## Abstract

Rotating rings of tetrahedra are well known from recreational mathematics. Rings of *N* tetrahedra with *N* even are analysed by symmetry-adapted versions of classical counting rules of mechanism analysis. For *N* ≥ 6, a single state of self-stress is found, together with *N*−5 symmetry-distinct mechanisms, which include the eponymous rotating mechanism. For *N*=4 in a generic configuration, a single mechanism remains together with three states of self-stress, but, uniquely in this case, the mechanism path passes through a bifurcation at which the number of mechanisms and states of self-stress is raised by one.

## 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. These rings can be assembled from planar nets or by origami (Mitchell 1997), and, with recent ‘micro-origami’ techniques, have been constructed on a millimetre scale as prototypes for micro-fabrication in 3D (Brittain *et al*. 2001). The rings are often associated with decorations of the plane with various patterns and are also known as kaleidocycles (Schattschneider & Walker 1977; Schattschneider 1988). The underlying mathematical objects are members of a family of cycles of edge-fused polyhedra having 2*N* vertices, 5*N* distinct edges and 4*N* triangular faces where the faces are those of *N* edge-sharing tetrahedra, and where each tetrahedron in the cycle is linked to its predecessor and successor at opposite edges. Certain members of this family display an ‘amusing and confusing’(Stalker 1933) motion in which each tetrahedron of the toroidal ring turns, the whole turning inside out like a smoke-ring. The shared edges act as hinges between rigid tetrahedral bodies. Rotating rings of tetrahedra were described in a patent in 1933 (Stalker), but objects of this form occur in the earlier mathematical literature (Brückner 1900). Animations of the motion are available at 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* is even and *N* is greater than or equal to six. For *N*=8, the motion is continuous, returning repeatedly to the starting configuration. For *N*=6, the range of movement is restricted by clashes between faces, but the underlying motion can be made continuous if the tetrahedra are transformed to an ‘isosceles’ shape by shrinking the shared edges; the system can be considered as a particular example of a ‘threefold symmetric Bricard linkage’ (Chen *et al*. 2005). The key feature common to equilateral and isosceles geometries is that successive shared edges remain mutually perpendicular. The present paper will concentrate on the cases of rings of *N* tetrahedra with this perpendicular-hinge geometry and where *N* is even.

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 the recently developed symmetry-extended versions of the classical tools of mechanism analysis, the mobility criterion (Guest & Fowler 2005) and the Maxwell 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 spherically jointed edge bars, where two opposite bars are shared with neighbouring tetrahedra. Each description implies a relationship between the symmetries of mechanisms, states of self-stress and structural components; this approach takes full advantage of the high point-group symmetry of the even-*N* rings of tetrahedra.

Counting and symmetry analysis for toroidal frameworks has been considered before in the context of toroidal deltahedra (Fowler & Guest 2002). There, it was shown that fully triangulated toroids have at least six states of self-stress of well-defined symmetry. Like the rotating rings, toroidal deltahedra have all triangular faces, but unlike the rings, the deltahedra are ‘toroidal polyhedra’ in that they enclose a single connected toroidal volume and all their edges are common to exactly two faces. The rotating rings, in which the enclosed tetrahedral volumes are disjoint and some edges are incident on four faces, are not polyhedral in this sense, and a different analysis is required.

## 2. Preliminary counting analysis

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

A generalized mobility rule is given in Guest & Fowler (2005) as(2.1)where *m* is the *mobility* (Hunt 1978), or number of mechanisms, of a mechanical linkage consisting of *N* bodies connected by *g* joints, where each joint *i* permits *f*_{i} relative freedoms, and *s* is the number of independent states of self-stress that the linkage can sustain. The parameter *s* can be considered equivalently as the number of *overconstraints*, independent geometric incompatibilities, or misfits, that are possible for the linkage. In the present case, *N* is the number of tetrahedra, and there are *g*=*N* hinge joints each permitting a single relative freedom; i.e. the revolute freedom between two adjacent tetrahedra. Thus,(2.2)

The generalized Maxwell count for a system of spherically jointed bars is given by Calladine (1978) as(2.3)where *j* is the number of spherical joints and *b* is the number of bars connecting them. In the present case, *j*=2*N* and *b*=5*N*, so that *m*−*s*=*N*−6 from equation (2.3), in agreement with the mobility criterion (2.2).

