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symmetry 2014, 6(2), 308-328; doi:10.3390/sym6020308

Article
The Almost Periodic Rigidity of Crystallographic Bar-Joint Frameworks
Ghada Badri , Derek Kitson * and Stephen C. Power
Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK; E-Mails: g.badri@lancaster.ac.uk (G.B.); s.power@lancaster.ac.uk (S.C.P.)
*
Author to whom correspondence should be addressed; E-Mail: d.kitson@lancaster.ac.uk; Tel.: +44-1524-59-4914.
Received: 25 February 2014; in revised form: 11 April 2014 / Accepted: 15 April 2014 /
Published: 24 April 2014

Abstract

: A crystallographic bar-joint framework, Symmetry 06 00308i1 in ℝd, is shown to be almost periodically infinitesimally rigid if and only if it is strictly periodically infinitesimally rigid and the rigid unit mode (RUM) spectrum, Ω( Symmetry 06 00308i1), is a singleton. Moreover, the almost periodic infinitesimal flexes of Symmetry 06 00308i1 are characterised in terms of a matrix-valued function, Φ Symmetry 06 00308i1 (z), on the d-torus, Symmetry 06 00308i2d, determined by a full rank translation symmetry group and an associated motif of joints and bars.
Keywords:
crystal framework; infinitesimal rigidity; almost periodic functions

1. Introduction

The rigidity of a crystallographic bar-joint framework, Symmetry 06 00308i1, in the Euclidean spaces, ℝd, with respect to periodic first order flexes is determined by a finite matrix, the associated periodic rigidity matrix. For essentially generic frameworks of this type in two dimensions, there is a deeper combinatorial characterisation, which is a counterpart of Laman's characterisation of the infinitesimal rigidity of generic placements of finite graphs in the plane. See Ross [1]. For related results and characterisations of other forms of periodic infinitesimal rigidity, see Borcea and Streinu [2,3], Connelly, Shen and Smith [4], Malestein and Theran [5], Owen and Power [6], Power [7,8] and Ross, Schulze and Whiteley [9].

There is also extensive literature in condensed matter physics concerning the nature and multiplicity of low-energy oscillations and rigid unit modes (RUMs) for material crystals in three dimensions. In this case, Bloch's theorem applies, and the excitation modes are periodic modulo a phase factor. The set of phase factors, or, equivalently, the set of reduced wave vectors for the modes, provides what may be viewed as the RUM spectrum of the crystal. See Dove et al. [10], Giddy et al. [11] and Wegner [12], for example. In Owen and Power [6] and Power [8], the RUM spectrum was formalised in mathematical terms as a subset, Ω( Symmetry 06 00308i1), of the d-torus, Symmetry 06 00308i2d, or as an equivalent subset of [0, 1)d, which arises from a choice of translation group Symmetry 06 00308i3. This set of multi-phases is determined by a matrix-valued function, Φ Symmetry 06 00308i1(z), on Symmetry 06 00308i2d with the value at z = 1̂ = (1, 1, … , 1) providing the corresponding periodic rigidity matrix for Symmetry 06 00308i3.

In the present article, we move beyond periodicity and consider infinitesimal flexes of a crystallographic bar-joint framework, which are almost periodic in the classical sense of Bohr. Such flexes are independent of any choice of translation group and so are intrinsic to Symmetry 06 00308i1 as an infinite bar-joint framework. It is shown that Symmetry 06 00308i1 is almost periodically infinitesimally rigid if and only if for some choice of translation group, it is periodically infinitesimally rigid, and the corresponding RUM spectrum is the minimal set {1̂}. More generally, we show how the almost periodic infinitesimal flexes of Symmetry 06 00308i1 are determined in terms of the matrix function, Φ Symmetry 06 00308i1(z).

An ongoing interest in the analysis of low-energy modes in material science is to quantify the implications of symmetry and local geometry for the set of RUM wave vectors. See, for example, Kapko et al. [13], where the phenomenon of extensible flexibility is related to the maximal symmetry and minimal density forms of an idealised zeolite crystal framework. Here, the term, extensive flexibility, corresponds to a maximal rigid unit mode spectrum Ω( Symmetry 06 00308i1) = Symmetry 06 00308i23. We show that for crystal frameworks whose RUM spectrum decomposes into a finite union of linear components, there is a corresponding vector space decomposition of the almost periodic flex space. The flexes in these subspaces are periodic in specific directions associated with certain symmetries of the crystallographic point group.

In Section 4, we give a small gallery of crystal frameworks, which display a variety of periodic and almost periodic flexibility properties.

2. Crystal Frameworks and the RUM Spectrum

A bar-joint framework in the Euclidean space, ℝd, is a pair consisting of a simple undirected graph G = (V, E) and an injective map p : V → ℝd. A (real) infinitesimal flex of (G, p) is a field of velocities, or velocity vectors, u(v), assigned to the joints, p(v), such that, for every edge, vwE,

( p ( v ) p ( w ) ) ( u ( v ) u ( w ) ) = 0
If the above condition holds for all pairs v,wV, then u is a trivial infinitesimal flex of (G, p). For convenience, we let p(E) denote the set of open line segments pe = (p(v), p(w)) with e = vwE.

Definition 1

A crystal framework, Symmetry 06 00308i1, is a bar-joint framework (G, p) for which there exist finite subsets, Fvp(V) and Fep(E), and a full rank translation group, Symmetry 06 00308i3, such that:

p ( V ) = { T ( p v ) : p v F v , T T } p ( E ) = { T ( p e ) : p e F e , T T }

The pair (Fv, Fe) is called a motif for Symmetry 06 00308i1. The elements of Fv are called motif vertices, and the elements of Fe are called motif edges. The translation group, Symmetry 06 00308i3, is necessarily of the form:

T = { j = 1 d k j a j : k j }
where a1, a2,…, ad are linearly independent vectors in ℝd. The translation x x + j = 1 d k j a j is denoted Tk for each k = (k1, … , kd) ∈ ℤd. For each motif vertex, p(v) ∈ Fv, and each k ∈ ℤd, we denote by (v, k) the unique vertex for which p(v, k) = Tk(p(v)). For each motif edge, peFe, with pe = (p(v,l),p(w,m)), we let (e,k) denote the unique edge for which p(e,k) = (p(v, l+ k),p(w,m+k)). In Section 4, we provide a number of illustrative examples of crystal frameworks with natural choices of the motif and translation group.

