symmetry analysis of mechanisms in rotating rings of tetrahedra

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A symmetry analysis of mechanismsin rotating rings of tetrahedra

BY P. W. FOWLER1 AN D S. D. GUEST2

1Department of Chemistry, University of Exeter, Stocker Road,Exeter EX4 4QD, UK

2Department of Engineering, University of Cambridge, Trumpington Street,Cambridge CB2 1PZ, UK

([email protected])

Rotating rings of tetrahedra are well known from recreational mathematics. Rings of Ntetrahedra with Neven are analysed by symmetry-adapted versions of classical countingrules of mechanism analysis. For NR 6, a single state of self-stress is found, together withNK5 symmetry-distinct mechanisms, which include the eponymous rotating mechanism.For NZ4 in a generic configuration, a single mechanism remains together with threestates of self-stress, but, uniquely in this case, the mechanism path passes through abifurcation at which the number of mechanisms and states of self-stress is raised by one.

Keywords: symmetry; mechanism; mobility

1. Introduction

Rotating rings of tetrahedra are well known from recreational mathematics(Rouse Ball 1939; Cundy & Rollett 1981). An example is shown in figure 1. Theserings can be assembled from planar nets or by origami (Mitchell 1997), and, withrecent micro-origami techniques, have been constructed on a millimetre scale asprototypes for micro-fabrication in 3D (Brittain et al. 2001). The rings are oftenassociated with decorations of the plane with various patterns and are alsoknown as kaleidocycles (Schattschneider & Walker 1977; Schattschneider 1988).The underlying mathematical objects are members of a family of cycles of edge-

fused polyhedra having 2N vertices, 5N distinct edges and 4N triangular faceswhere the faces are those of N edge-sharing tetrahedra, and where eachtetrahedron in the cycle is linked to its predecessor and successor at oppositeedges. Certain members of this family display an amusing and confusing(Stalker 1933) motion in which each tetrahedron of the toroidal ring turns, thewhole turning inside out like a smoke-ring. The shared edges act as hingesbetween rigid tetrahedral bodies. Rotating rings of tetrahedra were described in apatent in 1933 (Stalker), but objects of this form occur in the earliermathematical literature (Bruckner 1900). Animations of the motion are availableat a number of sites on the World Wide Web; e.g. Stark (2004).

Attention has centred on the case where the tetrahedra are equilateral, N iseven and N is greater than or equal to six. For NZ8, the motion is continuous,returning repeatedly to the starting configuration. For NZ6, the range of

Proc. R. Soc. A (2005) 461, 18291846

doi:10.1098/rspa.2004.1439

Published online 24 May 2005

Received 19 May 2004Accepted 6 November 2004

1829 q 2005 The Royal Society

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movement is restricted by clashes between faces, but the underlying motion canbe made continuous if the tetrahedra are transformed to an isosceles shape byshrinking the shared edges; the system can be considered as a particular example

of a threefold symmetric Bricard linkage (Chen et al. 2005). The key featurecommon to equilateral and isosceles geometries is that successive shared edgesremain mutually perpendicular. The present paper will concentrate on the casesof rings of N tetrahedra with this perpendicular-hinge geometry and where N iseven.

Our aim is to provide a general analysis of the mechanisms and states of self-stress in this subset of rotating rings of tetrahedra. For this purpose we use therecently developed symmetry-extended versions of the classical tools ofmechanism analysis, the mobility criterion (Guest & Fowler 2005) and theMaxwell counting rule (Fowler & Guest 2000). The mobility criterion treats each

tetrahedron as a rigid object, constrained by hinges along two opposite edges;Maxwell counting considers each tetrahedron to be formed from six sphericallyjointed edge bars, where two opposite bars are shared with neighbouring

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Figure 1. The finite motion of the rotating ring of six tetrahedra, showing one quarter of a completecycle: (a) D3h high symmetry point (the standardconfiguration); (b) generic C3v symmetry; (c) D3dhigh symmetry point; (d) generic C3v symmetry; (e) D3h high symmetry point. The tetrahedra arenumbered, and the shared (hinge) edges between tetrahedra have been marked with a dashed line.Note that, in travelling from configuration (a) to configuration (e), the structure has interchangedhorizontal and vertical sets of shared (hinge) edges.

P. W. Fowler and S. D. Guest1830

Proc. R. Soc. A (2005)



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tetrahedra. Each description implies a relationship between the symmetries ofmechanisms, states of self-stress and structural components; this approach takesfull advantage of the high point-group symmetry of the even-N rings oftetrahedra.

