EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 3079-3092
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Infinitesimal Rigidity Analysis of a Bar-Joint
Framework With Connected Braced Triangles

Ghada Matooq Badri

Mathematics Department, Faculty of Sciences, Umm Al-Qura University, Makkah,
Saudi Arabia

Abstract. The aim of this paper is to provide detailed infinitesimal rigidity analysis of a pla-
nar infinite bar-joint framework consisting of connected braced triangles. This is achieved in a
purely mathematical manner, using the infinitesimal flex condition, as we identify the non trivial
infinitesimal flexes of the finite framework consisting of n connected triangles. The result is then
generalized to the infinite case leading to the identification of a base for the space of all infinitesimal
flexes.

2020 Mathematics Subject Classifications: 52C25, 51M05, 47N50

Key Words and Phrases: Bar-joint framework, infinitesimal rigidity, non trivial flex

1. Introduction

There has been increasing interest in the analysis of mathematical bar-joint frame-
works, both in the finite and infinite sense. The realization of a framework in the Eu-
clidean space enables us to mathematically investigate properties of physical structures
and crystalline materials. Applications of rigidity now extend beyond physics and struc-
ture engineering finding their way to molecule analysis [8], [13], robotics [26] and much
more. Formally, a pair (G, p) in the Euclidean plane R2 is a mathematical bar-joint frame-
work G where G = (V,E) represents a simple graph, and p = (p1, p2, p3, . . . ) represents
a placement of the graph’s vertices in R2, with pi ̸= pj if (vi, vj) is an edge. The line
segments [pi, pj ] connected to the edges of G are the framework edges.

Definition 1. Let G = (V,E) be a graph with p ∈ V . The number |E(p)| of edges in the
graph with the vertex p as an endpoint is the degree of p, or d(p). The minimum degree of
G is δ(G) := min{d(p), p ∈ V }, and the maximum degree is ∆(G) := max{d(p), p ∈ V }.
G = (V,E) is said to be n-regular if all of the vertices have the same degree n.

DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5290

Email address: gmbadri@uqu.edu.sa (G. M. Badri)

https://www.ejpam.com 3079 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



G. M. Badri / Eur. J. Pure Appl. Math, 17 (4) (2024), 3079-3092 3080

Any displacement of the framework that preserves the distances between each pair of
framework vertices is called a rigid body motion. This transition, known as a flexing of
the structure, occurs when the lengths between vertices that are not connected by an edge
change. The outcome is a new configuration that differs from the previous one. In the
Euclidean plane R2, it is evident that rigid body motions result form linear combinations
of rotations and translations in either coordinate direction.

Definition 2. Let G = (G, p) be a finite framework with |V | = n in R2. A vector u =

(u1, . . . , un) in the vector space Hv(G) =
n⊕

i=1
R2 is called an infinitesimal flex if and only

if the orthogonality relation

⟨pi − pj , ui − uj⟩ = 0

is held for any edge e = [pi, pj ].

A framework G is infinitesimally rigid if every infinitesimal flex of G is trivial, and
infinitesimally flexible otherwise. If, however, all pairs of framework vertices, not just those
that form edges, satisfy the above condition, then u is considered a trivial infinitesimal
flex, or an infinitesimal rigid body motion.

Hv(G) will be used to represent the vector space containing all velocity vectors allocated
to the vertices of the framework. A vector subspace of Hv(G), which includes the subspace
of all infinitesimal rigid motions Hrig(G), is the space Hfl(G) of all infinitesimal flexes of G.

Definition 3. In R2, let G = (G, p) be an infinite framework. A vector in Hv(G) =∏
V R2 = R2

⊕
R2
⊕

+ . . . is an infinitesimal flex of G for which, just like in the finite
case,

⟨ui − uj , pi − pj⟩ = 0

is held for every edge e = [pi, pj ].

Let Hfl(G) be the linear space of all infinitesimal flexes, adapting the same notation as
in the finite case. This includes the three dimensional space of rigid body motions Hrig(G)
which is spanned by two translations and one rotation. See [5], [11], [14] and [24] for a
thorough introduction to bar-joint frameworks.