In the consideration of infinitesimal rigidity relative to general periodic flexes, or almost periodic flexes, it is convenient and natural to consider complex velocity vectors u : Fv × ℤd → ℂd. Indeed, such vectors are infinitesimal flexes if and only if their real and imaginary parts are infinitesimal flexes. A velocity vector u : Fv × ℤd → ℂd for Symmetry 06 00308i1 is said to be

  • local if u(v, k) ≠ 0 for at least one and at most finitely many (v, k) ∈ Fv × ℤd;

  • strictly periodic if u(v, k) = u(v, 0) for all (v, k) ∈ Fv × ℤd;

  • supercell periodic if u(v, k) = u(v, 0) for each motif vertex vFv and for all k in a full rank subgroup of ℤd of the form m1ℤ × ⋯ × mdℤ.

2.1. The Symbol Function and Rigidity Matrix

We now define the symbol function, Φ Symmetry 06 00308i1(z), of a crystal framework and the rigidity matrix, R( Symmetry 06 00308i1), from which it is derived. The d-dimensional torus is Symmetry 06 00308i2d = {(z1, …, zd) ∈ ℂd : |z1| = ⋯ = |zd| = 1}. For kZd, the associated monomial function, f : Symmetry 06 00308i2d → ℂ, given by f ( z 1 , , z d ) = z 1 k 1 z d k d, is written simply as zk.

Definition 2

Let Symmetry 06 00308i1 be a crystal framework ind with motif (Fv, Fe), and for e = (v, k)(w, l) ∈ Fe, let p(e) = p(v, k) − p(w, l). Then, Φ Symmetry 06 00308i1(z) is a matrix-valued function on Symmetry 06 00308i2d, which assigns a finite |Fe| × d|Fv| matrix to each z Symmetry 06 00308i2d. The rows of Φ Symmetry 06 00308i1(z) are labelled by the edges of Fe, and the columns are labelled by the vertex-coordinate pairs in Fv × {1,…, d}. The row for an edge e = (v, k)(w, l) with vw takes the form,

e [ 0 0 p ( e ) z ¯ k v 0 0 p ( e ) z ¯ l w 0 0 ]
while if v = w, it takes the form,
e [ 0 0 p ( e ) ( z ¯ k z ¯ l ) v 0 0 ]

Definition 3

Let Symmetry 06 00308i1 be a crystal framework ind with motif (Fv, Fe), and for e = vwFe, let p(e) = p(v) − p(w). Then, R( Symmetry 06 00308i1) is the infinite matrix, whose rows are labelled by the edges, (e, k) in Fe × ℤd, and whose columns are labelled by the pairs, (v, k) in Fv × ℤd. The row for an edge (e, k) = (v, l + k)(w, m + k), with e = (v, l)(w, m) ∈ Fe, takes the form:

( e , k ) [ 0 p ( e ) ( v , l + k ) 0 0 p ( e ) ( w , m + k ) 0 ]

It follows from this definition that a velocity vector u : Fv × ℤd → ℝd is an infinitesimal flex for Symmetry 06 00308i1 if and only if R( Symmetry 06 00308i1) u = 0.

2.2. The RUM Spectrum

Let ω = (w1,…, ωd) ∈ Symmetry 06 00308i2d. A velocity vector u : Fv × ℤd → ℂd is said to be ω-phase-periodic if u(v, k) = ωku(v, 0) for all (v, k) ∈ Fv × ℤd. Here, ωk is the product ω 1 k 1 ω d k d. We also write u = beω for this vector, where b is the vector (u(v, 0))v in ℂd|Fv| and eω is the multi-sequence (ωk)k∈ℤd. We refer to ω as a multi-phase for u. The velocity vector u is said to be phase-periodic if it is ω-phase-periodic for some multi-phase ω ∈ Symmetry 06 00308i2d.

To explain this terminology, note that a multi-phase is a d-tuple of unimodular complex numbers ω = (e1, … , ed). The k-th entry corresponds to the phase variation of the velocity vector, u, in the direction of the k-th lattice direction. The d-tuple of real numbers θ = (θ1, … , θd) can be taken to have entries in the interval [0, 2π), in which case, θ can be considered as the reduced wave vector of the velocity vector, u (see [8] for further remarks on this dynamical connection).

The following theorem is given in [8,14].

Theorem 1

Let Symmetry 06 00308i1 be a crystal framework ind; let ω Symmetry 06 00308i2d and let u = beω, with bFv × ℂd, be a ω-phase-periodic velocity vector. Then, the following conditions are equivalent:

(i)

R( Symmetry 06 00308i1)u = 0;

(ii)

Φ Symmetry 06 00308i1(ω̅)b = 0.

The RUM spectrum, ( Symmetry 06 00308i1), of Symmetry 06 00308i1 is defined to be the set of multi-phases, ω, for which there exists a nonzero phase-periodic infinitesimal flex for Symmetry 06 00308i1 as an infinite-bar-joint framework, or, equivalently, as the set of multi-phases for which the rank of the matrix, Φ Symmetry 06 00308i1(ω̅), is less than d|Fv|.

Note that the strictly periodic velocity vectors for Symmetry 06 00308i1 are, by definition, precisely the phase-periodic velocity vectors, which have multi-phase 1̂ = (1,…, 1) ∈ Symmetry 06 00308i2d. Evidently, the RUM spectrum must always contain the point, 1̂ ∈ Symmetry 06 00308i2d, as every constant velocity vector, u, is a phase-periodic infinitesimal flex with multi-phase 1̂. Thus, the RUM spectrum of Symmetry 06 00308i1 is a singleton (i.e., it contains only the single point, 1̂ if and only if every phase-periodic infinitesimal flex for Symmetry 06 00308i1 is strictly periodic.

