## Abstract

We formulate a concise deformation theory for periodic bar-and-joint frameworks in *R*^{d} and illustrate our algebraic–geometric approach on frameworks related to various crystalline structures. Particular attention is given to periodic frameworks modelled on silica, zeolites and perovskites. For frameworks akin to tectosilicates, which are made of one-skeleta of *d*-dimensional simplices, with each vertex common to exactly two simplices, we prove the existence of a space of periodicity-preserving infinitesimal flexes of dimension at least . However, these infinitesimal flexes need not come from genuine flexibility, as shown by rigid examples. The changes implicated in passing from a given lattice of periods to a sublattice of periods are illustrated with frameworks modelled on perovskites.

## 1. Introduction

Periodic frameworks are geometric structures related to crystalline materials and infinite trusses. What they retain as essential are an infinite graph *G* of bonds or rigid links *and* a large enough automorphism group *Γ* that marks the envisaged periodicity of *G* and is represented, in each realization, by a discrete lattice of translations of maximal rank. Large enough means that *G* modulo *Γ* is finite.

While precise definitions will be given as we proceed, it may be emphasized from the very beginning that our considerations refer to pairs (*G*,*Γ*) and not the infinite graph *G* alone. In particular, upon relaxing the periodicity requirements from *Γ* to a subgroup , the pair may have sensibly different properties by comparison with the pair (*G*,*Γ*).

Our principal aim in this paper is to formulate a deformation theory for periodic frameworks that is both natural and close in character to the existing theory for finite frameworks.

We present a mathematical set-up valid in arbitrary dimension *d* and based primarily on notions of algebraic geometry. It may be observed that, by periodicity, we have to contend only with a finite number of constraints and these constraints are expressed by polynomial, in fact quadratic, equations. For full rigour, we have to retain the corresponding defining ideal, since set-theoretical solutions may well have multiplicities that affect the dimensional count at the infinitesimal level. Another important feature is the fact that the lattice of translations representing *Γ* is allowed to vary as the framework deforms.

Historically, reasons to investigate deformation properties of periodic frameworks, that is, flexibility or rigidity, are mostly connected with studies of crystalline materials. In Bragg & Gibbs (1925) and Gibbs (1926), we may find early examples of heuristic considerations, based on a certain one-parameter framework deformation, assisting in the structure determination of *α*, or low quartz, from the solved structure of the more symmetric *β*, or high quartz. Pauling (1930) explicitly remarks that the zeolite frameworks corresponding to natrolite and scapolites have geometric flexibility. The image of a ‘cooperative tilting of the tetrahedra’ is often used when suggesting flexibility properties of silica and zeolite frameworks (Baur 1992), but precise geometric possibilities have not been pursued beyond some intuitively accessible examples.

An abundant family of minerals, known as *perovskites*, is related to deformations of a single type of periodic framework, made of vertex-sharing octahedra (Megaw 1973). Various representative structures are commonly described in terms of Glazer’s ‘tilt systems’ (Glazer 1972; Woodward 1997). We show in §5 that, for the appropriate choice of the periodicity group (figure 7*b*) the deformation space of the framework is three dimensional. On the other hand, the same graph with an enhanced periodicity group (figure 7*a*) becomes *rigid*, that is, there is no non-trivial deformation preserving the indicated periodicity.

Thus, our deformation theory identifies the underlying geometrical space for all transformations preserving the relevant bond (or link) lengths and periodicity.

Displacive phase transitions in certain minerals may well be considered in this perspective. Indeed, the *rigid unit mode model* proposed in a series of papers (Giddy *et al.* 1993; Dove 1997; Bieniok & Hammonds 1998) assumes the preservation of all bond distances implicated in the framework graph. It must be emphasized, however, that this theory operates under additional physical assumptions.

Our strictly geometrical approach can also be applied to infinite periodic trusses and may be contrasted with the engineering methods of Deshpande *et al.* (2001) or the Bloch-wave theory used in Hutchinson & Fleck (2006).

The kinematics developed for finite frameworks after Maxwell (1864) cannot be applied without adaptations to infinite frameworks.

For repetitive structures, the heuristic arguments of Guest & Hutchinson (2003) encounter a ‘paradox’ when comparing conditions of static determinacy with conditions of kinematic determinacy. It is suggested that the solution of the paradox lies in the ‘atypical’ character of periodic structures. This point of view, pursued in Kapko *et al.* (2009) and illustrated in dimension 2 with several examples, finds that periodic structures made of vertex-sharing triangles in dimension 2 or vertex-sharing tetrahedra in dimension 3 should have spaces of infinitesimal flexes of dimension at least 1, respectively, 3. The dependence of flexibility on the choice of *unit cell* is also noticed.