Definition 4. R(G, p) is the rigidity matrix of the infinite framework G = (G,P ). In R2,
it consists of rows indexed by the framework edges and columns labelled by the vertices but
with multiplicity two, i.e., vx1 , v

y
1 , v

x
2 , v

y
2 , . . . . The entries xi − xj , xj − xi, yi − yj , yj − yi

occur in the row labelled e = (vi, vj) with respective column labels vxi , v
y
i , v

x
j , v

y
j and equal

zero elsewhere.

R(G, p) defines a linear transformation from the space Hv(G) =
∏

V R2 to He(G) =∏
E R and it can be seen that a vector u in Hv(G) is an infinitesimal flex if and only if

R(G, p)u = 0.



G. M. Badri / Eur. J. Pure Appl. Math, 17 (4) (2024), 3079-3092 3081

Definition 5. A countably infinite framework G = (G, p) is edge vanishing if the sequence
(dei)

∞
i=1 formed by all bar lengths has no lower bound. G is edge unbounded if (dei)

∞
i=1 has

no upper bound. G is distance-regular if (dei)
∞
i=1 is bounded.

Theorem 1. [20] For a distance regular framework in R2 where the degrees of the vertices
are uniformly bounded, the rigidity matrix determines a bounded Hilbert space transfor-
mation.

One of the earliest contributions towards rigidity theory is Due to Laman, [16]. Laman’s
Theorem characterizes the rigidity of a generic framework as purely combinatorial, regard-
less of its geometry. For three dimensional frameworks, Euler conjectured that “A closed
spatial figure allows no changes, as long as it is not ripped apart”, see [14]. For convex
polyhedra, this conjecture was later answered by Cauchy [9], who proved that “If there
is an isometry between the faces of two strictly convex polyhedra which is an isometry
on each of the faces, then the two polyhedra are congruent”. A corollary of Cauchy’s
Theorem is that all convex polyhedrons are in fact rigid. The formal identification of a
mathematical framework as a set of bars and joints was first introduced by Asimov and
Roth [3], [4], [22] where each vertex corresponds to a joint and each bar represents an
edge. Finally, Euler’s conjecture was proven wrong by Connelly [10], who showed that
there exists a polyhedron that is not rigid.

Recent developments in rigidity theory include the formal identification of an infinite
bar-joint framework, crystallographic bar-joint frameworks and the rigidity matrix by
Owen and Power, [20]. The investigation of various types of flexes admitted by such
frameworks was carried out in [18] and [19]. Different forms of rigidity were also introduced,
for example, dimensional rigidity [1], global rigidity [12], almost periodic rigidity [6] and
bearing rigidity [17]. For more rigidity considerations, one can refer to [2], [15] and [25].

2. The Connected Braced Triangles Finite Bar-Joint Framework

Let F2tri be the finite bar-joint framework constructed by connecting two braced tri-
angles, with the planar placement suggested by Figure 1. Although any triangle in the
Euclidean plane R2 is rigid, the presence of a vertex on one edge implies the existence of
a non trivial infinitesimal flex of the triangle. In this case, the addition of the internal
horizontal edge would prevent any flexing of that edge. Now, let ui = (uxi , u

y
i ) be the

infinitesimal flex applied to the vertex pi = (pxi , p
y
i ). Assuming that all the flexes at the

base vertices equal zero, this would prevent any rigid body motion and any flexing of the
structure would be non trivial. The following theorem proves that this framework admits
a non trivial infinitesimal flex uniquely determined by the velocity applied at vertex p1.

Theorem 2. Let F2tri be the finite framework with two braced triangles, then F2tri admits
a one dimensional space of non trivial infinitesimal flexes.



G. M. Badri / Eur. J. Pure Appl. Math, 17 (4) (2024), 3079-3092 3082

p3

p2

p1

p4

p7

p6 p8

p5

x

y

α1

β1

α2

β2

Figure 1: The finite connected braced triangles framework F2tri

Proof. The proof makes direct use of the infinitesimal flex condition. Let
u = (u1, . . . , u8) denote a flex of F2tri with the component ui = (uxi , u

y
i ) being the velocity

applied at vertex pi. Subtracting appropriate multiples of rigid body motions we can
arrange for the vertices p3 and p7 to have zero velocity vectors. It follows immediately
that vertices p1, p4, p5 and p8 all have have zero velocity components in the y direction.