Corollary 1

Let Symmetry 06 00308i1 be a crystal framework ind with translation group Symmetry 06 00308i3. Then, the following statements are equivalent:

(i)

The Symmetry 06 00308i3-periodic real infinitesimal flexes of Symmetry 06 00308i1 are trivial;

(ii)

The Symmetry 06 00308i3-periodic complex infinitesimal flexes of Symmetry 06 00308i1 are trivial;

(iii)

The periodic rigidity matrix Φ Symmetry 06 00308i1(1,…, 1) has rank equal to d|Fv| − d.

We may view phase-periodic infinitesimal flexes of a crystal framework as the pure flexes with (possibly incommensurate) oscillatory variation relative to the periodicity lattice. Note that finite linear combinations of such flexes need not be phase-periodic, but will in fact be almost periodic in the sense of Definition 4 below Our main theorem may be viewed as an almost periodic flex variant of the general principle that general motions can be synthesised by combinations of purely oscillatory motions.

3. Almost Periodic Rigidity

In this section, we first outline the proof of the fundamental approximation theorem for uniformly almost periodic functions and its counterpart for almost periodic sequences. These ensure that a function (or sequence), which is almost periodic in the sense of Bohr, is approximable by trigonometric functions (or sequences) that are obtained in an explicit manner from convolution with Bochner-Fejér kernels. A convenient self-contained exposition of this fact for function approximation is given in Partington [15]. (See also [16,17]). The direct arguments there can be extended to almost periodic vector-valued functions on ℤd, and this embraces the setting of velocity fields relevant to the almost periodic rigidity of crystal frameworks (see Definition 4). The constructive approximation theorem that we require is given in Theorem 2. This theorem together with Lemmas 1 and 2 leads to the almost periodic rigidity theorem.

3.1. Almost Periodic Sequences

First, we recall the classical theory for univariable functions on ℝ. The Fejér kernel functions are given by:

K n ( x ) = | m | n + 1 ( 1 | m | n + 1 ) e imx , x
The positivity of the Kn and their approximate identity property under convolution with a continuous periodic functions feature in a standard proof that a continuous 2π-periodic function, f(x), on the real line is uniformly approximable by the explicit trigonometric functions:
g n ( x ) = 0 2 π f ( s ) K n ( s x ) d s 2 π
Almost periodic functions on the real line in the sense of Bohr are similarly uniformly approximable by an explicit sequence of trigonometric polynomials that are determined by convolution with certain Bochner-Fejér kernels.

Note first that it is elementary that the functions, gn(x), has the form gn(x) = [f, Rx Kn], where [·, ·] is the mean inner product:

[ f 1 , f 2 ] = lim T 1 2 T T T f 1 ( s ) f 2 ( s ) ¯ d s
and RxKn(s) = Kn(sx). It is classical that for a function, f(x) in AP(ℝ, ℂ), there is a sequence of Bochner-Fejér kernels K n , which provide, by the same formula, a uniformly approximating sequence of trigonometric polynomials, gn(x). In this case, the frequencies, λ, in the nonzero terms, aeiλx, of these approximants appear in a countable set derived (by rational division) from the spectrum, Λ(f), of f defined by:
Λ ( f ) = { λ : [ f ( x ) , e i λ x ] 0 }
It follows from a Parceval inequality for almost periodic functions that this spectrum, which we refer to as the Bohr spectrum, is a well-defined finite or countable set.

Similar considerations apply to the space, AP(ℤ, ℂ), of almost periodic sequences. The approximants are general trigonometric sequences, that is, sequences, (h(k))k∈ℤ, that have a finite sum form:

h ( k ) = ω = e i λ : λ F a λ ω k
so that, in our earlier notation,
h = ω = e i λ : λ F a λ e ω
where F is a finite subset of ℝ. The Bohr spectrum of a sequence, h in AP(ℤ, ℂ), is defined to be the set:
Λ ( h ) = { λ : [ h , e ω ] 0 , for ω = e i λ }
where
[ h 1 , h 2 ] = lim N 1 2 N + 1 | k | N h 1 ( k ) h 2 ( k ) ¯
This spectrum of ω values is now a subset of Symmetry 06 00308i2. Partial counterparts of the Fejér kernel functions, Kn (x), are given by the Fejér sequences:
K ( n , λ ) = | m | n + 1 ( 1 | m | n + 1 ) e ω m
associated with a single frequency ω = eiλ. In particular:
K ( n , λ ) ( k ) = | m | n + 1 ( 1 | m | n + 1 ) ω m k

The following fundamental approximation theorem indicates the explicit construction of the Bochner-Fejér kernels K n for h as coordinate-wise products of appropriate Fejér sequences.

Theorem 2

Let h be a sequence in AP (ℤ, ℂ); let α1,α2, … be a maximal subset of the Bohr spectrum Λ(h), which is independent over ℚ, and for n = 1, 2,… let:

K n = k = 1 n K ( n . n ! 1 , α k / n ! )
Then, h is the uniform limit of the sequence g1, g2, … of trigonometric sequences in AP(ℤ, ℂ) given by:
g n ( k ) = [ h , R k K n ] , k

The main ingredient in the proof of the theorem is that the Bohr spectrum is nonempty if h ≠ 0, and the arguments for this depend on the equivalence of Bohr almost periodicity with the Bochner condition that the set of translates of h is precompact for the uniform norm (see [15]).

The arguments leading to Theorem 2 can be generalised to obtain an exact counterpart theorem for AP(ℤd, ℂr). The approximating trigonometric sequences, g, now have a finite sum form:

g = ω F T d a ω e ω
where aω ∈ ℂr and eω, for ω = (ω1,…, ωd) in Symmetry 06 00308i2d, is the pure frequency sequence eω1 ⊗ ⋯ ⊗ eωd in AP(ℤd, ℂ) with:
e ω ( k ) = ω 1 k 1 ω d k d
The Bohr spectrum, Λ(h), is similarly defined and is a countable subset of points ω in Symmetry 06 00308i2d, which we freely identify with a countable subset of points, λ, in [0, 2π)d. For notational convenience, we state the general theorem only in the case d = 2.