Counting and symmetry analysis for toroidal frameworks has been consideredbefore in the context of toroidal deltahedra (Fowler & Guest 2002). There, it wasshown that fully triangulated toroids have at least six states of self-stress of well-defined symmetry. Like the rotating rings, toroidal deltahedra have all triangularfaces, but unlike the rings, the deltahedra are toroidal polyhedra in that theyenclose a single connected toroidal volume and all their edges are common toexactly two faces. The rotating rings, in which the enclosed tetrahedral volumesare disjoint and some edges are incident on four faces, are not polyhedral in thissense, and a different analysis is required.

2. Preliminary counting analysis

We begin with classical counting analyses, using both the mobility rule and theMaxwell count.

A generalized mobility rule is given in Guest & Fowler (2005) as

mKsZ 6NK1K6gCXg

iZ1

fi; (2.1)

where m is the mobility(Hunt 1978), or number of mechanisms, of a mechanicallinkage consisting of N bodies connected by g joints, where each joint i permitsfi relative freedoms, and s is the number of independent states of self-stress thatthe linkage can sustain. The parameter s can be considered equivalently as thenumber of overconstraints, independent geometric incompatibilities, or misfits,that are possible for the linkage. In the present case, N is the number oftetrahedra, and there are gZN hinge joints each permitting a single relativefreedom; i.e. the revolute freedom between two adjacent tetrahedra. Thus,

mKsZNK6: (2.2)

The generalized Maxwell count for a system of spherically jointed bars is givenby Calladine (1978) as

mKsZ 3jKbK6; (2.3)

where j is the number of spherical joints and b is the number of bars connectingthem. In the present case, jZ2N and bZ5N, so that mKsZNK6 from equation(2.3), in agreement with the mobility criterion (2.2).

As both m and s are non-negative integers, simple counting has thereforeestablished the existence of at least one mechanism for NO6, and conversely, atleast one state of self-stress for N!6. From the counting result, the smallest ring

of tetrahedra for which a mechanism must exist is that with NZ7, and indeed,such a mechanism has been remarked in this case and described as having anentire lack of symmetry (Rouse Ball 1939).

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For the case NZ6, counting alone does not demonstrate the existence of theknown mechanism, but it does establish that if such a mechanism exists, then somust a state of self-stress.

In fact, a separate kinematic argument can be used to demonstrate that, in a

generic configuration, sZ1 for all even NR6. The argument runs as follows.Consider a ring in a standard position where the centres of the hinges define aplanar N-gon, with N/2 of the hinges lying in the plane of the polygon, and thesame number lying perpendicular to it. Cutting the ring along a single joint givesa chain of linked tetrahedra. This cut chain cannot sustain a state of self-stress:equilibrium of a terminal tetrahedron implies that the joint to the nexttetrahedron in the chain is unstressed, and the same argument can be extendedto the next in the chain, and so on for each of the other tetrahedra in turn. Thus,any state of self-stress of the ring is contingent on restoration of the original joint,and is generated by a geometric misfit at the restored joint. Detailed

consideration of the five possible misfits (corresponding to the five independentkinematic constraints imposed by a revolute joint) shows that four can beaccommodated by rotation of the remaining NK1R5 joints. The only type ofmisfit that cannot be so accommodated is twisting, and a misfit of this sort leadsto a single twisting state of self-stress. Hence, sZ1 in the standard position. Asthe same argument can be advanced for nearby configurations, sZ1 in anygeneric configuration.

Given that sZ1, it follows from equation (2.2) that the number of mechanismsfor even NR6 is given by

mZNK5: (2.4)

Hence, the number of mechanisms grows linearly with N, although the countingapproach gives no indication of the nature of these additional mechanisms;further insight into this aspect is given by considering the symmetry of thesystem.

3. Symmetry analysis: NZ6

This section will treat the case NZ6 in explicit detail, as a preliminary to a

general analysis for all even N derived in 4. The symmetry extension of theMaxwell counting rule is used; exactly equivalent results are given by thecorresponding extension of the mobility rule.

The symmetry extension of Maxwells rule (Fowler & Guest 2000) is

GmKGsZGj!GTKGbKGTKGR; (3.1)

where G(m), G(s), G(b) and G(j) are the representations of the m mechanisms, sstates of self-stress, b bars and j joints, and GT and GR are the translational androtational representations (Atkins et al. 1970), all within the point group of theinstantaneous configuration.

We will consider the ring of six tetrahedra in each of the three symmetry-distinctconfigurations that it can assume. In the standard position (figure 1a), the sixtetrahedra are arranged with D3h point symmetry. The centres of the six hinges





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This result can be verified by deleting columns in the tabular calculation above,or applying the descent in symmetry correlation (Atkins et al. 1970):

A01; A002D3h/A1C3v;

A02; A001D3h/A2C3v:

In C3v the mechanism is totally symmetric, and as there is no equisymmetricstate of self-stress, the local linear analysis that we have carried out here issufficient to show that the mechanism for the ring of six tetrahedra is finite(Kangwai & Guest 1999), i.e. even for finite displacements, there is a continuousmechanism path in the configuration space of the ring of tetrahedra.