Applying the flex condition to the edge [p2, p3] we have:

⟨p2 − p3, u2⟩+ ⟨p3 − p2, u3⟩ = 0

(px2 − px3)u
x
2 + (py2 − py3)u

y
2 + (px3 − px2)u

x
3 + (py3 − py2)u

y
3 = 0

ux2 =

(
α1

γ1

)
uy2

Applying the flex condition to the edge [p1, p2]:

⟨p1 − p2, u1⟩+ ⟨p2 − p1, u2⟩ = 0

(px1 − px2)u
x
1 + (py1 − py2)u

y
1 + (px2 − px1)u

x
2 + (py2 − py1)u

y
2 = 0

γ1u
x
1 − γ1u

x
2 − β1u

y
2 = 0

Substituting ux2 :

ux1 =

(
α1 + β1

γ1

)
uy2

The latter expression being equivalent to:



G. M. Badri / Eur. J. Pure Appl. Math, 17 (4) (2024), 3079-3092 3083

ux2 =

(
1

1 + β1

α1

)
ux1

The application of the flex condition to both edges [p2, p4] and [p4, p5] implies that:

ux2 = ux4 = ux5 =

(
1

1 + β1

α1

)
ux1

Applying the flex condition to the edge [p5, p6]:

⟨p5 − p6, u5⟩+ ⟨p6 − p5, u6⟩ = 0

(px5 − px6)u
x
5 + (py5 − py6)u

y
5 + (px6 − px5)u

x
6 + (py6 − py5)u

y
6 = 0

Substituting ux6 it follows that:

γ2u
x
5 − α2u

y
6 − β2u

y
6 = 0

γ2u
x
5 = (α2 + β2)u

y
6

ux5 =

(
α2 + β2

γ2

)
uy6

ux5 =

(
1 +

β2
α2

)
ux6

Equivalently,

ux6 =

(
1

1 + β2

α2

)
ux5 =

(
1

1 + β1

α1

)(
1

1 + β2

α2

)
ux1

Applying the flex condition to the edge [p6, p7]:

⟨p6 − p7, u6⟩+ ⟨p7 − p6, u7⟩ = 0

(px6 − px7)u
x
6 + (py6 − py7)u

y
6 + (px7 − px6)u

x
7 + (py7 − py6)u

y
7 = 0

Therefore:

uy6 =

(
γ2
α2

)
ux6

uy6 =

(
γ2
α2

)(
1

1 + β1

α1

)(
1

1 + β2

α2

)
ux1

uy6 =

(
1

1 + β1

α1

)(
γ2

α2 + β2

)
ux1



G. M. Badri / Eur. J. Pure Appl. Math, 17 (4) (2024), 3079-3092 3084

Finally, applying the flex condition to the edge [p6, p8]:

⟨p6 − p8, u6⟩+ ⟨p8 − p6, u7⟩ = 0

(px6 − px8)u
x
6 + (py6 − py8)u

y
6 + (px8 − px6)u

x
8 + (py8 − py6)u

y
8 = 0

ux8 = ux6

The argument above, together with the appropriate substitutions we can identify the
non trivial infinitesimal flex u = (u1, u2, . . . , u8) of F2tri with the velocity components:

u1 = (ux1 , 0)

u2 =

((
1

1 + β1

α1

)
ux1 ,

(
γ1

α1 + β1

)
ux1

)
u3 = (0, 0)

u4 =

((
1

1 + β1

α1

)
ux1 , 0

)
u5 = (0, 0)

u6 =

((
1

1 + β1

α1

)(
1

1 + β2

α2

)
ux1 ,

(
1

1 + β1

α1

)(
γ2

α2 + β2

)
ux1

)
u7 = (0, 0)

u8 =

((
1

1 + β1

α1

)(
1

1 + β2

α2

)
ux1 , 0

)

Clearly, u is uniquely determined by the initial velocity applied at u1 and the conclusion
follows.

Corollary 1. Let Hfl(F2tri) denote the linear space of all infinitesimal flexes of F2tri.
Then dim(Hfl(F2tri)) = 4.

We now consider the finite framework constructed by connecting three braced triangles,
F3tri, suggested by Figure 2. In addition to the flex determined by the velocity applied at
p1, the following theorem identifies a new infinitesimal flex of this framework.

Theorem 3. Let F3tri be the finite framework with three braced triangles. Let u =
(u1, u2, . . . , u12) be the flex of F3tri with zero velocity components at all the vertices of
the first and third triangle. Then F3tri admits a non trivial infinitesimal flex uniquely
determined by uy7.