The metric that is appropriate in our context for the approximation of velocity fields is the uniform metric or norm ∥ · ∥; for velocity fields h, h′, we have:

h h = sup k , κ { h ( k , κ ) h ( k , κ ) 2 }

Theorem 3

Let h be a sequence in AP(ℤ2, ℂr), let (α1, β1), (α2, β2),… be a maximal subset of Λ (h) ⊂ [0,2π)2, which is independent over ℚ, and for n = 1,2,… let:

K n ( 2 ) = ( k = 1 n K ( n . n ! 1 , α k / n ! ) ) ( k = 1 n K ( n . n ! 1 , β k / n ! ) ) A P ( 2 , )
Then, h is the uniform limit of the sequence g1, g2,… of trigonometric sequences in AP(ℤ2,ℂr) given by:
g n ( k ) = [ h , R k K n ( 2 ) ] 2 , k 2

Note that here, [·, ·]2 is the natural well-defined sesquilinear map from AP(ℤ2, ℂr) × AP(ℤ2, ℂ) to ℂr.

We remark that the theory of uniformly almost periodic multi-variable functions on ℝd and on ℤd is part of the abstract theory of almost periodic functions on locally compact abelian groups, due to Bochner and von Neumann [18]. For further details, see also Levitan and Zhikov [19], Loomis [20] and Shubin[21].

3.2. Almost Periodic Rigidity

We now characterise when a crystal framework admits no nontrivial almost periodic infinitesimal flexes, and in this case, we say that it is almost periodically rigid. This is evidently a form of rigidity, which is independent of any choice of translation group.

For a crystal framework, Symmetry 06 00308i1, with full rank translation group Symmetry 06 00308i3, various linear transformations may be associated with the rigidity matrix, R( Symmetry 06 00308i1). These transformations are restrictions of the induced linear transformation:

R ( C ) : d × | F v | d d × | F e |
where ℂd×|Fv| ⊗ ℂd is the vector space of all velocity fields on p(V), that is, the space of functions h : ℤd × |Fv| → ℂd. The codomain of R( Symmetry 06 00308i1) is the vector space of complex-valued functions on the set of edges.

The right shift operators on the domain and codomain of R( Symmetry 06 00308i1) for the integral vector, l in ℤd, are denoted by R l V and R l E, respectively. Here, the right shift of a sequence, h(k, κ), by l is the sequence, h(kl, κ). We note that:

R ( C ) R l V = R l E R ( C )

Definition 4

Let h : ℤd × Fv → ℂd be a velocity field.

  • An integral vector, l ind, is an ε-translation vector for h if R l V ( h ) h < .

  • The velocity field, h, is Bohr almost periodic if for every ∊ > 0, the set of ∊-translation vectors, l, is relatively dense ind.

Lemma 1

Let g be the vector-valued trigonometric multi-sequence with finite sum representation:

g = ω F T d a ω e ω
with nonzero coefficients, aω ind|Fv|. If g is a nonzero infinitesimal flex for Symmetry 06 00308i1, then each component sequence aωeω is a nonzero ω-phase-periodic infinitesimal flex.

Proof

For ω = (ω1, … , ωd) ∈ Symmetry 06 00308i2d and N ∈ ℕ, let RV (ω, N) be the linear map on the normed space ℓ(ℤd × Fv, ℂd) given in terms of the right shift operators, R k V, k ∈ ℤd, by:

R V ( ω , N ) = 1 ( N + 1 ) d k : 0 k i N ω ¯ k R k V
Similarly, let RE(ω, N) be the linear map on ℓ(ℤd × Fe, ℂ) given by,
R E ( ω , N ) = 1 ( N + 1 ) d k : 0 k i N ω ¯ k R k E
The sequence R (ω′, N)(aωeω) converges uniformly to aωeω, when ω′ = ω, and to the zero sequence otherwise, since for each (l, vκ) ∈ ℤd × Fv,
lim N R V ( ω , N ) ( a ω e ω ) ( l , v κ ) = lim N 1 ( N + 1 ) d k : 0 k i N ω ¯ k R k V ( a ω e ω ) ( l , v κ ) = lim N 1 ( N + 1 ) d k : 0 k i N ω ¯ k ω l + k a ω = ( lim N 1 ( N + 1 ) d k : 0 k i N ω ¯ k ω k ) ω l a ω
and
lim N 1 ( N + 1 ) d k : 0 k i N ω ¯ k ω k = { 1 if ω = ω 0 if ω ω
Note also that R( Symmetry 06 00308i1) is a bounded linear transformation from ℓ(ℤd × Fv, ℂd) to ℓ(ℤd × Fe, ℂ), which commutes with the right shift operators. Thus, if ω is a multi-frequency for g, then:
R ( C ) ( a ω e ω ) = lim N R ( C ) ( R V ( ω , N ) g ) = lim N R E ( ω , N ) R ( C ) g = 0

Lemma 2

Let K be a trigonometric polynomial in AP(ℤd × Fv, ℂ), and let h be an infinitesimal flex for Symmetry 06 00308i1 in AP(ℤd × Fv, ℂd). Then, the mean convolution multi-sequence g : ℤd × Fv → ℂd given by g(k, vκ) = [h, Rk(K)]ℤd is an infinitesimal flex for Symmetry 06 00308i1.

Proof

By linearity, it suffices to assume that K is the elementary multi-sequence, eω, so that K(k) = ωk for k ∈ ℤd. Then, g is the uniform limit of the sequence (gN), where:

g N ( k , v κ ) = 1 ( N + 1 ) d 1 s i N ω ¯ s ( R s h ) ( k , v κ )
To see this, note that the convergence is uniform if h is a trigonometric sequence. Since the linear maps, hgN, are contractive for the uniform norm, uniform convergence holds for a general almost periodic velocity sequence. Thus, since the vector space of infinitesimal flexes is invariant under translation, it follows that R( Symmetry 06 00308i1)gN = 0 for each N and, hence, that R( Symmetry 06 00308i1)g = 0.