Our approach has no paradox to explain away and obtains the infinitesimal flexes detected by these authors as a simple consequence of parameter counting. The result (theorem 4.2) holds in arbitrary dimension *d* and gives at least infinitesimal flexes for all *tectosimplicial* frameworks, that is, periodic frameworks constructed from the one-skeleta of vertex-sharing *d*-dimensional simplices.

The claim that this infinitesimal information would be enough to infer actual ‘deformation mechanisms’ is not valid in general, as demonstrated by our construction of infinitesimally flexible yet rigid examples in §4. Moreover, the theory presented in this paper answers the questions left open in Kapko *et al.* (2009) by providing, for any given periodic framework (with a specified periodicity), a linear system for computing the space of infinitesimal deformations and a quadratic system for describing the deformation space.

## 2. Definitions and notations

### Definition 2.1.

A *d*-periodic graph is a pair (*G*,*Γ*), where *G*=(*V*,*E*) is a simple infinite graph with vertices *V* , edges *E* and finite degree at every vertex, while is a free Abelian group of automorphisms that has rank *d*, acts without fixed points and has a finite number of vertex orbits.

Thus, we have a finite quotient multigraph *G*/*Γ* since both *V*/*Γ* and *E*/*Γ* are finite. *Γ* is isomorphic with *Z*^{d}, but we do not consider any particular isomorphism as part of the structure. The action of *Γ* on *V* will be indicated as *v*↦*γv*, meaning that the automorphism *γ*∈*Γ* takes the vertex *v*∈*V* to *γv*∈*V* . We may also refer to *Γ* as the *periodicity group* or *period lattice* of *G*, and elements *γ*∈*Γ* may be called *periods* of *G*. Since edges are not oriented, the pairs (*u*,*v*) and (*v*,*u*) will indicate the same edge between vertices *u* and *v*.

The group of translations in the Euclidean space *R*^{d} will be denoted by 𝒯 (*R*^{d}). It is naturally identified with *R*^{d} since any translation has the form
but the distinction in notation will emphasize that we mean the isometry , rather than the vector *t*∈*R*^{d}.

### Definition 2.2.

A periodic placement, or simply placement, of a *d*-periodic graph (*G*,*Γ*) in *R*^{d} is defined by two functions
with *p* assigning points in *R*^{d} to the vertices of *G* and *π*, a faithful representation that is, an injective homomorphism of *Γ* into the group of translations. These two functions must satisfy
2.1
that is, must render commutative the following diagram
for all *γ*∈*Γ*.

If we write *π*(*γ*)(*x*)=*x*+*γ**, with *γ**∈*R*^{d} the corresponding translation vector, the placement condition (2.1) takes the form

### Note

Placements of periodic graphs have been investigated by Delgado-Friedrichs (2005).

A placement that does not allow the endpoints of any edge to have the same image defines a * d-periodic bar-and-joint framework* in

*R*

^{d}, with edges (

*u*,

*v*)∈

*E*corresponding to bars (rigid segments) [

*p*(

*u*),

*p*(

*v*)] and vertices corresponding to (spherical) joints. The length of a bar becomes a positive weight assigned to the edge 2.2 and we obtain a

*weighted*(

*d*-periodic graph*G*,

*Γ*,ℓ) with a particular

*realization*(

*p*,

*π*).

### Definition 2.3.

A realization of the weighted *d*-periodic graph (*G*,*Γ*,ℓ) in *R*^{d} is a placement that induces the given weights.

Realizations that differ by an isometry of *R*^{d} will be considered as the same *configuration*, hence the *configuration space* of (*G*,*Γ*,ℓ) is the *quotient space* of all realizations by the group *E*(*d*) of all isometries of *R*^{d}.

### Commentary

For a finite weighted graph, the realization space is made of the real points of an affine algebraic variety. Infinite weighted graphs may lead to realization spaces that are complicated inverse limits. As a simple example, indicative of the fact that, in this general setting, realization spaces need not be of finite dimension, one may consider the weighted graph given in *R*^{2} by the square tiling with vertices in *Z*^{2}. For every pair of sequences *a*_{n}∈{0,1} and *α*_{n}∈(0,*π*/2), *n*=1,2,…, row ±*n* will deform the squares ‘to the right’ for *a*_{n}=0 or ‘to the left’ for *a*_{n}=1, into rhombi with base angle *α*_{n}. This provides uncountably many subsets of infinite dimension in the realization space. The assumption of a *d*-periodic structure as defined above safeguards a finite-dimensional treatment and the use of differential or algebraic–geometric notions.