G. M. Badri / Eur. J. Pure Appl. Math, 17 (4) (2024), 3079-3092 3085

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

p11

p12

α1

β1

γ1

α2

β2

γ2

α3

β3

γ3

σ1 σ2

Figure 2: The finite connected braced triangles framework F3tri

Proof. With the first and third framework vertices all having zero velocity components,
u1 = u2 = u3 = u4 = u9 = u10 = u11 = u12 = (0, 0). From which it immediately follows
that u7 only admits a velocity component in the y direction, the velocity in the x direction,
ux7 , being zero.

Applying the flex condition to the edge [p4, p5] we have:

⟨p4 − p5, u4⟩+ ⟨p5 − p4, u5⟩ = 0

(px4 − px5)u
x
4 + (py4 − py5)u

y
4 + (px5 − px4)u

x
5 + (py5 − py4)u

y
5 = 0

σ1u
x
5 + (α2 + β2 − α1)u

y
5 = 0

Which implies:

ux5 =

(
α1 − (α2 + β2)

σ1

)
uy5

Applying the flex condition to the edge [p7, p8]:

⟨p7 − p8, u8⟩+ ⟨p8 − p7, u8⟩ = 0

(px7 − px8)u
x
7 + (py7 − py8)u

y
7 + (px8 − px7)u

x
8 + (py8 − py7)u

y
8 = 0

uy7 = uy8

Applying the flex condition to the edge [p5, p8]:

⟨p5 − p8, u5⟩+ ⟨p8 − p5, u8⟩ = 0

(px5 − px8)u
x
5 + (py5 − py8)u

y
5 + (px8 − px5)u

x
8 + (py8 − py5)u

y
8 = 0

uy5 = uy8



G. M. Badri / Eur. J. Pure Appl. Math, 17 (4) (2024), 3079-3092 3086

Applying the flex condition to the edge [p8, p9]:

⟨p8 − p9, u8⟩+ ⟨p9 − p8, u9⟩ = 0

(px8 − px9)u
x
8 + (py8 − py9)u

y
8 + (px9 − px8)u

x
9 + (py9 − py8)u

y
9 = 0

(−σ1)u
x
8 + (α2 − (α3 + β3))u

y
8 = 0

And it follows that

ux8 =

(
α2 − (α3 + β3)

σ2

)
uy8

Applying the flex condition to the edge [p6, p7]:

⟨p6 − p7, u6⟩+ ⟨p7 − p6, u7⟩ = 0

(px6 − px7)u
x
6 + (py6 − py7)u

y
6 + (px7 − px6)u

x
7 + (py7 − py6)u

y
7 = 0

(−γ2)u
x
6 = α2u

y
6

uy6 =

(
γ2
α2

)
ux6

Applying the flex condition to the edge [p6, p8]:

⟨p6 − p8, u6⟩+ ⟨p8 − p6, u8⟩ = 0

(px6 − px8)u
x
6 + (py6 − py8)u

y
6 + (px8 − px6)u

x
8 + (py8 − py6)u

y
8 = 0

ux6 = ux8

Finally, applying the flex condition to the edge [p5, p6]:

⟨p5 − p6, u5⟩+ ⟨p6 − p5, u6⟩ = 0

(px5 − px6)u
x
5 + (py5 − py6)u

y
5 + (px6 − px5)u

x
6 + (py6 − py5)u

y
6 = 0

From which it follows that:

γ2u
x
5 + β2u

y
5 = γ2u

x
6 + β2u

y
6

Substituting uy6:

γ2u
x
5 + β2u

y
5 = γ2u

x
6 + β2

(
γ2
α2

)
ux6

ux5 =

(
1 +

β2
α2

)
ux6 −

(
β2
γ2

)
uy5



G. M. Badri / Eur. J. Pure Appl. Math, 17 (4) (2024), 3079-3092 3087

Substituting ux6 = ux8 and uy8 = uy5:

ux5 =

(
1 +

β2
α2

)(
α2 + (α3 + β3)

σ2

)
uy8 −

(
β2
γ2

)
uy5

=

[(
1 +

β2
α2

)(
α2 + (α3 + β3)

σ2

)
− β2

γ2

]
uy5

With the appropriate substitutions we arrive at an infinitesimal flex of F3tri with the
velocity components:

u5 =

([(
1 +

β2
α2

)(
α2 + (α3 + β3)

σ2

)
− β2

γ2

]
uy7, u

y
7

)
u6 =

((
α2 − (α3 + β3)

σ2

)
uy7,

(
γ2
α2

)(
α2 − (α3 + β3)

σ2

)
uy7

)
u7 = (0, uy7)

u8 =

((
α2 − (α3 + β3)

σ2

)
uy7, u

y
7

)
uniquely determined by uy7 and with zero velocities elsewhere.