If the finite mechanism is followed further along the displacement coordinatethe structure passes through a second point of high symmetry, a D3dconfiguration (figure 1c), where alternative hinges lie an equal angle above andbelow the horizontal plane. In this antiprism configuration

D3d GmKGsZA2uKA1u: (3.4)

Continuation of the motion leads through C3v configurations (figure 1d) backto a D3h position (figure 1e), where horizontal and vertical hinges have beenexchanged with respect to the initial setting. The cyclic nature of this finitemechanism is apparent.

Note that although its formulation across the three particular point groupsuses different representation labels, the symmetry of the mobility excess for thering of six tetrahedra is always compatible with the single expression,

GmKGsZGzKG3; (3.5)

where Gz is the symmetry of a translation along the principal axis and G3 is therepresentation of a pseudo-scalar (a quantity whose sign is preserved underproper, and reversed under improper, operations). The mechanism is always ananapole rotation of an underlying torus, and the state of self-stress always followsa counter-rotating pattern on the torus, irrespective of the particular symmetryof the instantaneous configuration. Figure 2 shows how Gzand G3 link to patternsof vectors on the torus.

To summarize, use of the Maxwell counting rule in its symmetry-adapted formhas enabled us to give a complete account of the interesting static and kinematicbehaviour for NZ6. Use of the symmetry-adapted mobility criterion (Guest &Fowler 2005) gives the same results. This case has only one mechanism, thecharacteristic anapole rotation, but, as we have seen from pure counting, moremechanisms emerge for larger N. These too are amenable to a symmetrytreatment, as the following section will demonstrate.

4. Symmetry analysis: the general case

The previous analysis can be generalized to cover all even values ofN. Again, we

concentrate on the Maxwell analysis, although all results reported here could alsobe obtained with the symmetry version of the mobility criterion. We considerthe same sequence of standard D(N/2)h, generic rotated C(N/2)v and alternative





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high-symmetry D(N/2)dantiprism configurations. For these groups we will use thenotation C(f) for the symmetry operation of rotation through f about theprincipal axis, and S(f) for the corresponding improper operation. C(f) andC(Kf) belong to the same class, as do S(f) and S(Kf). It turns out to be

convenient to consider doubly odd cases, NZ4pC2, and doubly even cases, NZ4p, separately. The case NZ4p is further split into NZ4p, pO1 and NZ4, pZ1,as the standard notation for point groups C2v, D2h, D2d (pZ1) differs from thatfor C2pv, D2ph, D2pd (pO1). In addition to this technicality, the case pZ1 alsopresents some special features that justify a separate treatment, given in 5.

(a) Rings with NZ4pC2

In the standard position, the Ntetrahedra are arranged with D(N/2)hZD(2pC1)hpoint symmetry. The centres of the hinges define a planar N-gon; hinges liealternatively within, and perpendicular to, theshmirror plane.The normalthroughthe centre of the hexagon defines the principal axis. A C02 axis is defined by eachhorizontal hinge and its opposite perpendicular partner; the same pair of hingesdefines a svmirror plane. The structure has (2pC1) C

02 axes, and (2pC1) svplanes.

The Maxwell symmetry analysis is

D(2pC1)h E 2C(f) (2pC1)C02 sh 2S(f) (2pC1)sv

G(j) 8pC4 0 2 4pC2 0 4!GT 3 c

CK1 1 cK 1

24pC12 0 K2 4pC2 0 4KGTKGR K6 K2cC 2 0 0 0

24pC6 K2cC 0 4pC2 0 4KG(b) K20pK10 0 K2 K4pK2 0 K2

G(m)KG(s) 4pK4 K2cC K2 0 0 2

where cGZG1C2 cos f. To reduce G(m)KG(s) to a tractable form, we note thata representation GL defined by

GLZG

mKG

sCG

TCG

RKG

zCG

3;

(4.1)would have only integer characters

D(2pC1)h E 2C(f) (2pC1)C02 s(h) 2S(f) (2pC1)sv

GL 4pC2 0 K2 0 0 0

and specifically has a character under the identity that is equal to half the orderof the point group. The structure ofGL is most easily understood by consideringthe limit N/N, D(2pC1)h/DNh. In DNh, GL reduces to an angular-momentum

type expansion

GLZSKg CS

CuCPgCPuCDgCDuCFgCFuC/ (4.2)





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with leading terms GTZSCuCPu, GRZS

KgCPg followed by symmetries of

scalar (DgCFuC/) and vector (DuCFgC/) cylindrical harmonics.The symmetries in the series PgCDgCFgC/ and PuCDuCFuC/ are

compactly written as ELgand ELu, respectively, where (LZ1)hP, (LZ2)hD,.