The following theorem shows that a crystal framework is almost periodically rigid if and only if it is periodically rigid and the RUM spectrum is trivial.

Theorem 4

Let Symmetry 06 00308i1 be a crystallographic bar-joint framework ind. The following statements are equivalent:

(i)

Every almost periodic infinitesimal flex of Symmetry 06 00308i1 is trivial;

(ii)

Every strictly periodic infinitesimal flex of Symmetry 06 00308i1 is trivial, and Ω( Symmetry 06 00308i1) = {1̂}.

Proof

(i) ⇒ (ii) This follows, since every phase-periodic infinitesimal flex is also an almost periodic infinitesimal flex.

(ii) ⇒ (i) Let u be an almost periodic infinitesimal flex. Then, by Theorem 3, for d dimensions, u is a uniform limit of the sequence, (gn), of trigonometric sequences in AP(ℤd × Fv, ℂd) given by:

g n ( k , v κ ) = [ h , R k ( K n ( d ) ) ] d , k d
where K n ( d ), n = 1,2,…, is the sequence of Bochner-Fejér kernels for u. By Lemma 2, each trigonometric sequence, gn, is an infinitesimal flex of Symmetry 06 00308i1, and so, by Lemma 1, each gn is a finite linear combination of phase-periodic infinitesimal flexes of Symmetry 06 00308i1. By hypothesis, the RUM spectrum, Ω( Symmetry 06 00308i1), contains the single multi-phase, 1̂, and so, every phase-periodic infinitesimal flex is strictly periodic. In particular, gn is strictly periodic. It follows that u is strictly periodic and, hence, trivial, as desired.

We note that it follows from the proof of the theorem that any almost periodic infinitesimal flex is approximable by a sequence of finite linear combinations of phase-periodic flexes, with approximation in the sense of uniform convergence for the uniform norm given before in Theorem 3. For supercell periodic flexes, one can be more specific, as in the following proposition, which essentially follows from Theorem 1 and Lemma 1 (see also [8]).

Proposition 5

Let Symmetry 06 00308i1 be a crystallographic bar-joint framework ind. Then, the space of supercell periodic infinitesimal flexes for m-fold periodicity with m = (m1,…, md) is equal to the linear span of:

{ b e ω : Φ ( ω ¯ ) b = 0 , ω Ω m ( C ) }
wherem( Symmetry 06 00308i1) is the finite subset of the RUM spectrum given by the multi-phases ω, whose k-th component is an mk–th root of unity.

In particular, every supercell periodic infinitesimal flex for Symmetry 06 00308i1 is an almost periodic infinitesimal flex for Symmetry 06 00308i1.

The Bohr spectrum of an almost periodic infinitesimal flex, u, of the crystal framework, Symmetry 06 00308i1, is the finite or countable set given by:

Λ ( u , C ) = { λ [ 0 , 1 ) d : [ u , e ω ] d 0 , for ω = e 2 π i λ T d }
It follows from Theorem 3, Lemmas 1 and 2, as in the proof above, that Λ(u, Symmetry 06 00308i1) is contained in the RUM spectrum of Symmetry 06 00308i1. Furthermore, since phase-periodic flexes are almost periodic, it follows that the RUM spectrum as a subset of [0, 1)d is the union of the Bohr spectra of all almost periodic infinitesimal flexes. Note that the spectra here depend on the translation group in the following manner. If Symmetry 06 00308i1′ has the same underlying bar-joint framework as Symmetry 06 00308i1, but full rank translation group Symmetry 06 00308i3′ ⊆ Symmetry 06 00308i3, then the infinitesimal flex, u, is represented anew as a sequence in A P ( d × F v , d ) , where F v is a vertex motif for Symmetry 06 00308i3′. The Bohr spectrum, Λ(u, Symmetry 06 00308i1), is then the image of Λ(u, Symmetry 06 00308i1) under the surjective map, Symmetry 06 00308i2d Symmetry 06 00308i2d, induced by the inclusion, Symmetry 06 00308i3′ ⊆ Symmetry 06 00308i3. This follows the same relationship as that for the RUM spectrum noted in [8]. It also follows from this that the dimension of ( Symmetry 06 00308i1), as a topological space or as an algebraic variety, is independent of the translation group, and we refer to this integer, which takes values between zero and d, as the RUM dimension of Symmetry 06 00308i1.

We see in the next section that Ω( Symmetry 06 00308i1), as a subset of [0, 1)d, often decomposes as a union of linear components. This is the case, for example, in two dimensions if Φ Symmetry 06 00308i1(z) is a square matrix function, whose determinant polynomial, det Φ Symmetry 06 00308i1(z), either vanishes identically or factorises into simple factors of the form (zn − λwm) with |λ| = 1. It follows that each almost periodic flex, u, of Symmetry 06 00308i1 admits a finite sum decomposition u1 + ⋯ + ur in which each component, ui, is an almost periodic flex, whose Bohr spectrum lies in the i-th linear component. Such component flexes are partially periodic, being periodic in certain directions of translational symmetry.

4. Gallery of Crystal Frameworks

We now exhibit a number of illustrative examples. The first two of these show two extreme cases: firstly, where the RUM spectrum is a singleton, and secondly, where the RUM spectrum is Symmetry 06 00308i2d.

Example 1

Let Symmetry 06 00308i1 = (G, p) be the crystallographic bar-joint framework with motif (Fv, Fe) and translation group Symmetry 06 00308i3 indicated in Table 1. Simplifying earlier notation, the motif vertex is labelled v and the motif edges are labelled e0 = v(0, 0)v(1, 0), e1 = v(1, 0)v(0, 1) and e2 = v(0, 0)v(0, 1). The translation group is Symmetry 06 00308i3 = {k1a1 + k2a2 : k1,k2 ∈ ℤ}, where a1 = (1, 0) and a 2 = ( 1 2 , 3 2 ). The symbol function for Symmetry 06 00308i1 is,

Symmetry 06 00308e1
Note that Φ Symmetry 06 00308i1(z, w) has rank two unless z = w = 1, and so, the RUM spectrum of Symmetry 06 00308i1 is the singleton, (1, 1) ∈ Symmetry 06 00308i22. Furthermore, there are no non-trivial strictly periodic infinitesimal flexes of Symmetry 06 00308i1, and so, by Theorem 4, Symmetry 06 00308i1 is almost periodically infinitesimally rigid.