Corollary 2. Let Hfl(F3tri) denote the linear space of all infinitesimal flexes of F3tri.
Then dim(Hfl(F3tri)) = 5.

Corollary 3. Let Hfl(Fntri) denote the linear space of all infinitesimal flexes of the finite
framework with n connected braced triangles. Then

dim(Hfl(Fntri)) = n+ 2.

3. The Connected Braced Triangles Infinite Bar-Joint Framework

Let Gtri be the infinite strip bar-joint framework in R2 constructed by joining copies
of the connecting braced triangles finite framework along in the positive x coordinate
direction, as suggested by Figure 3.

. . .

Figure 3: The infinite connected braced triangles framework Gtri



G. M. Badri / Eur. J. Pure Appl. Math, 17 (4) (2024), 3079-3092 3088

This section is dedicated to the identification of a base for the space of all infinitesimal
flexes of the infinite connected braced triangles framework. In fact, such bases do exist for
infinite frameworks and this was thoroughly investigated in [7]. For more considerations
of infinite strip frameworks see [20], [21] and [23].

Similar to the finite case, with the base vertices having zero velocity components, the
action implied by the velocity at vertex p1 does in fact extend to the rest of the framework
in the infinite case.

Theorem 4. Let Gtri be the connected braced triangles infinite bar-joint framework in R2.
Let u be the non trivial infinitesimal flex of Gtri, with the velocity component un,i applied
at vertex pn,i, and with un,3 = (0, 0) for all n ∈ N. Then:

un,1 =

(
n−1∏
i=1

(
1

1 + βi

αi

)
ux1,1, 0

)

. . . . . .

pn,1

pn+1,1

pn,2 pn,4

pn,3

αn

βn

γn

pn+1,2

pn+1,3

pn+1,4

βn+1

αn+1

γn+1

σn

Figure 4: Labelling of the infinite connected braced triangles framework Gtri

Proof. The proof proceeds by Mathematical Induction. The case n = 2 follows im-
mediately from the proof of Theorem 2. We now assume that this is true for n, that
is

un,1 =

(
n−1∏
i=1

(
1

1 + βi

αi

)
ux1,1, 0

)
.

The proof for n+1 makes direct use of the flex condition. It is obvious that each flex un,1
has a zero velocity component in the y direction since un,3 = (0, 0) for all n. To find the



G. M. Badri / Eur. J. Pure Appl. Math, 17 (4) (2024), 3079-3092 3089

velocity component in the x direction we first start by applying the flex condition to the
edge [pn,2, pn,3]:

⟨pn,2 − pn,3, un,2⟩+ ⟨pn,3 − pn,2, un,3⟩ = 0

(pxn,2 − pxn,3)u
x
n,2 + (pyn,2 − pyn,3)u

y
n,2 + (pxn,3 − pxn,2)u

x
n,3 + (pyn,3 − pyn,2)u

y
n,3 = 0

−γnu
x
n,2 + αnu

y
n,2 = 0

From which it follows that:

uyn,2 =

(
γn
αn

)
uxn,2

Applying the flex condition to the edge [pn,1, pn,2]:

⟨pn,1 − pn,2, un,1⟩+ ⟨pn,2 − pn,1, un,2⟩ = 0

(pxn,1 − pxn,2)u
x
n,1 + (pyn,1 − pyn,2)u

y
n,1 + (pxn,2 − pxn,1)u

x
n,2 + (pyn,2 − pyn,1)u

y
n,2 = 0

γnu
x
n,1 − γnu

x
n,2 − βnu

y
n,2 = 0

Substituting uyn,2:

γnu
x
n,1 − γnu

x
n,2 − βn

(
γn
αn

)
uxn,2 = 0

uxn,2 =

(
1

1 + βn

αn

)
uxn,1

Applying the flex condition to the edge [pn,2, pn,4]:

