1;�1;�1g P6 : f�1;�1;�1gP7 : f�1;1;�1g P8 : f1;1;�1g

Applying Algorithm 2 yields thirteen symmetry axes (Fig. 9a)and their symmetry operations as follows (all 13 axes pass throughthe centre of symmetry O):

A. Zingoni / Computers and Structures 94–95 (2012) 34–44 41

3 axes coinciding with the coordinate axes x, y, z: symmetry ele-

ments: 4r;C4;C�14 ;C2

n o.

Each axis is associated with four reflection planes (r), a 2p/4clockwise rotation, a 2p/4 anticlockwise rotation, and a 2p/2rotation (Fig. 9(b)).

4 axes passing through the points P1, P2, P3, P4: symmetry ele-

ments: 3r;C3;C�13

n o.

Each axis is associated with three reflection planes (r), a 2p/3clockwise rotation, and a 2p/3 anticlockwise rotation (Fig. 9(c)).

8 axes passing through the points M12, M23, M34, M14, M15, M48

(where Mij is the midpoint of the line joining points Pi and Pj): sym-metry elements: {2r,C2}

Each axis is associated with two reflection planes (r) and a 2p/2rotation (Fig. 9(d)).

Not all the above symmetry operations are distinct. Some havethe same effect as others. The total number of distinct symmetryoperations (rotations and reflections) for the constellation is 48,including the identity element e (which is equivalent to a rotationthrough 2p about any of the symmetry axes). From the database ofTable 1, the symmetry group of the constellation is deduced as the

START

Choose a reference spherical surface of radius R and centred on O

Set 0=φ( r initially coincides with the z axis)

Search for reflection planes and rotation symmetries of the constellation with respect to the z axis,

using the algorithm for 2-D configurations

φφφ Δ+=

Is 2

πφ > ? Yes END

(search completed)

No

Set 0=θ

Search for reflection planes and rotation symmetries of the constellation with respect to the ( )θφ ,r axis,

using the algorithm for 2-D configurations

θθθ Δ+=

No Is πθ 2= ?

Yes

Fig. 8. Algorithm 2: Searching for symmetry in 3-dimensional space.

octahedral group Oh. If the points Pi (i = 1,2, . . . ,8) are plotted in thexyz space, they are seen to lie at the corners of a cube.

In the schemes presented in previous work [9,11,13,19,20,26],the symmetry groups of the configurations were manually as-signed. A procedure of the type outlined in this paper could haveautomatically yielded the symmetry groups of highest order, aswell as all other possible sub-groups. Consider the layered spacegrids depicted in Fig. 10 [11]. Since the symmetry properties ofeach of these configurations (a–d) are obvious from a visual inspec-tion, the formal search procedure of present considerations israther superfluous, but for the purposes of validating the algo-rithms, the constellations of nodes of the triple-layer configura-tions (appearing as the lower of the two elevations shown foreach space grid) were subjected to the 3-D search algorithm. Hav-ing correctly identified the centre of symmetry on the basis of Eqs.(1)–(4), the z axis in these diagrams was taken as the first trial axis,and with respect to this axis, the following sets of distinct symme-try properties were established:

Grid (a): C3; C�13 ; S3; S�1

3 ; C12; C2

2; C32; r1; r2; r3;rh.

Grid (b): C6; C�16 ; C3; C�1

3 ; C2; S6; S�16 ; S3; S�1

3 ; i; Ca2; Cb

2; Cc2;

C12; C2

2; C32; ra; rb; rc; r1; r2; r3; rh.

Grid (c): C2; i; Cx2; Cy

2; rx; ry; rh.Grid (d): C4; C�1

4 ; C2; S4; S�14 ; i; Cx

2; Cy2;C

12; C2

2; rx; ry; r1; r2;

rh.

If we add the identity element to each of these sets, we canreadily recognise the ensuing sets, with the help of Table 1, asthe sets of symmetry elements for the dihedral groups D3h, D6h,D2h and D4h, respectively. In fact, once a ‘‘strong’’ axis of symmetryhad been identified as the z axis, the accelerated procedure ofSection 5 was invoked to identify the symmetry properties aboutthis axis more quickly, bypassing the computational effortnormally required for the more general search.