Thus, a local one-parameter deformation of the realization (*p*,*π*) may be defined as a differentiable family of realizations (*p*_{t},*π*_{t}), *t*∈(−*ϵ*,*ϵ*), with (*p*_{0},*π*_{0})=(*p*,*π*). It is important to retain the fact that the injective homomorphism may and usually does vary with *t*.

## 3. Periodic frameworks

In this paper, the expression *periodic framework* or, more precisely, *d*-*periodic framework*, will refer to a *connected* weighted *d*-periodic graph (*G*,*Γ*,ℓ) together with a particular realization (*p*,*π*) in *R*^{d}. Obviously, the space of realizations and the configuration space is determined by (*G*,*Γ*,ℓ) alone, but the particular realization will allow us to speak about the rigidity or flexibility of the framework according to the local topology of the configuration space at the corresponding configuration. Since ℓ can be inferred from *p*, a periodic framework will be usually denoted by (*G*,*Γ*,*p*,*π*).

### (a) Realizations

Our main global result is concentrated in the system of *quadratic equations (3.4 )* of our next theorem. It shows that the realization space of a weighted periodic graph has a structure that can be approached via algebraic geometry. For all concepts implicated here, we refer to Shafarevich (1994).

### Theorem 3.1.

*The realization space of a weighted d-periodic graph (G,Γ,ℓ) is naturally described by the real points of a quasi-projective algebraic variety.*

### Proof.

Let *a*=|*V*/*Γ*| and *b*=|*E*/*Γ*| denote the number of vertex and edge orbits of *Γ* in *G*. Let *v*_{1},…,*v*_{a} be a complete set of representatives for *V*/*Γ* and (*v*_{i},*γ*_{β}*v*_{j}) a complete set of representatives of *E*/*Γ*, with *b* appropriate pairs of indices 1≤*i*=*i*(*β*)≤*j*=*j*(*β*)≤*a* and periods *γ*(*β*)∈*Γ*, *β*=1,…,*b*. We put ℓ_{β} for the weight of the corresponding edge.

Let us choose an isomorphism *Γ*≈*Z*^{d} and denote the corresponding standard basis of periods *γ*_{k},*k*=1,…,*d*. We shall use the additive notation in the periodicity group *Γ*. Then, the above periods *γ*(*β*) have unique expressions
3.1

Consider now the Zarisky-open subset of the affine space . By definition, a Zarisky-open subset is the complement of a closed algebraic subvariety. A point (*x*,*M*)=(*x*_{1},…,*x*_{a},*μ*_{1},…,*μ*_{d}) will have coordinates *x*_{i}∈*R*^{d} for placing *v*_{i} and the columns in the matrix *M*∈*GL*(*d*) give period vectors *μ*_{k}∈*R*^{d} for representing the translation vectors for our basis of periods in *Γ*. In accordance with equation (3.1), we put
3.2
where *c*_{β}∈*Z*^{d} is the column vector with components .

The realization space of (*G*,*Γ*,ℓ) may be identified with the solutions in (*R*^{d})^{a}×*GL*(*d*) of the following system of quadratic equations:
3.3
which may also be written in the form
3.4
where the dependence of the indices *i*=*i*(*β*)≤*j*=*j*(*β*) on *β* is recalled.

A solution (*x*,*M*) of this system gives a placement for *v*_{i}, *i*=1,…,*a*, and thereby a placement following the periodicity rule
(with additive notation in *Γ*).

The injective morphism extends the correspondence between the basis of periods *γ*_{k} and the translation vectors *μ*_{k}. This yields the realization (*p*,*π*).

The converse direction from a realization to a solution of the system is obvious. ■

### Corollary 3.2.

*Regarded as a complex quasi-projective variety, the realization space will have all irreducible components of complex dimension greater or equal to ad+d^{2}−b*.

### Proposition 3.3.

*The relation between the numbers a=|V/Γ| and b=|E/Γ| for a d-periodic graph (G,Γ) is given by the formula*
3.5

*where*

*v*_{1},…,*v*_{a}is a complete set of representatives for*V*/*Γ*and*deg*(*v*_{k}) denotes the degree, or valency, of the vertex*v*_{k}, that is, the number of edges emanating from*v*

_{k}.