⟨pn,2 − pn,4, un,2⟩+ ⟨pn,4 − pn,2, un,4⟩ = 0

(pxn,2 − pxn,4)u
x
n,2 + (pyn,2 − pyn,4)u

y
n,2 + (pxn,4 − pxn,2)u

x
n,4 + (pyn,4 − pyn,2)u

y
n,4 = 0

−γnu
x
n,2 + γnu

x
n,4 = 0

Hence,

uxn,4 = uxn,2 =

(
1

1 + βn

αn

)
uxn,1

Applying the flex condition to the edge [pn,4, pn+1,1]:

⟨pn,4 − pn+1,1, un,4⟩+ ⟨pn+1,1 − pn,4, un+1,1⟩ = 0

(pxn,4 − pxn+1,1)u
x
n,4 + (pyn,4 − pyn+1,1)u

y
n,4 + (pxn+1,1 − pxn,4)u

x
n+1,1 + (pyn+1,1 − pyn,4)u

y
n+1,1 = 0

uxn+1,1 = uxn,4



G. M. Badri / Eur. J. Pure Appl. Math, 17 (4) (2024), 3079-3092 3090

Finally, using the induction hypothesis:

uxn+1,1 = uxn,4

=

(
1

1 + βn

αn

)
uxn,1

=

(
1

1 + βn

αn

)(
n−1∏
i=1

(
1

1 + βi

αi

))
ux1,1

=
n∏

i=1

(
1

1 + βi

αi

)
ux1,1

and the result follows.

Definition 6. Let G = (G, p) be an infinite framework in R2. An infinitesimal flex
u = (un) of G is called local if un = (0, 0) for all but finitely many values of n.

Corollary 4. Let Gtri be the connected braced triangles infinite bar-joint framework in R2.
If the vertices of all but one triangle have zero velocity components, then Gtri admits the
local flex identified in Theorem 3.

Theorem 5. Let Gtri be the connected braced triangles infinite bar-joint framework in R2.
Then B = {wx, wy, r}

⋃
{v}

⋃
{un : n ∈ N} is a base for Hfl(Gtri), where wx, wy, r are

the three planar rigid body motions, v is the non trivial flex in Theorem 4 and un is the
local flex with non zero velocity components at one triangle and zero velocities elsewhere.

Proof. The proof proceeds with an exhaustion argument where appropriate multiples
are subtracted from an arbitrary flex until we achieve zero flexing of the structure. Let
s = (sn) =

(
(sn,i)

4
i=1

)∞
n=1

be an arbitrary flex of Gtri. Subtracting sx1,3w
x + sy1,3w

y from
the flex s one can arrange for vertex p1,3 to have a zero velocity. Subtracting sy2,3r, vertex
p2,3 now has a zero velocity implying that any further flexing of Gtri would be non trivial.
Proceeding in the same manner, subtracting sx1,1v results in a zero velocity at vertex p1,1
from which it follows that vertex p2,1 is also assigned a zero velocity. To this end, vertices
p1,1, p1,2, . . . , p2,3, p2,4 of the first two triangles are all fixed. Finally, successive subtraction
of syn,3un, assigns zero velocities to the points pn,3. This results in every vertex of Gtri

admitting a zero velocity and the conclusion follows.

4. Conclusion

In this paper we determined the non trivial flexes of the planar finite frameworks
with n connected braced triangles. This lead to the identification of a bases for the
spaces of all infinitesimal flexes, Hfl(Fntri). The nature of the non trivial base elements
emphasises how the flexibility of a framework is impacted by its unique geometry. This
knowledge can be taken forward to provide detailed rigidity analysis of infinite frameworks



REFERENCES 3091

by understanding the rigidity of their finite subframeworks placed in Euclidean spaces. In
this specific case, the infinite framework admits similar flexing properties to those of the
finite frameworks. Additionally, we found that the action of the non trivial infinitesimal
flex with zero velocities at the points pn,3 extends to the rest of the framework and is
uniquely determined by the x component of the velocity initially applied at vertex p1,1.

Furthermore, such analysis can be applied to crystal frameworks for example, a class
of bar-joint frameworks known for their translational symmetry. This also can be used
to identify special types of flexes such as strictly periodic, phase periodic and supercell
periodic flexes.

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