As both *m* and *s* are non-negative integers, simple counting has therefore established the existence of at least one mechanism for *N*>6, and conversely, at least 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 *N*=7, and indeed, such a mechanism has been remarked in this case and described as having ‘an entire lack of symmetry’ (Rouse Ball 1939).

For the case *N*=6, counting alone does not demonstrate the existence of the known mechanism, but it does establish that if such a mechanism exists, then so must a state of self-stress.

In fact, a separate kinematic argument can be used to demonstrate that, in a generic configuration, *s*=1 for all even *N* ≥ 6. The argument runs as follows. Consider a ring in a *standard position* where the centres of the hinges define a planar *N*-gon, with *N*/2 of the hinges lying in the plane of the polygon, and the same number lying perpendicular to it. Cutting the ring along a single joint gives a 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 next tetrahedron in the chain is unstressed, and the same argument can be extended to 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 independent kinematic constraints imposed by a revolute joint) shows that four can be accommodated by rotation of the remaining *N*−1 ≥ 5 joints. The only type of misfit that cannot be so accommodated is *twisting*, and a misfit of this sort leads to a single ‘twisting’ state of self-stress. Hence, *s*=1 in the standard position. As the same argument can be advanced for nearby configurations, *s*=1 in any *generic* configuration.

Given that *s*=1, it follows from equation (2.2) that the number of mechanisms for even *N* ≥ 6 is given by(2.4)Hence, the number of mechanisms grows linearly with *N*, although the counting approach gives no indication of the nature of these additional mechanisms; further insight into this aspect is given by considering the symmetry of the system.

## 3. Symmetry analysis: *N*=6

This section will treat the case *N*=6 in explicit detail, as a preliminary to a general analysis for all even *N* derived in §4. The symmetry extension of the Maxwell counting rule is used; exactly equivalent results are given by the corresponding extension of the mobility rule.

The symmetry extension of Maxwell's rule (Fowler & Guest 2000) is(3.1)where *Γ*(*m*), *Γ*(*s*), *Γ*(*b*) and *Γ*(*j*) are the representations of the *m* mechanisms, *s* states of self-stress, *b* bars and *j* joints, and *Γ*_{T} and *Γ*_{R} are the translational and rotational representations (Atkins *et al*. 1970), all within the point group of the instantaneous configuration.

We will consider the ring of six tetrahedra in each of the three symmetry-distinct configurations that it can assume. In the standard position (figure 1*a*), the six tetrahedra are arranged with *D*_{3h} point symmetry. The centres of the six hinges define a planar hexagon; hinges lie alternatively in, and perpendicular to, this *σ*_{h} mirror plane. The normal through the centre of the hexagon defines the axis for the threefold proper and improper rotations *C*_{3} and *S*_{3}. A *C*_{2} axis is defined by each horizontal hinge and passes through the centre of the opposite, perpendicular, hinge. Each *σ*_{v} mirror plane contains one vertical and one horizontal hinge-line.

The Maxwell symmetry analysis can be laid out in tabular form as

which reduces to(3.2)

Hence, symmetry analysis has predicted a mechanism of *A*_{2}″ symmetry, accompanied by a state of self-stress of *A*_{1}″ symmetry. *A*_{2}″ is the symmetry of the *anapole* rotation of a torus (figure 2*a*) and describes the eponymous ‘rotating’ motion of this ring of tetrahedra. *A*_{1}″ is the symmetry of a *counter*-*rotating* pattern on a torus (figure 2*b*) that describes the state of self-stress that would be generated by a twisting mismatch at each hinge.