### Proof.

We consider all the edges emanating from the vertices *v*_{1},…,*v*_{a}. We obtain a set of edges of the form (*v*_{i},*γv*_{j}). Each edge orbit is represented by two edges in our set, since (*v*_{i},*γv*_{j}) and (*γ*^{−1}*v*_{i},*v*_{j})≡(*v*_{j},*γ*^{−1}*v*_{i}) are equivalent under *Γ*. The formula follows. ■

### Remark 3.4.

The advantage of using the notion of quotient graph in crystallography is pointed out by Klee (2004).

As already observed, the isometry group *E*(*d*) of *R*^{d} acts naturally on the realization space and all orbits are of dimension =*dim*(*E*(*d*)).

### Definition 3.5.

We denote by ℜ=ℜ(*G*,*Γ*,ℓ) the realization space and by 𝒞= 𝒞(*G*,*Γ*,ℓ)=ℜ/*E*(*d*) the configuration space of (*G*,*Γ*,ℓ). The realization space will be endowed with the topology induced from the Euclidean space , which is independent of the choices made in theorem 3.1. The configuration space will be endowed with the resulting quotient topology. The connected component of the framework (*G*,*Γ*,*p*,*π*) is called the deformation space of the framework.

A periodic framework (*G*,*Γ*,*p*,*π*) is called rigid when the corresponding configuration is an isolated point in 𝒞, that is, the deformation space is reduced to one point.

### (b) Infinitesimal deformations

For the infinitesimal theory, we consider the ideal generated by the equations in equation (3.3) as part of the structure of the realization space ℜ. We then retain the real tangent space and factor out the trivial infinitesimal motions, that is, those induced from the action of *E*(*d*).

### Definition 3.6.

The space of infinitesimal deformations of a periodic framework (*G*,*Γ*,*p*,*π*) is the real tangent space *T*_{(p,π)}ℜ. The space of infinitesimal flexes is its quotient space by the -dimensional subspace of trivial infinitesimal motions.

When the space of infinitesimal flexes is trivial, the framework will be called infinitesimally rigid.

We note that infinitesimal rigidity implies rigidity, but the converse does not hold. By corollary 3.2, the space of infinitesimal flexes is of dimension at least *ad*+*d*^{2}−*b*− =*ad*−*b*+ .

The tangent space *T*_{(p,π)}ℜ can be described as the kernel of the linear map obtained by differentiating the system (3.3) at the point *x*_{i}=*p*(*v*_{i}), *μ*_{k}= translation vector of *π*(*γ*_{k}). Thus, tangent vectors (*y*_{1},…,*y*_{a},*ν*_{1},…,*ν*_{d}) are solutions of the linear system
3.6
where .

The matrix corresponding to this linear system is called the *rigidity matrix* at the given realization. As observed, its rank is at most *b*− and the dimension of the kernel is at least *ad*−*b*+ .

Recalling that the indices *i*≤*j* implicated in each row of the above system depend on 1≤*β*≤*b*, i.e. *i*=*i*(*β*) and *j*=*j*(*β*), we shall denote by
3.7
the *edge vectors* corresponding in the given realization (*p*,*π*) to the *b* classes of edges in *E*/*Γ*. If we denote by *N* the *d*×*d* matrix with column vectors *ν*_{i}, equations (3.6) take the form
and show that (for an adequate ordering of the entries in *N*) the ‘rigidity matrix’ at the realization (*p*,*π*) has the form
3.8
This is a *b*×(*ad*+*d*^{2}) matrix whose null-space gives the space of infinitesimal deformations of the framework (*G*,*Γ*,*p*,*π*).

A simple consequence of the linear system defining infinitesimal deformations is the following:

### Proposition 3.7.

*Let ( p,π) be a placement in R^{d} of a d-periodic graph (G,Γ) and let be an affine transformation. By composing p with A and π with the linear part dA(0) of A, we obtain a placement denoted A(p,π). Let ℓ and A(ℓ) denote the corresponding weights. Then, the realizations (p,π) for (G,Γ,ℓ) and A(p,π) for (G,Γ,A(ℓ)) have the same dimension of infinitesimal flexes*.