In fact, Symmetry 06 00308i1 is sequentially infinitesimally rigid in the sense that there exists an increasing chain of finite subgraphs G1G2 ⊂ ⋯ of G, such that every vertex of G is contained in some Gn and each sub-framework (Gn, p) is infinitesimally rigid. For example, for each n, take Gn to be the vertex-induced subgraph on {v(k1, k2) : (k1, k2) ∈ ℤ2, |ki| ≤ n}. It follows that Symmetry 06 00308i1 admits no nontrivial infinitesimal flexes and, so, is (absolutely) infinitesimally rigid as a bar-joint framework. In [22], we obtain a general characterisation of countable simple graphs, G, whose locally generic placements are infinitesimally rigid in this sense. The condition is that G should contain a vertex-complete chain of (2, 3)-tight subgraphs. The crystal framework, Symmetry 06 00308i1, may be viewed as a non-generic placement of such a graph, which remains infinitesimally rigid despite the crystallographic symmetry.

If the symbol function, Φ Symmetry 06 00308i1(z), is a square matrix, then the determinant of Φ Symmetry 06 00308i1(z) gives rise to the crystal polynomial, p Symmetry 06 00308i1(z) (see [8]). It is shown in [6] that such a crystal framework has a local infinitesimal flex if and only if the crystal polynomial, p Symmetry 06 00308i1(z), is identically zero.

Example 2

Consider the crystal framework, Symmetry 06 00308i1, with the motif and translation group shown in Table 2. The motif vertices are v0 = (0, 0) and v 1 = ( 1 2 , 1 2 ). The motif edges are e0 = v0(0, 0)v0(1, 0), e1 = v0(0, 0)v0(0, 1), e2 = v0(0, 1)v1(0, 0) and e3 = v0(1, 0)v1(0, 0). Note that the symbol function, Φ Symmetry 06 00308i1(z,w), is a square matrix,

Symmetry 06 00308e2
The determinant of Φ Symmetry 06 00308i1(z, w) vanishes identically, and so, the RUM spectrum of Symmetry 06 00308i1 is Symmetry 06 00308i22. A local infinitesimal flex of Symmetry 06 00308i1 is evident by defining u(v1) = (1, 1) and uv = 0 for all vv1. A phase-periodic infinitesimal flex of Symmetry 06 00308i1 for ω = (ω1, ω2) is obtained by taking u(v0, k) = 0 and u ( v 1 , k ) = ω 1 k 1 ω 2 k 2 ( 1 , 1 ) for each k = (k1, k2) ∈ ℤ2. In particular, any finite linear combination of such phase-periodic flexes will be an almost periodic infinitesimal flex for Symmetry 06 00308i1.

Recall that a velocity vector u is supercell periodic for a crystal framework, Symmetry 06 00308i1, if u(vκ, 0) = u(vκ, k) for each motif vertex, vκ, and all k in a full rank subgroup of ℤd.

Example 3

Let Symmetry 06 00308i1 be the crystallographic bar-joint framework with motif (Fv, Fe) and translation group Symmetry 06 00308i3 indicated in Table 3. Note that Symmetry 06 00308i1 has symbol function,

Symmetry 06 00308e3
The RUM spectrum of Symmetry 06 00308i1 is Ω( Symmetry 06 00308i1) = {(1, 1), (−1, 1)}. Note that Symmetry 06 00308i1 does not admit any non-trivial infinitesimal flexes, which are strictly periodic with respect to Symmetry 06 00308i3. However, Symmetry 06 00308i1 does admit non-trivial supercell periodic infinitesimal flexes, which may be constructed from the motif and RUM spectrum Assign velocity vectors u(v0, 0) and u(v1, 0) to the motif vertices and consider the multi-phase ω = (ω1, ω2) = (−1, 1) ∈ Ω( Symmetry 06 00308i1). Define for each k = (k1, k2) ∈ ℤ2,
u ( v 0 , k ) = ω k u ( v 0 , 0 ) = ( 1 ) k 1 u ( v 0 , 0 ) u ( v 1 , k ) = ω k u ( v 1 , 0 ) = ( 1 ) k 1 u ( v 1 , 0 )
Then, u is supercell periodic with respect to the full rank subgroup 2ℤ × ℤ. If, for example, we set u(v0,0) = (1,−1) and u(v1,0) = (−1,−1), then u is also an infinitesimal flex for Symmetry 06 00308i1. Note that u consists of alternating rotational motions. In the notation of Proposition 5, u has m-fold periodicity, where we have taken m = (2, 1); the multi-phase ω = (−1, 1) is contained in,
Ω m ( C ) = { ω Ω ( C ) : ω 1 2 = 1 , ω 2 = 1 }
and u is the ω-phase-periodic velocity vector, beω, where b = (u(v0,0), u(v1, 0)) ∈ ker Φ Symmetry 06 00308i1(ω̅).

Example 4

Let Symmetry 06 00308i1 be the crystallographic bar-joint framework in2, which is indicated in Table 4. The motif vertices are p(v0) = (0, 0) and p ( v 1 ) = ( 1 3 , 2 3 ). The symbol function, Φ Symmetry 06 00308i1(z, w) is,

Symmetry 06 00308e4
The RUM spectrum, Ω( Symmetry 06 00308i1), is the singleton, (1, 1) ∈ Symmetry 06 00308i22. However, every strictly periodic velocity field, u, with (p(v0)−p(v1))·(u(v0)−u(v1)) = 0 and u(v0) = u(v1) is a non-trivial strictly periodic infinitesimal flex of Symmetry 06 00308i1.