If we displace the structure along the pathway of the *A*_{2}″ mechanism, the symmetry of the whole object falls to *C*_{3v}, with loss of *C*_{2}, *σ*_{h} and *S*_{3} symmetry elements. In the reduced symmetry of this generic *C*_{3v} configuration (figure 1*b*), the Maxwell calculation gives(3.3)This result can be verified by deleting columns in the tabular calculation above, or applying the ‘descent in symmetry’ correlation (Atkins *et al*. 1970):In *C*_{3v} the mechanism is totally symmetric, and as there is no equisymmetric state of self-stress, the local linear analysis that we have carried out here is sufficient 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 continuous mechanism path in the configuration space of the ring of tetrahedra.

If the finite mechanism is followed further along the displacement coordinate the structure passes through a second point of high symmetry, a *D*_{3d} configuration (figure 1*c*), where alternative hinges lie an equal angle above and below the horizontal plane. In this ‘antiprism’ configuration(3.4)

Continuation of the motion leads through *C*_{3v} configurations (figure 1*d*) back to a *D*_{3h} position (figure 1*e*), where horizontal and vertical hinges have been exchanged with respect to the initial setting. The cyclic nature of this finite mechanism is apparent.

Note that although its formulation across the three particular point groups uses different representation labels, the symmetry of the mobility excess for the ring of six tetrahedra is always compatible with the single expression,(3.5)where *Γ*_{z} is the symmetry of a translation along the principal axis and *Γ*_{ϵ} is the representation of a pseudo-scalar (a quantity whose sign is preserved under proper, and reversed under improper, operations). The mechanism is always an anapole rotation of an underlying torus, and the state of self-stress always follows a counter-rotating pattern on the torus, irrespective of the particular symmetry of the instantaneous configuration. Figure 2 shows how *Γ*_{z} and *Γ*_{ϵ} link to patterns of vectors on the torus.

To summarize, use of the Maxwell counting rule in its symmetry-adapted form has enabled us to give a complete account of the interesting static and kinematic behaviour for *N*=6. Use of the symmetry-adapted mobility criterion (Guest & Fowler 2005) gives the same results. This case has only one mechanism, the characteristic anapole rotation, but, as we have seen from pure counting, more mechanisms emerge for larger *N*. These too are amenable to a symmetry treatment, as the following section will demonstrate.

## 4. Symmetry analysis: the general case

The previous analysis can be generalized to cover all even values of *N*. Again, we concentrate on the Maxwell analysis, although all results reported here could also be obtained with the symmetry version of the mobility criterion. We consider the same sequence of standard *D*_{(N/2)h}, generic rotated *C*_{(N/2)v} and alternative high-symmetry *D*_{(N/2)d} antiprism configurations. For these groups we will use the notation *C*(*ϕ*) for the symmetry operation of rotation through *ϕ* about the principal axis, and *S*(*ϕ*) for the corresponding improper operation. *C*(*ϕ*) and *C*(−*ϕ*) belong to the same class, as do *S*(*ϕ*) and *S*(−*ϕ*). It turns out to be convenient to consider doubly odd cases, *N*=4*p*+2, and doubly even cases, *N*=4*p*, separately. The case *N*=4*p* is further split into *N*=4*p*, *p*>1 and *N*=4, *p*=1, as the standard notation for point groups *C*_{2v}, *D*_{2h}, *D*_{2d} (*p*=1) differs from that for *C*_{2pv}, *D*_{2ph}, *D*_{2pd} (*p*>1). In addition to this technicality, the case *p*=1 also presents some special features that justify a separate treatment, given in §5.

### (a) Rings with *N*=4*p*+2

In the standard position, the *N* tetrahedra are arranged with *D*_{(N/2)h}=*D*_{(2p+1)h} point symmetry. The centres of the hinges define a planar *N*-gon; hinges lie alternatively within, and perpendicular to, the *σ*_{h} mirror plane. The normal through the centre of the hexagon defines the principal axis. A *C*′_{2} axis is defined by each horizontal hinge and its opposite perpendicular partner; the same pair of hinges defines a *σ*_{v} mirror plane. The structure has (2*p*+1) *C*′_{2} axes, and (2*p*+1) *σ*_{v} planes.