### Proof.

It is enough to verify the case when *A* is a linear transformation *L*. The correspondence of infinitesimal deformations is given by
3.9
■

### Corollary 3.8 (Affine invariance of infinitesimal rigidity).

The realizations (*p*,*π*) for (*G*,*Γ*,ℓ) is infinitesimally rigid if and only if the realization *A*(*p*,*π*) for (*G*,*Γ*,*A*(ℓ)) is infinitesimally rigid.

### Definition 3.9

A minimally rigid *d*-periodic graph (*G*,*Γ*) is a *d*-periodic graph with *b*=*ad*+ that allows for some weights ℓ an infinitesimally rigid realization.

The simplest example of a minimally rigid graph presented in *R*^{d} is obtained from the one-skeleton of a *d*-dimensional simplex translated by the lattice generated by its edge vectors. In this case, *a*=1 and *b*= =*d*+ .

Using the previous example as a rigid background, we may consider now an ordinary minimally rigid graph (Borcea & Streinu 2004) presented in *R*^{d} with *n* vertices, use edges to attach it rigidly at vertices of the background and then repeat this operation by periodicity. The result will be a minimally rigid *d*-periodic graph with *a*=*n*+1 and *b*= +*nd*=(*n*+1)*d*+ . This construction shows that the *periodic rigidity theory* cannot be simpler than the ordinary *rigidity theory* for finite frameworks.

### (c) Periodic self-stresses

In this section, we introduce the notion of periodic self-stress and obtain the periodic analogue of formulas known for finite frameworks.

We use the form (3.8) obtained for the rigidity matrix of the framework (*G*,*Γ*,*p*,*π*) and consider this *b*×(*ad*+*d*^{2}) matrix as made of two matrices of sizes *b*×*ad* and *b*×*d*^{2}
3.10
By transposition, we obtain an (*ad*+*d*^{2})×*b* matrix
3.11

### Definition 3.10.

A periodic self-stress or simply a self-stress for the framework (*G*,*Γ*,*p*,*π*) is an assignment of weights *ω*_{β}∈*R*, *β*=1,…,*b*, to the equivalence classes of edges in *E*/*Γ*, such that the resulting column vector *ω* satisfies the linear system
3.12
Thus, the space of self-stresses is given by the null-space of the transposed rigidity matrix.

As an assignment of weights on the quotient multi-graph *G*/*Γ*, this definition is not affected by the choices of vertex representatives and ordering implicated in producing . This fact can be observed as follows. The first *ad* conditions on *ω*, that is,
3.13
have the invariant meaning that, at any vertex, the sum of the outgoing edge vectors multiplied with the assigned weights vanishes. This is entirely in keeping with the traditional finite case. However, in the periodic case, we have another *d*^{2} conditions, namely
3.14
Nevertheless, it is easy to see that upon changing, say, the representative *v*_{i} with the equivalent *γv*_{i},*γ*∈*Γ*, the system (3.14) only changes by terms already null because of equation (3.13).

The relation between the dimensions and is obtained exactly as in the finite case. With , we have This implies the following proposition.

### Proposition 3.11.

*Let ( G,Γ,p,π) be a d-periodic framework. Then, the dimensions s and m of the spaces of periodic self-stresses and periodic infinitesimal deformations are related by*
3.15

Excluding the trivial infinitesimal deformations, we find
3.16
where *f*=*m*− denotes the dimension of the space of infinitesimal flexes.

### (d) The case *a*=1

We shall discuss here the case , which corresponds with a transitive action of *Γ* on the vertices of *G*, that is *a*=1.

Indeed, the vanishing of means that all edge vectors are periods and our assumption of connectivity for *G* implies that all vertices are equivalent under *Γ*. The vectors *c*_{β} are simply the expression of the edge vectors *e*_{β} with respect to the chosen fundamental periods.

Our interest will focus on minimally rigid periodic graphs in this class. By definition 3.9, we must have
and we want infinitesimally rigid realizations, that is *f*=0. By equation (3.16), this is equivalent with *s*=0. Thus, must have maximal rank.

The columns in this matrix can be written, in a more compact form, as tensors *c*_{β}⊗*e*_{β}∈*R*^{d2}. Either in view of corollary 3.8 or observing directly that rank considerations are not affected by using the basis given by the fundamental periods, infinitesimal rigidity is seen to be equivalent to the linear independence of the *b*= symmetric tensors *c*_{β}⊗*c*_{β}.

It fact, the subspace of symmetric tensors has precisely dimension and our inquiry has a simple geometric answer.