As we have noted, the RUM dimension of a crystal framework, Symmetry 06 00308i1, is the dimension of the RUM spectrum, Ω( Symmetry 06 00308i1), as a real algebraic variety (see [8]).

Example 5

The crystal framework, Symmetry 06 00308i1, illustrated in Table 5 has motif vertices v0 = (0,0), v 1 = ( 1 2 , 3 2 ) and v2 = (0, 1). The symbol function is a square matrix,

Symmetry 06 00308e5

The crystal polynomial factors in to linear parts,

p c ( z , w ) = ( z 1 ) ( w 1 ) ( z w )
and so, the RUM spectrum is a proper subset of Symmetry 06 00308i22, whose representation in [0, 1)2 consists of the points (s,t) in the line segments given by:
s = 0 , t = 0 , s = t
In particular, Symmetry 06 00308i1 is almost periodically infinitesimally flexible, but has no local infinitesimal flexes. Furthermore, the RUM dimension of Symmetry 06 00308i1 is one. It follows that every almost periodic infinitesimal flex decomposes as a sum u1 + u2 + u3 of three almost periodic flexes corresponding to this ordered decomposition. Furthermore, u1, with the Bohr spectrum in the line s = 0, is periodic in the direction of the period vector a1 = (1, 0), while u2, with the Bohr spectrum in the line t = 0, is periodic in the direction of the period vector a 2 = ( 1 / 2 , ( 2 + 3 ) / 2 ), and u3, with the Bohr spectrum, in the line s = t is periodic in the direction a1a2.

Example 6

Let Symmetry 06 00308i1 be the crystal framework illustrated in Table 6. The symbol function, Φ(z, w), is the square matrix,

Symmetry 06 00308e6
The crystal polynomial is:
p c ( z , w ) = ( z 1 ) ( z + 1 ) ( w 1 ) ( w + 1 )
As in the last example, the RUM spectrum decomposes as a union of linear subsets. In the [0, 1)2 representation, it yields two horizontal and two vertical lines. From this, it follows that any almost periodic infinitesimal flex decomposes as a sum of two flexes, each of which is supercell periodic in one of the axial directions.

The following two examples, shown in Tables 7 and 8, have the same underlying graph and the same crystallographic point group, the dihedral group, Symmetry 06 00308i12v. However, the former has RUM dimension one and is linearly indecomposable while the latter has RUM dimension zero. Note that four-regular crystal frameworks such as these have a square symbol function.

Example 7

Let Symmetry 06 00308i1 be the crystal framework with |Fe| = d|Fv| illustrated in Table 7. The symbol function, Φ Symmetry 06 00308i1(z,w), is,

[ 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 2 1 2 3 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 2 1 2 3 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 2 1 2 3 2 0 0 0 0 0 0 1 2 3 2 0 0 0 0 0 0 0 0 1 2 3 2 0 0 0 0 0 0 1 2 3 2 0 0 0 0 0 0 0 0 0 0 1 2 3 2 0 0 0 0 0 0 1 2 3 2 0 0 0 0 0 0 0 0 1 2 3 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 z ¯ 0 0 0 0 0 0 0 1 2 3 2 0 0 0 0 0 0 0 0 z ¯ 2 3 z ¯ 2 0 0 0 0 0 0 1 2 3 2 0 0 0 0 0 0 0 0 0 0 z ¯ 2 3 z ¯ 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 z ¯ 0 0 0 0 0 0 0 1 2 3 2 0 0 0 0 w ¯ 2 3 w ¯ 2 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 2 0 0 w ¯ 2 3 w ¯ 2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 2 3 2 0 0 0 0 1 2 3 2 0 0 0 0 0 0 0 0 0 0 1 2 3 2 0 0 1 2 3 2 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ]
The crystal polynnomial factorizes in to linear factors,
p c ( z , w ) = ( z + 1 ) ( z 1 ) 3

Example 8

Let Symmetry 06 00308i1 be the crystal framework illustrated in Table 8. The framework motif satisfies |Fe| = d|Fv|, and so, the symbol function, Φ Symmetry 06 00308i1(z,w), is a square matrix,

[ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b a b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b a b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b a b a 0 0 0 0 0 0 b a 0 0 0 0 0 0 0 0 b a 0 0 0 0 0 0 1 2 3 2 0 0 0 0 0 0 0 0 0 0 1 2 3 2 0 0 0 0 0 0 1 2 3 2 0 0 0 0 0 0 0 0 1 2 3 2 0 0 0 0 0 0 a b 0 0 0 0 0 0 0 0 0 0 a z ¯ b z ¯ 0 0 0 0 0 0 1 2 1 2 0 0 0 0 0 0 0 0 z ¯ 2 z ¯ 2 0 0 0 0 0 0 1 2 1 2 0 0 0 0 0 0 0 0 0 0 z ¯ 2 z ¯ 2 0 0 0 0 0 0 a b 0 0 0 0 0 0 0 0 a z ¯ b z ¯ 0 0 0 0 0 0 1 2 3 2 0 0 0 0 w ¯ 2 3 w ¯ 2 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 2 0 0 w ¯ 2 3 w ¯ 2 0 0 0 0 0 0 0 0 0 0 0 0 a b 0 0 0 0 0 0 a b 0 0 0 0 0 0 0 0 0 0 1 2 1 2 0 0 0 0 1 2 1 2 0 0 0 0 0 0 0 0 0 0 1 2 1 2 0 0 1 2 1 2 0 0 a b 0 0 0 0 0 0 0 0 0 0 0 0 a b 0 0 ]
where a = 3 + 1 2 2 and b = 3 1 2 2. The crystal polynomial takes the form,
p c ( z , w ) = z 4 w 1 3 z 3 w 2 + ( 3 2 2 ) z 3 w 1 2 3 z 3 + ( 1 2 + 1 2 3 ) z 2 w 2 + 1 2 3 z 2 w 1 3 z 2 1 2 z w 2 + ( 3 2 1 3 ) z w + 1 3 z 1 2 w
which leads to a finite RUM spectrum.