(Altmann & Herzig 1994). Representations EL(g/u) are defined by their characterunder the operations of DNh as

DNh E 2C(Nf) C2 NC02 i 2S(Nf) s(h) Nsv

ELg 2 2 cos LF 2(K1)L

0 2 2(K1)L cos LF 2(K1)L 0

ELu 2 2 cos LF 2(K1)L

0 K2 K2(K1)L cos LF K2(K1)L 0

from which it is seen that the characters of ELg and ELu are equal under properoperations, and equal but opposite in sign under improper operations.

In the compact notation, equation (4.2) becomes

DNh GLZA2gCA1uCXp

LZ1

ELgCELu; (4.3)

SKg hA2g;SCu hA1u which on descent back to D(2pC1)h is

D2pC1h GLZA02CA

002CXp

LZ1

E0LCE00L; (4.4)

which can be written as

D2pC1h GLZGTCGRCXp

LZ2

E0LCE00L; (4.5)

where GTZA002CE

01, GRZA

02CE

001 , giving the final form ofG(m)KG(s) as

D2pC1h GmKGsZA002KA

001CXp

LZ2

E0LCE00L; (4.6)

where A002ZGz and A001ZG3 in this group. Note that the result (4.6) reduces to

equation (3.2) for D3h, NZ6, pZ1, where the summation term would disappear.

Displacement along the Gzanapole mechanism takes the ring of tetrahedra to aC(2pC1)v point on the rotation pathway. The horizontal plane, C

02 axes and S(f)

improper axes are then no longer symmetry elements. The appropriate forms ofequations (4.4) and (4.6) are

C2pC1v GLZA1CA2C2Xp

LZ1

EL; (4.7)

C2pC1v GmKGsZA1KA2C2Xp

LZ2

EL: (4.8)

Again, the results for NZ6, pZ1 are recovered by deleting the summation terms.At the intermediate antiprism hinge configuration, the ring of tetrahedra

has D(2pC1)d symmetry. In this point group, all perpendicular C02 axes fall into





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a single class, each axis passing though the centres of opposite tetrahedra andthrough the mid-points of four non-hinge bars. The sd mirror planes alsoconstitute a single class, and each plane contains two opposite hinge bars.Equations (4.4) and (4.6) become

D2pC1d GLZA2gCA2uCXp

LZ1

ELgCELu; (4.9)

D2pC1d GmKGsZA2uKA1uCXp

LZ2

ELgCELu: (4.10)

Again, the results for NZ6, pZ1 are recovered by deleting the summation terms.

(b) Rings with NZ4p, pO1

In the standard position, the N tetrahedra are arranged with D(N/2)hZD(2p)hpoint group symmetry. In D(2p)h, there are two classes of binary rotation axis(C02, C

002) perpendicular to the principal axis, and two classes of reflection plane

(sv, sd) that contain the principal axis. We choose C02 to bisect two vertical hinges

and C002 to contain two horizontal hinges. Consequently, sv contains two verticalhinges, and sd contains two horizontal hinges.

The Maxwell symmetry calculation in tabular form is

D(2p)h E 2C(f) C2 pC02 pC

02 i 2S(f) sh psd psv

G(j) 8p 0 0 0 4 0 0 4p 4 4!GT 3 c

CK1 K1 K1 K3 c

K 1 1 1

24p 0 0 0 K4 0 0 4p 4 4KGTKGR K6 K2c

C2 2 2 0 0 0 0 0

24pK6 K2cC 2 2 K2 0 0 4p 4 4KG(b) K20p 0 0 K2 K2 0 0 K4p K2 K2

G(m)KG(s) 4pK6 K2cC 2 0 K4 0 0 0 2 2

where again cGZG1C2 cos f. To reduce G(m)KG(s), we again invoke GL(equation (4.1)), which now has the integer characters

D(2p)h E 2C(f) C2 pC02 pC

02 i 2S(f) s(h) psd psv

GL 4p 0 0 0 K4 0 0 0 0 0

and again has the form of an angular momentum expansion in the DNh

supergroup.

As there is no distinction between C02 and C

002 in DNh, there is a subtlety in the

termination of the expansion when N is doubly odd: GL is of the order 4p, andtherefore can include only half of the four combinations comprised within





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the final pair of degenerate representations EpgCEpu. In D(2p)h, pO1, EpgCEpureduces to B1gCB2gCB1uCB2u and inspection of characters under E and C

002

shows that the half to be retained is B1gCB1u, the part of EpgCEpu withcharacter K1 under C002. Thus, the form ofGL in D(2p)h, pO1, is

D2ph GLZA2gCA2uCB1gCB1uCXpK1

LZ1

ELgCELu: (4.11)

(Note the change in labelling for GzZSCu on descent from DNh to D(2p)h: in DNh,

the AltmannHerzig convention is GzZA1u but in D(2p)h with 1!p!N,GzZA2u.) From equation (4.11) GL can be written for pO1 as