### Theorem 3.12.

*Let (G,Γ) be a d-periodic graph with a=|V/Γ|=1 and b=|E/Γ|=. Then, (G,Γ) is a minimally rigid graph. More precisely, the necessary and sufficient condition for a realization in R*^{d} *to be infinitesimally rigid is that the directions of the edges do not lie on one and the same quadric.*

### Proof.

The argument is straightforward in the language of projective geometry. The directions of the edges are considered as projective points with homogeneous coordinates *c*_{β}∈*P*_{d−1}. The tensors *c*_{β}⊗*c*_{β} then represent the images of these points under the quadratic Veronese embedding (Shafarevich 1994)

The linear dependence of *c*_{β}⊗*c*_{β} would mean that all these points lie on a hyperplane in the target space, which would mean that all points *c*_{β} lie on a quadric in the source space. ■

### Commentary

This result may be compared with inquiries conducted from an engineering point of view in Deshpande *et al.* (2001). We re-emphasize here that our theory is for *pairs* (*G*,*Γ*) and the deformations under consideration are only those preserving the *Γ*-periodicity of the structure. Upon relaxing the periodicity requirements by replacing the lattice *Γ* with a sublattice of finite index , the deformation properties of may be significantly different from those of (*G*,*Γ*). Obviously, there is an inclusion of realization spaces , but minimal rigidity is lost upon transition to a proper sublattice of periods. Our section on frameworks modelled on perovskites will address specifically such issues.

## 4. Periodic frameworks modelled on silica and zeolites

The frameworks considered in this section are modelled on *tectosilicates* , which include the silica polymorphs cristobalite and quartz and the aluminosilicates known as zeolites (Megaw 1973). The oxygen atoms in these crystalline structures are organized as vertex-sharing tetrahedra, and this is the feature we generalize to arbitrary dimension *d*.

### Definition 4.1.

In what follows, a *tectosimplicial framework* in *R*^{d} will be a particular type of *d*-periodic framework, namely one that may be envisaged as made of one-skeleta of *d*-dimensional simplices in such a way that any vertex is common to exactly two simplices. The corresponding graphs will be 2*d*-regular, that is, with all vertices of degree 2*d*.

Abstractly, the periodic graph (*G*,*Γ*) of a tectosimplicial framework is obtained from a *d*-periodic graph (*S*,*Γ*) with all vertices of degree (*d*+1), which may be called the ‘graph of simplices’. The vertices of *G* are taken to be the edges of *S* and two vertices in *G* form an edge if and only if, as edges of *S*, they have a vertex in common.

For this class of frameworks, the continuous parameters corresponding to trivial deformations can be conveniently eliminated by fixing or pinning down the position of a *marked* simplex, say *σ*. In other words, *p* is prescribed on the marked simplex, and the space of realizations (*G*,*Γ*,ℓ,*p*.*π*) with such *p* will be called the *marked realization space*. Obviously, it will be a closed subvariety of the full realization space of codimension since obtained by adding as many independent equations to the system (3.4) defining ℜ. Since trivial deformations are eliminated, the marked realization space ℜ_{σ} becomes isomorphic with the configuration space .

As a first ‘example’, we describe a tectosimplicial framework in *R*^{d} that may be considered as a generalized *d*-dimensional version of the Kagome framework in the plane. For illustrations and various discussions of the equilateral planar case, we refer to Guest & Hutchinson (2003), Hutchinson & Fleck (2006) and Kapko *et al.* (2009).

Our illustration for *d*=3, given in figure 1, will emphasize the periodicity under consideration and point to the source of the geometric flexibility inherent in the structure.

We start with a simplex *σ* with vertices at 0,*s*_{1},…,*s*_{d}, joined at the origin by a simplex *τ* with vertices at 0,*t*_{1},…,*t*_{d}. We denote by *S* the matrix with column vectors *s*_{k} and by *T* the matrix with column vectors *t*_{k}. We assume *det*(*T*−*S*)≠0 and consider as fundamental periods the vectors *λ*_{k}=*t*_{k}−*s*_{k}, *k*=1,…,*d*. We define *Γ* to be the group generated by these vectors under addition. When we translate our pair of simplices by all elements of *Γ*, we obtain a tectosimplicial framework made of the one-skeleta of all these translated simplices. The underlying infinite graph *G* has and our *d*-periodic graph (*G*,*Γ*) comes with the tautological placement. We shall consider the weights ℓ determined by this placement. We note that with *a*=|*V*/*Γ*|=*d*+1 and *b*=|*E*/*Γ*|=2 =*ad*, corollary 3.2 predicts at least infinitesimal flexes.