The Maxwell symmetry analysis is

where *c*^{±}=±1+2 cos *ϕ*. To reduce *Γ*(*m*)−*Γ*(*s*) to a tractable form, we note that a representation *Γ*_{Λ} defined by(4.1)would have only integer characters

and specifically has a character under the identity that is equal to half the order of the point group. The structure of *Γ*_{Λ} is most easily understood by considering the limit *N*→∞, *D*_{(2p+1)h}→*D*_{∞h}. In *D*_{∞h}, *Γ*_{Λ} reduces to an angular-momentum type expansion(4.2)with leading terms , followed by symmetries of scalar (*Δ*_{g}+*Φ*_{u}+⋯) and vector (*Δ*_{u}+*Φ*_{g}+⋯) cylindrical harmonics.

The symmetries in the series *Π*_{g}+*Δ*_{g}+*Φ*_{g}+⋯ and *Π*_{u}+*Δ*_{u}+*Φ*_{u}+⋯ are compactly written as *E*_{Lg} and *E*_{Lu}, respectively, where (*L*=1)≡*Π*, (*L*=2)≡*Δ*,… (Altmann & Herzig 1994). Representations *E*_{L(g/u)} are defined by their character under the operations of *D*_{∞h} as

from which it is seen that the characters of *E*_{Lg} and *E*_{Lu} are equal under proper operations, and equal but opposite in sign under improper operations.

In the compact notation, equation (4.2) becomes(4.3) which on descent back to *D*_{(2p+1)h} is(4.4)which can be written as(4.5)where , , giving the final form of *Γ*(*m*)−*Γ*(*s*) as(4.6)where *A*″_{2}=*Γ*_{z} and *A*″_{1}=*Γ*_{ϵ} in this group. Note that the result (4.6) reduces to equation (3.2) for *D*_{3h}, *N*=6, *p*=1, where the summation term would disappear.

Displacement along the *Γ*_{z} anapole mechanism takes the ring of tetrahedra to a *C*_{(2p+1)v} point on the rotation pathway. The horizontal plane, *C*′_{2} axes and *S*(*ϕ*) improper axes are then no longer symmetry elements. The appropriate forms of equations (4.4) and (4.6) are(4.7)(4.8)Again, the results for *N*=6, *p*=1 are recovered by deleting the summation terms.

At the intermediate ‘antiprism’ hinge configuration, the ring of tetrahedra has *D*_{(2p+1)d} symmetry. In this point group, all perpendicular *C*′_{2} 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 *σ*_{d} mirror planes also constitute a single class, and each plane contains two opposite hinge bars. Equations (4.4) and (4.6) become(4.9)(4.10)Again, the results for *N*=6, *p*=1 are recovered by deleting the summation terms.

### (b) Rings with *N*=4*p*, *p*>1

In the standard position, the *N* tetrahedra are arranged with *D*_{(N/2)h}=*D*_{(2p)h} point group symmetry. In *D*_{(2p)h}, there are two classes of binary rotation axis (*C*′_{2}, *C*″_{2}) perpendicular to the principal axis, and two classes of reflection plane (*σ*_{v}, *σ*_{d}) that contain the principal axis. We choose *C*′_{2} to bisect two vertical hinges and *C*″_{2} to contain two horizontal hinges. Consequently, *σ*_{v} contains two vertical hinges, and *σ*_{d} contains two horizontal hinges.

The Maxwell symmetry calculation in tabular form is

where again *c*^{±}=±1+2 cos *ϕ*. To reduce *Γ*(*m*)−*Γ*(*s*), we again invoke *Γ*_{Λ} (equation (4.1)), which now has the integer characters

and again has the form of an angular momentum expansion in the *D*_{∞h} supergroup.