D2ph GLZGTCGRCB1gCB1uCXpK1

LZ2

ELgCELu; (4.12)

and the final form ofG(m)KG(s) for pO1 is then

D2ph GmKGsZA2uKA1uCB1gCB1uCXpK1

LZ2

ELgCELu; (4.13)

where, as noted above, A2uZGz and A1uZG3 in this group.Displacement along the Gzanapole mechanism takes the ring of tetrahedra to a

C(2p)v point on the rotation pathway. The horizontal plane, C02 and C

002 axes and

S(f) improper axes are then no longer symmetry elements. The appropriateforms of equations (4.11) and (4.13) are

C2pv GLZA1CA2CB1CB2C2XpK1

LZ1

EL; (4.14)

C2pv GmKGsZA1KA2CB1CB2C2XpK1

LZ1

EL: (4.15)

Finally, at the intermediate antiprism hinge configuration, the ring oftetrahedra has D(2pC1)d symmetry. In this point group, all perpendicular C

02

axes fall into a single class, each axis passing though the centres of opposite

tetrahedra, and through the mid-points of four non-hinge bars. The sd mirrorplanes also constitute a single class, and each contains two opposite hinge bars.Equations (4.11) and (4.13) become

D2pd GLZA2CB2CX2pK1

LZ1

EL; (4.16)

D2pd GmKGsZB2KB1CX2pK2

LZ2

EL: (4.17)

(c) Interpretation

We have derived explicit formulae for the mobility excess of rings oftetrahedra with doubly odd and doubly even N in each of three distinct





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symmetry groups. All six formulae (4.6), (4.8), (4.10), (4.13), (4.15) and (4.17)can be subsumed in one general expression

GmKGsZGzKG3CGLKGTKGR; (4.18)

where Gz is the non-degenerate representation of a one-parameter anapolemechanism transforming in the same way as simple translation along the mainaxis of the underlying torus, and G3 is the non-degenerate representation of theunique state of self-stress. The final term (GLKGTKGR) includes contributionswith negative weights, but as GL contains complete copies of GT and GR forNR6, the bracketed term is well defined: it is either vanishing (NZ6), orpositive (NO6).

Given that (GLKGTKGR) is a reducible representation with non-negativeweights, and given the kinematic result sZ1, the mobility excess (4.18) resolvesinto separate expressions for the symmetries spanned by the states of self-stress

and mechanisms:GsZG3; (4.19)

GmZGzCGLKGTKGR; (4.20)

valid for generic configurations of rings with even NR 6.The angular-momentum description of GL (equation (4.3)) gives a physical

picture of the sets of mechanisms for NO 6 that are additional to thecharacteristic anapole rotation. As noted earlier, (GLKGTKGR) has terms oftwo types. In DN

h, the representations E

2g, E

3u, E

4g,. are those of scalar

cylindrical harmonics, which can be visualized with appropriate patterns ofshading on the torus (figure 3c,e,g,i). A given En(g/u) in this series describes a pairof functions that are interconverted on rotation ofp/2n about the main toroidalaxis, each having nnodal planes containing that axis. Both members of the pairare symmetric with respect to reflection in the horizontal mirror plane. Thealternative representations E2u, E3g, E4u,. are those of vector cylindricalharmonics, and describe pairs of vector fields on the torus, again interconvertingunder rotation by p/2n and again having n nodal planes containing the verticalcylinder axis, but now antisymmetric with respect to reflection in the horizontalplane; each vector symmetry is related to a scalar harmonic symmetry through

multiplication by Gz (Eng!A1uZEnu). The vector harmonics can also bevisualized with appropriate shading of the torus (figure 3d,f,h,j).

Physical models of the mechanisms of the ring of tetrahedra follow from thevisualizations of figure 3; for simplicity, we shall consider these in the standardsetting. The case NZ12 is shown in the standard setting in figure 4. The ring ofNtetrahedra in the standard setting has N/2 vertical and N/2 horizontal hinges.A full description of the mechanisms is given if the freedoms of these two sets ofhinges are treated separately, considering in turn one set to be locked and theother free to move.

Consider initially the case where the horizontal hinges are locked. The

freedoms of a planar cycle of rigid bodies connected pairwise by N/2perpendicular revolute hinges can be represented by sets of N/2 scalars (C foropening, K for closing, say). If these scalars are considered using an angular





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momentum description (for NZ12 this is shown in figure 5), then neither theconcerted opening motion (LZ0) nor the pair of motions with a single verticalplane of antisymmetry (LZ1) correspond to mechanisms. The remainder of the

Figure 3. Representation of the single state of self-stress (a) and mechanisms (b)(j), of rings ofrotating tetrahedra, shown as decorations of a torus. For NZ6: (a) shows A002; (b) shows A

001. For

NZ8: (a) A1u; (b) A2u; (c) B1g; (d) B1u. For NZ10: (a) A002; (b) A

001; (c) and (e) E

02; (d) and (f) E

002.