8. Computational implications

For 2-dimensional configurations, the computational effort in-volved in searching for symmetry is directly related to the size ofthe angular increments Dw chosen. The smaller the size of theseincrements, the more refined will be the search, but the greaterwill be the number p of J planes to be searched. In general, thesearch has to sweep through a total angle of p in order to captureall the existing planes of symmetry (if any) of a constellation. Forgood accuracy, let us assume an increment size of half-a-degree(that is, p/360 radians). The total number of J planes to be searchedis given by (here we assume the worst case where an acceleratedsearch based on the b of Fig. 5 is not possible)

p ¼ pDw

ð6Þ

which for a Dw of p/360, gives p = 360. If m is the total number ofpoints in the constellation, the total number of symmetry opera-tions to be performed on the system is equal to m (one reflectionoperation per point), for each setting of the J plane. Thus the totalnumber M of symmetry operations involved in the entire searchis given by

M ¼ pm ð7Þ

Considering a constellation of 100 points as an example, and takingp as 360, we obtain a value of 36,000 for the parameter M. The com-putational effort implied is clearly not trivial.

For 3-dimensional configurations, the computational effort re-quired is much greater. The search field for axes of symmetry in-

(a)

(b) (c) (d)

Fig. 9. Symmetry axes of the 8-point cubic constellation: (a) isometric view showing all 13 axes; (b) view along x, y or z axis; (c) view towards O and through point P1, P2, P3 orP4; (d) view towards O and through point M12, M23, M34, M14, M15 or M48.

42 A. Zingoni / Computers and Structures 94–95 (2012) 34–44

volves increments of size D/ in the / direction (0 6 / 6 p/2), andincrements of size Dh in the h direction (0 6 h < 2p). (Refer toFig. 7.) Denoting the total number of these increments by n and grespectively, we have

n ¼ ðp=2ÞD/

; g ¼ 2pDh

ð8Þ

The total number of the �r axes (refer to Fig. 7) to be checked forsymmetry, denoted by N, is therefore given by

N ¼ ng ð9Þ

If D/ and Dh are each chosen as 0.5� (that is, p/360 radians) forgood accuracy, the parameter N becomes 129,600.

For each �r axis, we need to perform up to M symmetry opera-tions (as per the algorithm for 2-dimensional configurations), in or-der to find all the symmetries (reflections and rotations) about thataxis. The total number Q of symmetry operations for a 3-dimen-sional constellation having m points is therefore given by

Q ¼ MN ¼ pmng ð10Þ

Considering our example of a constellation of 100 points, but thistime distributed randomly in 3-dimensional space, and adopting0.5� increments for Dw, D/ and Dh (implying that p = 360,n = 180, g = 720), the total number of symmetry operations is ob-tained as

Q ¼ 360� 100� 180� 720 ¼ 4;665;600;000

Clearly searching for symmetry in 3-dimensional constellations re-quires huge computational effort, even for constellations with a rel-atively small number of points.

9. Concluding remarks

A procedure for the search and identification of symmetry insystems of points or nodes distributed in 2-dimensional and 3-dimensional spaces has been proposed. The procedure is intendedfor incorporation into the group-theoretic computational frame-work, where the first step is always the identification of the sym-metry properties (and hence the symmetry group) of a physicalsystem. Automatic recognition of symmetry is a desirable featureof any general group-theoretic program, since for complex systemswith a large number of structural members, nodes or joints, or anyother physical systems with a large number of points arranged in aseemingly random manner, we cannot rely on simple inspectionalone for the identification of all symmetry properties.

We began with the premise that if a physical system has a cen-tre of symmetry, then this must coincide with the centroid of thepoints describing the system. The search for symmetry in oneplane (i.e. in 2 dimensions) then amounted to identifying all rota-tion operations about the axis perpendicular to the plane of thesystem and passing through the centre of symmetry (i.e. the axisof rotational symmetry), and all reflection operations in planes per-pendicular to the plane of the system and containing the axis ofrotational symmetry.

In the case of 3-dimensional systems, the strategy consisted insearching for axes of symmetry over an imaginary spherical surfacecentred about the centre of symmetry of the system. This involvessubjecting each position of the trial axis to a search for symmetryoperations in 2 dimensions (i.e. in the plane perpendicular to theaxis and containing the centre of symmetry), and noting the sym-metries (reflection planes and rotations) about the axis. The consol-

Fig. 10. Examples of layered space grids [11].

A. Zingoni / Computers and Structures 94–95 (2012) 34–44 43

idated set of all axes of symmetry and their associated symmetryelements then allows the symmetry group for the system to beidentified from the database of Table 1. Even for systems with arelatively small number of points such as 100, the computationaleffort involved in a high-resolution 3-dimensional search for sym-metry is substantial. Clearly, the procedure requires high comput-ing power for effective implementation.

Acknowledgements

The author would like to acknowledge the assistance of Mr.Angus Rule of the University of Cape Town, who prepared the illus-trations. The financial assistance of the National Research Founda-tion of South Africa is gratefully acknowledged.

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