As there is no distinction between *C*′_{2} and *C*″_{2} in *D*_{∞h}, there is a subtlety in the termination of the expansion when *N* is doubly odd: *Γ*_{Λ} is of the order 4*p*, and therefore can include only half of the four combinations comprised within the final pair of degenerate representations *E*_{pg}+*E*_{pu}. In *D*_{(2p)h}, *p*>1, *E*_{pg}+*E*_{pu} reduces to *B*_{1g}+*B*_{2g}+*B*_{1u}+*B*_{2u} and inspection of characters under *E* and *C*″_{2} shows that the half to be retained is *B*_{1g}+*B*_{1u}, the part of *E*_{pg}+*E*_{pu} with character −1 under *C*″_{2}. Thus, the form of *Γ*_{Λ} in *D*_{(2p)h}, *p*>1, is(4.11)(Note the change in labelling for on descent from *D*_{∞h} to *D*_{(2p)h}: in *D*_{∞h}, the Altmann–Herzig convention is *Γ*_{z}=*A*_{1u} but in *D*_{(2p)h} with 1<*p*<∞, *Γ*_{z}=*A*_{2u}.) From equation (4.11) *Γ*_{Λ} can be written for *p*>1 as(4.12)and the final form of *Γ*(*m*)−*Γ*(*s*) for *p*>1 is then(4.13)where, as noted above, *A*_{2u}=*Γ*_{z} and *A*_{1u}=*Γ*_{ϵ} in this group.

Displacement along the *Γ*_{z} anapole mechanism takes the ring of tetrahedra to a *C*_{(2p)v} point on the rotation pathway. The horizontal plane, *C*′_{2} and *C*″_{2} axes and *S*(*ϕ*) improper axes are then no longer symmetry elements. The appropriate forms of equations (4.11) and (4.13) are(4.14)(4.15)

Finally, at the intermediate antiprism hinge configuration, the ring of tetrahedra has *D*_{(2p+1)d} symmetry. In this point group, all perpendicular *C*′_{2} 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 *σ*_{d} mirror planes also constitute a single class, and each contains two opposite hinge bars. Equations (4.11) and (4.13) become(4.16)(4.17)

### (c) Interpretation

We have derived explicit formulae for the mobility excess of rings of tetrahedra with doubly odd and doubly even *N* in each of three distinct 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(4.18)where *Γ*_{z} is the non-degenerate representation of a one-parameter anapole mechanism transforming in the same way as simple translation along the main axis of the underlying torus, and *Γ*_{ϵ} is the non-degenerate representation of the unique state of self-stress. The final term (*Γ*_{Λ}−*Γ*_{T}−*Γ*_{R}) includes contributions with negative weights, but as *Γ*_{Λ} contains complete copies of *Γ*_{T} and *Γ*_{R} for *N* ≥ 6, the bracketed term is well defined: it is either vanishing (*N*=6), or positive (*N* > 6).

Given that (*Γ*_{Λ}−*Γ*_{T}−*Γ*_{R}) is a reducible representation with non-negative weights, and given the kinematic result *s*=1, the mobility excess (4.18) resolves into separate expressions for the symmetries spanned by the states of self-stress and mechanisms:(4.19)(4.20)valid for generic configurations of rings with even *N* ≥ 6.

The angular-momentum description of *Γ*_{Λ} (equation (4.3)) gives a physical picture of the sets of mechanisms for *N *> 6 that are additional to the characteristic anapole rotation. As noted earlier, (*Γ*_{Λ}−*Γ*_{T}−*Γ*_{R}) has terms of two types. In *D*_{∞h}, the representations *E*_{2g}, *E*_{3u}, *E*_{4g},…are those of scalar cylindrical harmonics, which can be visualized with appropriate patterns of shading on the torus (figure 3*c*,*e*,*g*,*i*). A given *E*_{n(g/u)} in this series describes a pair of functions that are interconverted on rotation of π/2*n* about the main toroidal axis, each having *n* nodal planes containing that axis. Both members of the pair are symmetric with respect to reflection in the horizontal mirror plane. The alternative representations *E*_{2u}, *E*_{3g}, *E*_{4u},…are those of vector cylindrical harmonics, and describe pairs of vector fields on the torus, again interconverting under rotation by π/2*n* and again having *n* nodal planes containing the vertical cylinder axis, but now antisymmetric with respect to reflection in the horizontal plane; each vector symmetry is related to a scalar harmonic symmetry through multiplication by *Γ*_{z} (*E*_{ng}×*A*_{1u}=*E*_{nu}). The vector harmonics can also be visualized with appropriate shading of the torus (figure 3*d*,*f*,*h*,*j*).