For NZ12: (a) A1u; (b) A2u; (c) and (e) E2g; (d) and (f) E2u; (g) B1u; (h) B1g. For NZ14: (a) A002;

(b) A001; (c) and (e) E02; (d) and (f) E

002; (g) and (i) E

03; (h) and (j) E

003.





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N/2 independent combinations of hinge freedoms span exactly the series E2g, E3u,

E4g,. of the scalar cylindrical harmonics. When N/2 is even, the mechanismsoccur in pairs; when N/2 is odd, one function at the highest value ofL has nodesat all hinge positions and is dropped from the series.

Likewise, consider the case where the vertical hinges are locked. The freedomsof a planar cycle of rigid bodies connected pairwise by N/2 in-plane revolute hingescan be represented by sets ofN/2 vectors normal to the cycle plane (C phase in thehalf space of the hinge motion corresponding to approach of the connected bodies,say). If these vertical vectors are considered using an angular momentumdescription (for NZ12 this is shown in figure 6), the concerted motion of the hinges(LZ0) now corresponds to the anapole rotation of the ring of tetrahedra

instantaneously, at this configuration, the anapole mechanism requires rotationsabout only the horizontal hinges. The pair of motions with a single vertical plane ofantisymmetry (LZ1) again do not correspond to mechanisms. The remainder ofthe N/2 independent combinations of hinge freedoms spans exactly the series E2u,E3g, E4u,. of the vector cylindrical harmonics. When N/2 is even, themechanisms occur in pairs; when N/2 is odd, one function at the highest valueofL has nodes at all hinge positions and is dropped from the series.

Once the ring of tetrahedra moves along any of the mechanisms, it loses thesymmetry of the standard setting and mechanisms may begin to mix insymmetry, with loss of the distinction between horizontal and perpendicular

hinges; however, the cylindrical harmonics still give a qualitative picture of thenumber and types of mechanism.

5. The special case NZ4

All of the analysis so far has been restricted to the rings of NR6 tetrahedra.However, it is possible to assemble four suitably shaped tetrahedra in a ring. Thesteric constraints are severe, and, in particular, it is not possible to use physicalregular tetrahedra, but, for example, right angled quarter-tetrahedra formed byjoining two opposite edge mid-points and two vertices of a regular tetrahedra are

possible components. Much of the previous analysis applies in this case, althoughthe equivalence of the symmetry results is obscured by differences in notation forAbelian and non-Abelian groups.

Figure 4. A ring of 12 tetrahedra in the standard setting. The faces of the tetrahedra have beenshaded with the same colour scheme used in figure 1a, and shared (hinge) edges between tetrahedrahave been marked with a dashed line.





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The case NZ4 has one obvious difference from the larger rings in that there isa bifurcation in the path followed by the mechanism. This is most clearlyrevealed by starting, not at the standard (in this case D2h) configuration, but at

the alternative high-symmetry point, the D2d arrangement of four tetrahedra.Figure 7 shows the D2d arrangement of four skinny tetrahedra, and itsequivalence to a set of four cubes. Figure 8 illustrates the bifurcation of the

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Figure 6. An angular momentum expansion of the vectors representing the opening or closing of thesix horizontal hinges in a ring of 12 tetrahedra; each value is the magnitude of a vertical vector (notnormalized). The bars between the horizontal hinges correspond to pairs of tetrahedra joined by alocked vertical hinge. (a) LZ0; (b) LZ1; (c) LZ2; (d) LZ3. The dashed lines show nodal planes.The pair LZ1 does not correspond to mechanisms; LZ0 is the anapole rotation.

1 1

11

11

1 1

11

11

0 0

00

00

00

00

(b)(i)

(b)(ii)

(c)(i)

(c)(ii)

(d)(i)

(d)(ii)

(a)

1 1

1 1

1 1

1 1

22

1 1

1 1

22

1 1

1 1

Figure 5. An angular momentum expansion of scalar values (not normalized) representing theopening or closing of the six vertical hinges in a ring of 12 tetrahedra. The bars between the verticalhinges correspond to pairs of tetrahedra joined by a locked horizontal hinge. (a) LZ0; (b) LZ1;(c) LZ2; (d) LZ3. The dashed lines show nodal planes. LZ0 and LZ1 do not correspond tomechanisms.