Example 9

The crystal framework illustrated in Table 9 is based on a motif consisting of a regular octagon of equilateral triangles. The crystal polynomial is,

p c ( z , w ) = p 1 ( z , w ) p 2 ( z , w )
where
p 1 ( z , w ) = ( 3 2 ) z 2 w z w 2 + 2 ( 2 3 + 1 ) z w + ( 3 2 ) w z p 2 ( z , w ) = ( 3 + 2 ) z 2 w z w 2 2 ( 2 + 3 1 ) z w + ( 3 + 2 ) w z
The RUM spectrum consists of points (z, w) ∈ Symmetry 06 00308i22 that satisfy ℜ(w) = aℜ(z) + (1 − a) for either a = 3 2 or a = 3 + 2. This set is illustrated in Figure 1 as a subset of the torus [0, 1)2, which consists of four closed curves with the common intersection point (0,0).

Let Symmetry 06 00308i4 be the basic one-dimensional grid framework for the lattice, ℤ in ℝ. For any crystal framework, Symmetry 06 00308i1 in ℝd, one may construct a product framework Symmetry 06 00308i4 = Symmetry 06 00308i1 × Symmetry 06 00308i1 in ℝd+1, whose intersection with the hyperplanes ℝd × {n} are copies of Symmetry 06 00308i1 and where these copies are connected by the edges ((p(v), n), (p(v), n + 1)). In the case that Symmetry 06 00308i1 has square matrix symbol function Φ Symmetry 06 00308i1(z1,…,zd), it is straightforward to verify that:

p C ˜ ( z 1 , , z d + 1 ) = ( z d + 1 1 ) | F v | p C ( z 1 , , z d )
This leads readily to the identification of the RUM spectrum in Symmetry 06 00308i23 of such frameworks. Further, three-dimensional examples not of this product form may be found in Power [8] and Wegner [12].

Example 10

Let Symmetry 06 00308i4 be the three-dimensional framework derived from the regular octagon framework of Example 9. Then the crystal polynomial admits a three-fold factorisation and it follows that the RUM spectrum has the topological structure of four two-dimensional tori connected over the common circle of points (1, 1, z) in Symmetry 06 00308i23.

Further Work

The examples above and the foregoing analysis suggest a number of intriguing lines of investigation.

On the computational side, it would be of interest to explore the variation of the RUM spectrum with respect to parameters determining distinct placements, as in the pair given in Tables 7 and 8. Parameter curves of constant RUM dimension and their connections with crystallographic symmetry would be of particular interest.

On the theoretical side, it is natural now to seek characterisations of almost periodic infinitesimal rigidity and flexibility for frameworks that are almost periodic or quasicrystallographic. One class of such frameworks would require a vertex set that is an almost periodic perturbation (or even an incommensurate periodic perturbation) of a periodic framework vertex set, with the bar lengths adjusted accordingly.

D. Kitson and S. C. Power are supported by the Engineering and Physical Sciences Research Council (EPSRC) grant EP/J008648/1.

Conflicts of Interest

The authors declare no conflicts of interest.

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Symmetry 06 00308f1 1024
Figure 1. The RUM spectrum for Example 9.

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Figure 1. The RUM spectrum for Example 9.
Symmetry 06 00308f1 1024
Table Table 1. An infinitesimally rigid crystal framework.

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Table 1. An infinitesimally rigid crystal framework.
MotifTranslation groupCrystal framework
Symmetry 06 00308t1 Symmetry 06 00308t2 Symmetry 06 00308t3
Table Table 2. A crystal framework with the full rigid unit mode (RUM) spectrum.

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Table 2. A crystal framework with the full rigid unit mode (RUM) spectrum.
MotifTranslation groupCrystal framework
Symmetry 06 00308t4 Symmetry 06 00308t5 Symmetry 06 00308t6
Table Table 3. A crystal framework with the RUM spectrum Ω( Symmetry 06 00308i1) = {(1, 1), (−1, 1)}.

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Table 3. A crystal framework with the RUM spectrum Ω( Symmetry 06 00308i1) = {(1, 1), (−1, 1)}.
MotifTranslation groupCrystal framework
Symmetry 06 00308t7 Symmetry 06 00308t8 Symmetry 06 00308t9
Table Table 4. A crystal framework with Ω( Symmetry 06 00308i1) = {(1, 1)} and nontrivial strictly periodic flexes.

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Table 4. A crystal framework with Ω( Symmetry 06 00308i1) = {(1, 1)} and nontrivial strictly periodic flexes.
MotifTranslation groupCrystal framework
Symmetry 06 00308t10 Symmetry 06 00308t11 Symmetry 06 00308t12
Table Table 5. A crystal framework with RUM dimension one.

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Table 5. A crystal framework with RUM dimension one.
MotifTranslation groupCrystal framework
Symmetry 06 00308t13 Symmetry 06 00308t14 Symmetry 06 00308t15
Table Table 6. A crystal framework with supercell periodic flexes.

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Table 6. A crystal framework with supercell periodic flexes.
MotifTranslation groupCrystal framework
Symmetry 06 00308t16 Symmetry 06 00308t17 Symmetry 06 00308t18
Table Table 7. A crystal framework with the linearly decomposable RUM spectrum.

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Table 7. A crystal framework with the linearly decomposable RUM spectrum.
MotifTranslation groupCrystal framework
Symmetry 06 00308t19 Symmetry 06 00308t20 Symmetry 06 00308t21
Table Table 8. A crystal framework with finite RUM spectrum.

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Table 8. A crystal framework with finite RUM spectrum.
MotifTranslation groupCrystal framework
Symmetry 06 00308t22 Symmetry 06 00308t23 Symmetry 06 00308t24
Table Table 9. A crystal framework with the linearly indecomposable RUM spectrum.

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Table 9. A crystal framework with the linearly indecomposable RUM spectrum.
MotifTranslation groupCrystal framework
Symmetry 06 00308t25 Symmetry 06 00308t26 Symmetry 06 00308t27
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