Physical models of the mechanisms of the ring of tetrahedra follow from the visualizations of figure 3; for simplicity, we shall consider these in the standard setting. The case *N*=12 is shown in the standard setting in figure 4. The ring of *N* tetrahedra 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 of hinges are treated separately, considering in turn one set to be locked and the other 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*/2 perpendicular revolute hinges can be represented by sets of *N*/2 scalars (+ for opening, − for closing, say). If these scalars are considered using an angular momentum description (for *N*=12 this is shown in figure 5), then neither the concerted opening motion (*Λ*=0) nor the pair of motions with a single vertical plane of antisymmetry (*Λ*=1) correspond to mechanisms. The remainder of the *N*/2 independent combinations of hinge freedoms span exactly the series *E*_{2g}, *E*_{3u}, *E*_{4g},… of the scalar cylindrical harmonics. When *N*/2 is even, the mechanisms occur in pairs; when *N*/2 is odd, one function at the highest value of *Λ* has nodes at all hinge positions and is dropped from the series.

Likewise, consider the case where the vertical hinges are locked. The freedoms of a planar cycle of rigid bodies connected pairwise by *N*/2 in-plane revolute hinges can be represented by sets of *N*/2 vectors normal to the cycle plane (+ phase in the half space of the hinge motion corresponding to approach of the connected bodies, say). If these vertical vectors are considered using an angular momentum description (for *N*=12 this is shown in figure 6), the concerted motion of the hinges (*Λ*=0) now corresponds to the anapole rotation of the ring of tetrahedra—instantaneously, at this configuration, the anapole mechanism requires rotations about only the horizontal hinges. The pair of motions with a single vertical plane of antisymmetry (*Λ*=1) again do not correspond to mechanisms. The remainder of the *N*/2 independent combinations of hinge freedoms spans exactly the series *E*_{2u}, *E*_{3g}, *E*_{4u},…of the vector cylindrical harmonics. When *N*/2 is even, the mechanisms occur in pairs; when *N*/2 is odd, one function at the highest value of *Λ* 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 the symmetry of the standard setting and mechanisms may begin to mix in symmetry, with loss of the distinction between horizontal and perpendicular hinges; however, the cylindrical harmonics still give a qualitative picture of the number and types of mechanism.

## 5. The special case *N*=4

All of the analysis so far has been restricted to the rings of *N* ≥ 6 tetrahedra. However, it is possible to assemble four suitably shaped tetrahedra in a ring. The steric constraints are severe, and, in particular, it is not possible to use physical regular tetrahedra, but, for example, right angled ‘quarter-tetrahedra’ formed by joining two opposite edge mid-points and two vertices of a regular tetrahedra are possible components. Much of the previous analysis applies in this case, although the equivalence of the symmetry results is obscured by differences in notation for Abelian and non-Abelian groups.

The case *N*=4 has one obvious difference from the larger rings in that there is a bifurcation in the path followed by the mechanism. This is most clearly revealed by starting, not at the standard (in this case *D*_{2h}) configuration, but at the alternative high-symmetry point, the *D*_{2d} arrangement of four tetrahedra. Figure 7 shows the *D*_{2d} arrangement of four ‘skinny’ tetrahedra, and its equivalence to a set of four cubes. Figure 8 illustrates the bifurcation of the mechanism of the four-ring. Relative rotation (*a*) about one pair of collinear hinges leads from the *D*_{2d} configuration I, through a sequence of *C*_{2v} configurations (not shown) to the standard setting II; here the hinges have *D*_{2h} symmetry. Alternatively, relative rotation (*b*) about the other pair of collinear hinges leads to a distinct *D*_{2h} standard setting III. Each of the paths (*a*) and (*b*) are theoretically continuous, crossing at I, although when the ring is realized with the cubic blocks shown in the figure, steric clashes prevent continuation of the paths through the high symmetry point; this would not be the case with suitably skinny bodies. This is an example of a kinematotropic mechanism in the extended sense defined by Galletti & Fanghella (2001).