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mechanism of the four-ring. Relative rotation (a) about one pair of collinearhinges leads from the D2d configuration I, through a sequence of C2vconfigurations (not shown) to the standard setting II; here the hinges have D2hsymmetry. Alternatively, relative rotation (b) about the other pair of collinear

hinges leads to a distinct D2h standard setting III. Each of the paths (a) and (b)are theoretically continuous, crossing at I, although when the ring is realizedwith the cubic blocks shown in the figure, steric clashes prevent continuation ofthe paths through the high symmetry point; this would not be the case withsuitably skinny bodies. This is an example of a kinematotropic mechanism in theextended sense defined by Galletti & Fanghella (2001).

Counting equation (2.2) gives the mobility of the ring of four tetrahedra/cubesas mKsZK2. Symmetry analysis (3.1) gives the representation of the mobilityin the different point groups accessed by the mechanisms as

D2dI GmKGsZB2KB1KE; (5.1)

C2vI/II GmKGsZA1KA2KB1KB2; (5.2)

D2hII GmKGsZB1uKB3gKAuKB3u; (5.3)

C2vI/III GmKGsZA1KA2KB1KB2; (5.4)

D2hIII GmKGsZB1uKB2gKAuKB2u: (5.5)

The symmetry analysis in all configurations shows a single mechanism and threestates of self-stress. Analysis in the generic C2v configuration along one of the twobranches shows a totally symmetric mechanism with no blocking equisymmetricstate of self-stress; the mechanism is therefore finite (Kangwai & Guest 1999).

Neither the counting nor symmetry analysis gives any hint of the presence of asecond mechanism at the D2dbifurcation point. Indeed, this is clearly a geometricphenomenon, critically dependent on the simultaneous collinearity of two pairs ofhinges at this configuration. This is compatible with, but not implied by, D2dsymmetry; mechanisms which require special geometric configurations willalways escape a generic symmetry analysis, and require a specific analysis at thespecial geometry. A structure in which the meeting points of the collinear pairswere symmetrically displaced along the z-axis, leaving the outer ends ofthe hinges unshifted, would still belong to the D2d point group, but only thesingle mechanism predicted by the symmetry mobility analysis would remain;

such a system, however, would not be a ring of tetrahedra in the strict sensedefined in 1.

6. Conclusions

This paper provides a general symmetry analysis of a ring of even-Nregular tetrahedra. Symmetry reveals that generically, for even NR6, thecount mKsZNK6 comes from sZ1 states of self-stress, and a symmetricallydistinct mZNK5 mechanisms. The finite nature of these mechanisms is also

shown, including the eponymous rotating mechanisms. The treatment showsthe utility of a symmetry analysis in enriching the information available frompure counting.





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The NZ4 case has been considered separately. Here, generically, symmetryshows that the count mKsZNK6 comes from sZ3 states of self-stress, and mZ1mechanisms. Uniquely in this case, the rotating mechanism passes through

a particular geometric configuration where there is a bifurcation in themechanism path. Detection of the bifurcation is outside the symmetryclassification.

The analysis as presented considers only even-N cases that lie on the pathfollowed by the rotating mechanism. For NO6, there are many other mechanismpaths, leading to lower symmetry configurations, and other points of mechanismbifurcation that could be followed; we have not considered these other paths,although the general methodology remains valid for them.

It is possible to generalize the even-N ring of tetrahedra with mutuallyperpendicular hinges, as is considered in this paper. Odd-N rings can be

constructed.The case NZ7 was noted by Coxeter (Rouse Ball 1939), and we have

constructed an example using skinny tetrahedra. Generically, it has C1symmetry (no symmetry operation other than the identity), but passes thoughtwo distinct C2 high-symmetry configurations; in the generic C1 configuration,symmetry analysis reduces to simple counting.

The objects considered here all have a cylindrical topology: if a path aroundthe ring is followed, passing from one tetrahedron to the next along faces that areadjacent, either across an edge or a vertex, then eventually the face that was theinitial starting point is reached. For skinny tetrahedra, it is also possible to join

even-Nrings with a Mobius twist (Stark 2004). We have constructed examples ofrings of tetrahedra with mutually perpendicular hinges that have a Mobius twist,but found that accessible symmetries, at least for NZ8, are rather low.

hinge line

hinge line

Figure 7. A ring of four tetrahedra in a D2d configuration. The faces of the tetrahedra have beenshaded with the same colour scheme as used in figure 1c, shared (hinge) edges between tetrahedrahave been marked with a dashed line and hidden lines are shown in grey. The fine lines show a setof four cubes hinged together in the same way that has equivalent mobility; the mechanisms aredetermined by the arrangement of hinge lines rather than the details of the hinged bodies.





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Finally, recent investigations (Gan & Pellegrino 2003; Chen et al. 2005) haveshown that the mobility of rings may persist under relaxation of the conditionthat consecutive hinges are perpendicular. Many of these systems have potentialapplication as deployable structures.

S.D.G. acknowledges the support of the Leverhulme Trust.