First of all, we may consider the simplex *σ* in the role of the *marked* simplex. The marked realization space ℜ_{σ}=ℜ(*G*,*Γ*,ℓ)_{σ} and thereby the configuration space are fairly accessible geometrically and will illustrate the results of the algebraic description in theorem 3.1. Indeed, in any marked realization, the simplex *τ* must be placed as *Mτ* for some orthogonal transformation *M*∈*O*(*d*). The fundamental periods will then be represented by the vectors *μ*_{k}=*Mt*_{k}−*s*_{k}, *k*=1,…,*d*. Since they must be independent, we have the restriction *det*(*MT*−*S*)≠0. We conclude that the configuration space may be identified with *O*(*d*)∖{*M*: *det*(*MT*−*S*)=0}, a -dimensional manifold.

For *d*=3, the most symmetric case, when *σ*=−*τ* is a regular simplex, gives the ideal structure of high cristobalite (Megaw 1973).

The main result of this section is as follows.

### Theorem 4.2.

*Any tectosimplicial framework in R*^{d} *has at least* *infinitesimal flexes. However, rigid examples can be constructed.*

### Proof.

The lower bound on infinitesimal flexes is a simple consequence of the theory presented in the previous sections and the relation *ad*=*b* that holds for all tectosimplicial frameworks in *R*^{d}. Indeed, all vertices have degree 2*d*, and equation (3.5) gives the stated relation. ■

### Remark 4.3.

If the dimension for infinitesimal flexes is *exactly* , the dimension for infinitesimal deformations is exactly *d*^{2}, and by the implicit function theorem, the realization space will be smooth, of dimension *d*^{2}, in some neighbourhood of the given realization.

The remaining task is the construction of rigid examples. In fact, we shall describe tectosimplicial frameworks made of congruent simplices and rigid not only for the initially specified choice of period lattice, but also for any choice of sublattice of periods (of finite index).

Our first example is in dimension *d*=2 and is based on a simple idea for turning a finite flexible linkage into a rigid one, albeit still infinitesimally flexible. This is illustrated in figure 2.

The infinite periodic setting is illustrated in figure 3. It should be conceived as made of equilateral triangles of three different sizes, corresponding to edges of length 1, 2 and 8. Four small triangles and two medium triangles are aligned along an edge of a big triangle, which is itself an edge of the framework. In figure 3, this is the edge *AC* of triangle *ABC*. Thus, the pieces delineated in black or grey are rigid: indeed, if we assume that the triangle *ABC* is glued to the plane, then the two medium and four small triangles underneath edge *AC* remain in place, because the sum of their edges equals the length of *AC*. Since two rigid pieces with two points in common form a rigid assembly, the entire periodic framework is rigid as well. The lattice of periods is generated by the vectors defined by the edges of the highlighted fundamental parallelogram.

We shall elaborate on this idea and obtain rigid examples made of *congruent* regular triangles with no overlapping edges. We start with a subdivision lemma.

### Lemma 4.4.

*A regular d-dimensional simplex with edge length 2^{n} can be obtained as a rigid assemblage of (d+1)^{n} regular simplices of unit edge length, with all vertices other than the original ones belonging to exactly two simplices*.

### Proof.

Figure 4 illustrates the planar case. In arbitrary dimension *d*, it suffices to prove the case *n*=1. When a regular simplex is truncated at all vertices by hyperplanes passing through the midpoints of the edges incident to each given vertex, the remaining polytope is a standard second hypersimplex. The infinitesimal rigidity, and thus the rigidity of the one-skeleton of this structure, was proved by Borcea (2008). When the one-skeleta of the (*d*+1)-removed simplices are put back, one obtains a rigid regular simplex as claimed in the lemma. ■

Using lemma 4.4, we can turn the example given in figure 3 into one made entirely of congruent equilateral triangles. In order to eliminate overlapping along edges, we substitute the basic pieces as described in figure 5.

First we replace the basic triangle *ABC*, with an isosceles triangle, keeping the scaling factors of 1/2 and 1/4 for the smaller parts. In figure 5*b*, the equal edges of the isosceles triangles are solid and the third edge is dashed. Then, the isosceles triangle will be *replaced* by the framework (made of thick black edges) from figure 5*c*, made of two equilateral triangles with edge length 2 and one equilateral triangle of edge length 1. This structure is equivalent, for rigidity purposes, to the isosceles triangle *ABC*. Finally, applying lemma 4.4 yields a *rigid planar periodic framework* made of congruent equilateral triangles (figure 6).