Counting equation (2.2) gives the mobility of the ring of four tetrahedra/cubes as *m*−*s*=−2. Symmetry analysis (3.1) gives the representation of the mobility in the different point groups accessed by the mechanisms as(5.1)(5.2)(5.3)(5.4)(5.5)The symmetry analysis in all configurations shows a single mechanism and three states of self-stress. Analysis in the generic *C*_{2v} configuration along one of the two branches shows a totally symmetric mechanism with no blocking equisymmetric state of self-stress; the mechanism is therefore finite (Kangwai & Guest 1999).

Neither the counting nor symmetry analysis gives any hint of the presence of a second mechanism at the *D*_{2d} bifurcation point. Indeed, this is clearly a geometric phenomenon, critically dependent on the simultaneous collinearity of two pairs of hinges at this configuration. This is compatible with, but not implied by, *D*_{2d} symmetry; mechanisms which require special geometric configurations will always escape a generic symmetry analysis, and require a specific analysis at the special geometry. A structure in which the meeting points of the collinear pairs were symmetrically displaced along the *z*-axis, leaving the outer ends of the hinges unshifted, would still belong to the *D*_{2d} point group, but only the single mechanism predicted by the symmetry mobility analysis would remain; such a system, however, would not be a ring of tetrahedra in the strict sense defined in §1.

## 6. Conclusions

This paper provides a general symmetry analysis of a ring of even-*N* regular tetrahedra. Symmetry reveals that generically, for even *N* ≥ 6, the count *m*−*s*=*N*−6 comes from *s*=1 states of self-stress, and a symmetrically distinct *m*=*N*−5 mechanisms. The finite nature of these mechanisms is also shown, including the eponymous ‘rotating’ mechanisms. The treatment shows the utility of a symmetry analysis in enriching the information available from pure counting.

The *N*=4 case has been considered separately. Here, generically, symmetry shows that the count *m*−*s*=*N*−6 comes from *s*=3 states of self-stress, and *m*=1 mechanisms. Uniquely in this case, the ‘rotating’ mechanism passes through a particular geometric configuration where there is a bifurcation in the mechanism path. Detection of the bifurcation is outside the symmetry classification.

The analysis as presented considers only even-*N* cases that lie on the path followed by the rotating mechanism. For *N*>6, there are many other mechanism paths, leading to lower symmetry configurations, and other points of mechanism bifurcation 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 mutually perpendicular hinges, as is considered in this paper. Odd-*N* rings can be constructed.

The case *N*=7 was noted by Coxeter (Rouse Ball 1939), and we have constructed an example using skinny tetrahedra. Generically, it has *C*_{1} symmetry (no symmetry operation other than the identity), but passes though two distinct *C*_{2} high-symmetry configurations; in the generic *C*_{1} configuration, symmetry analysis reduces to simple counting.

The objects considered here all have a cylindrical topology: if a path around the ring is followed, passing from one tetrahedron to the next along faces that are adjacent, either across an edge or a vertex, then eventually the face that was the initial starting point is reached. For skinny tetrahedra, it is also possible to join even-*N* rings with a Mobius twist (Stark 2004). We have constructed examples of rings of tetrahedra with mutually perpendicular hinges that have a Mobius twist, but found that accessible symmetries, at least for *N*=8, are rather low.

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

## Acknowledgments

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

## Footnotes

- Received May 19, 2004.
- Accepted November 6, 2004.

- © 2005 The Royal Society