References

Altmann, S. L. & Herzig, P. 1994 Point-group theory tables. Oxford: Clarendon Press.Atkins, P. W., Child, M. S. & Phillips, C. S. G. 1970 Tables for group theory. Oxford: Clarendon

Press.Brittain, S. T., Schueller, O. J. A., Wu, H., Whitesides, S. & Whitesides, G. M. 2001 Micro-

origami: fabrication of small, three-dimensional, metallic structures. J. Phys. Chem. B 105,347350.

Bruckner, M. 1900 Vielecke und Vielflache. Leipzig: B.G. Teubner.Calladine, C. R. 1978 Buckminster Fullers Tensegrity structures and Clerk Maxwells rules for

the construction of stiff frames. Int. J. Solids Struct. 14, 161172.Ceulemans, A., Chibotaru, L. F. & Fowler, P. W. 1998 Molecular anapole rotations. Phys. Rev.

Lett. 80, 18611864.

Chen, Y., You, Z. & Tarnai, T. 2005 Threefold-symmetric Bricard linkages for deployablestructures. Int. J. Solids Struct. 42, 22872301. (doi:10.1016/j.ijsolstr.2004.09.014.)

Cundy, H. M. & Rollett, A. R. 1981 Mathematical models, 3rd edn. Diss: Tarquin Publications.

(a)

(b)

(I)

(II)

(III)

x

y

z

Figure 8. A set of four hinged cubes used to show the mobility of a ring of four tetrahedra: therelationship between the four cubes and the four tetrahedra is shown in figure 7. I shows the D2dbifurcation configuration: two mechanism paths emerge. Relative rotation (a) about one pair ofcollinear hinges leads from the D2d configuration I, through a sequence of C2v configurations (notshown) to the standard setting II; here the hinges have D2h symmetry. Alternatively, relative

rotation (b) about the other pair of collinear hinges leads to a distinct D2h standard setting III.





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Fowler, P. W. & Guest, S. D. 2000 A symmetry extension of Maxwells rule for rigidity of frames.Int. J. Solids Struct. 37, 17931804.

Fowler, P. W. & Guest, S. D. 2002 Symmetry and states of self stress in triangulated toroidalframes. Int. J. Solids Struct. 39, 43854393.

Galletti, C. & Fanghella, P. 2001 Single-loop kinematotropic mechanisms. Mech. Mach. Theory36,743761.Gan, W. W. & Pellegrino, S. 2003 Closed-loop deployable structures. In Proc. 44th

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference,710 April 2003, Norfolk, VA. AIAA 2003-1450.

Guest, S. D. & Fowler, P. W. 2005 A symmetry-extended mobility rule. Mech. Mach. Theory.(doi:10.1016/j.mechmachtheory.2004.12.017.)

Hunt, K. H. 1978 Kinematic geometry of mechanisms. Oxford: Clarendon Press.Kangwai, R. D. & Guest, S. D. 1999 Detection of finite mechanisms in symmetric structures. Int.

J. Solids Struct. 36, 55075527.Mitchell, D. 1997 Mathematical origami: geometrical shapes by paper folding. Diss: Tarquin

Publications.Rouse Ball, W. W. 1939 Mathematical recreations and essays, 11th edn. London: Macmillan.

Revised and extended by Coxeter, H. S. M..Schattschneider, D. 1988 Creating kaleidocycles and more. In Shaping space. A polyhedral

approach (ed. M. Senechal & G. Fleck), pp. 5760. Basel: Birkhauser.Schattschneider, D. & Walker, W. 1977 M.C. Escher kaleidocycles. New York: Ballantine Books.

Editions are available in 17 languages, including: M.C. Escher Kaleidozyklen. Cologne: Taschen,1992; M.C. Escher Kalejdocykler. Berlin: Taschen, 1990; M.C. Escher Caleidocycli. Cologne:Evergreen, 1992.

Stalker, R. M. 1933 Advertising medium or toy. US Patent 1,997,022, filed 27 April 1933 andissued 9 April 1935.

Stark, M. 2004 The kaleidohedron from the IsoAxis grid. http://www.ac-noumea.nc/maths/amc/

polyhedr/IsoAxis_htm (dated February 2000, updated 28 July 2004).



http://www.ac-noumea.nc/maths/amc/polyhedr/IsoAxis_htmhttp://www.ac-noumea.nc/maths/amc/polyhedr/IsoAxis_htmhttp://rspa.royalsocietypublishing.org/http://rspa.royalsocietypublishing.org/http://rspa.royalsocietypublishing.org/http://rspa.royalsocietypublishing.org/http://www.ac-noumea.nc/maths/amc/polyhedr/IsoAxis_htmhttp://www.ac-noumea.nc/maths/amc/polyhedr/IsoAxis_htm

symmetry analysis of mechanisms in rotating rings of tetrahedra

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