The example in figure 3 can be adapted in arbitrary dimension *d* as follows.

We start with a regular simplex with edge length 8 and vertices labelled *v*_{0},…,*v*_{d}. We use the facet [*v*_{1},…,*v*_{d}] as a ‘sole’. We imagine this facet as made of *d* regular simplices of dimension *d*−1 and edge length 4, as in lemma 4.4. Then, each of these simplices, except the one with vertex *v*_{1}, is imagined as made of *d* regular simplices of dimension *d*−1 and edge length 2. The simplex with *v*_{1} as vertex is imagined, in the same way, as made of *d*^{2} regular simplices of dimension *d*−1 and unit edge length. Over this pattern on the sole, we consider now, on the opposite side of vertex *v*_{0}, vertices completing the previously described (*d*−1)-dimensional regular simplices (with edge length 2 or 1) to *d*-dimensional regular simplices. Figuratively speaking, we have placed ‘crampons’ on the sole.

At this stage, we use the hyperplane defined by all the new vertices of the small simplices and produce the reflected copy of our structure with respect to this hyperplane. The reflected copy will be subject to an orthogonal transformation that preserves the reflecting hyperplane and has the effect of the cyclic permutation (12…*d*) on the assemblage of small reflected simplices. This construction, with one piece in black and the congruent copy in grey, provides a *d*-dimensional analogue of one black and grey pair in figure 3.

The lattice of periods will be the one generated by the vectors connecting *v*_{0} and the centres of the full *d*-dimensional size four black simplices (other than the one with *v*_{1} as vertex) with their grey counterparts. With that, the rigidity assessment is very simple. The black piece, as well as the grey copy, is obviously rigid. Their assembly is rigid since the small crampons fix one piece to the other. The translates corresponding to pairs of vertices from medium simplices (i.e. edge length 2) attach rigidly one rigid black–grey pair to another. Finally, these infinite ‘sheets’ assemble in a unique way.

### Commentary

The lower bound on infinitesimal flexes in our theorem is in agreement with the findings of Kapko *et al.* (2009), where, based on structural mechanics heuristic arguments (see also Guest & Hutchinson 2003), a space of infinitesimal flexes of dimension at least 1 for *d*=2 and at least 3 for *d*=3 is recognized for frameworks modelled on zeolites. However, our rigid examples show that the presence of actual ‘deformation mechanisms’ cannot be inferred from the infinitesimal information alone. From our point of view, the property of being *isostatic*, i.e. minimally rigid, is relative to a specified periodicity and zeolite frameworks cannot be isostatic for any choice of periodicity lattice.

## 5. Periodic frameworks modelled on perovskites

The essential features of the periodic frameworks discussed in this section are delineated in figure 7. The octahedra are assumed centrally symmetric and congruent as labelled. By proposition 3.7, for all infinitesimal considerations, we may assume in fact that the octahedra are regular.

The infinite graph *G* under consideration is the same for the two presentations and should be visualized (in either case) by placing copies of the octahedra for all translations generated by integral linear combinations of the marked translations.

Note that the five periods depicted on the right generate a sublattice of index 2 in the lattice *Γ* generated by the three periods depicted on the left. Moreover, the two dependencies
5.1
are not affected by a relative rotation of the two octahedra joined at *F*_{1}=*E*_{2}.

The infinitesimal rigidity of the left presentation of (*G*,*Γ*) follows immediately from the infinitesimal rigidity of the convex octahedron.

The three-dimensional deformation space of the right presentation of may be described via the rotation group *SO*(3). Except for degenerate positions, every posture with *F*_{1}=*E*_{2} of the second octahedron relative to the first (fixed) octahedron defines at the same time a deformation of the framework (with the marked representation for ) and a rotation (taking the second octahedron to the first).

## Acknowledgements

The interest of the authors in zeolite frameworks was stimulated by discussions with Michael Treacy, Simon Guest, Michael Thorpe and Vitaly Kapko at the workshop ‘Geometric constraints with applications to CAD and biology’, organized by the second author at Bellairs Research Institute, in January 2009.

Research on this paper was sponsored by a DARPA ‘23 Mathematical Challenges’ grant. All statements, findings or conclusions contained in this publication are those of the authors and do not necessarily reflect the position or policy of the Government. No official endorsement should be inferred.

## Footnotes

- Received December 27, 2009.
- Accepted February 19, 2010.

- © 2010 The Royal Society