Necessary conditions for the extendibility of a first-order flex of a polyhedron to its flex
Victor Alexandrov

Necessary conditions for the extendibility of a 1-st order flex]Necessary conditions 
for the extendibility of a first-order flex of a polyhedron to its flex

Sobolev Institute of Mathematics, Koptyug ave., 4, 
Novosibirsk, 630090, Russia and Department of Physics, 
Novosibirsk State University, Pirogov str., 2, Novosibirsk, 
630090, Russia
alex@math.nsc.ru
November 2, 2019

We derive fundamentally new equations that are satisfied by first-order flexes of a flexible polyhedron.
Moreover, we indicate two sources of such new equations.
These sources are the Dehn invariants and rigidity matrix.
The equations derived provide us with fundamentally new necessary conditions for the extendibility of 
a first-order flex of a polyhedron to its flex.
Keywords: Euclidean 3-space, flexible polyhedron, infinitesimal bending, Dehn invatiant, rigidity matrix.
Mathematics subject classification (2010):  52C25. 	


[
    Victor Alexandrov
    December 30, 2023
=====================





§ INTRODUCTION


A polyhedron (more precisely, a compact polyhedral boundary-free surface in ℝ^3)
is called flexible if its spatial form can be changed by a continuous deformation, 
in the course of which each face remains congruent to itself.
If some faces of a flexible polyhedron are not triangular, then one can triangulate them
in an arbitrary way and obtain a new flexible polyhedron with triangular faces only.
This is the reason why, in the theory of flexible polyhedra, it is customary to study
flexible polyhedra with triangular faces only.
Such a polyhedron is flexible, if and only if its spatial form can be
changed by a continuous motion of its vertices, during which the length of every edge remains
unaltered (so that the change in the spatial shape of the polyhedron results from the change of its dihedral angles only).

The first examples and properties of flexible polyhedra (more precisely, of flexible octahedra with self-intersections)
were discovered in the 19th century <cit.>. 
Later flexible octahedra were studied in <cit.> and <cit.>.
However, the real flowering of the theory of flexible polyhedra began in the mid-1970s when
Robert Connelly constructed a sphere-homeomorphic self-intersection free flexible polyhedron <cit.>.
Now we already know that any flexible polyhedron in Euclidean 3-space preserves the total mean curvature
(see <cit.>), enclosed volume (there are especially many articles devoted to this issue, so we point out
several of them in chronological order: 
<cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, and <cit.>),
and Dehn invariants (see <cit.>).
The reader can find more details about the theory of flexible polyhedra in the above mentioned
articles, as well as in the following review articles, which we list in chronological order:
<cit.>, <cit.>, <cit.>, <cit.>, and <cit.>.

However, there are many intriguing open problems in the theory of flexible polyhedra.
One of them reads as follows: given a polyhedron, recognize whether it is flexible or not.
One approach to this problem is to analyze the infinitesimal flexes of the given polyhedron.

Recall that an infinitesimal flex (more precisely, an nth order flex for some n⩾ 1) 
of a polyhedron with triangular faces
is a continuous motion of the vertices (depending on some parameter t, which can be
interpreted as time, with the initial polyhedron corresponding to t=0)
that changes every edge length by a value which is o(t^n) as t→ 0.
An nth order flex can be identified with the set of the derivatives of the position vectors of the vertices 
of orders 1,…, n of the initial polyhedron. 
An nth order flex satisfies some well-known equations.
Many geometers studied conditions under which an nth order flex can be extended to an (n+1)th order flex or to a flex.
The reader can find more details about the infinitesimal flexes of polyhedra and smooth surfaces,
for example, in the classic article <cit.>, review articles <cit.>, <cit.> and <cit.>,
as well as in the book <cit.>.

In the present article, we derive fundamentally new equations that are satisfied by first-order flexes of a flexible polyhedron.
Moreover, we indicate two sources of such new equations.
Namely in Section <ref>, we derive the desired equations using the fact that the Dehn invariants remain unaltered 
during the flex, a fact that was established for the first time by A.A. Gaifullin and L.S. Ignashchenko <cit.>.
In Section <ref>, we derive another set of desired equations using the notion of the rigidity matrix.
The equations derived provide us with fundamentally new necessary conditions for the extendibility of 
a first-order flex of a polyhedron to its flex.

Finally, we recall that, in <cit.>, sufficient conditions are found for the extendibility of an nth order flex
of a polyhedron, n⩾ 1, to its flex.
Those conditions are based on absolutely different ideas than those used in the present article.



§ NEW EQUATIONS FOR THE VELOCITY VECTORS OF THE VERTICES OF A FLEXIBLE POLYHEDRON, GENERATED BY THE DEHN INVARIANTS


Let P_0 be an oriented polyhedron in ℝ^3 with non-degenerate triangular faces
(the latter means that no straight line contains all three vertices of a face).
Moreover, suppose that P_0 is a member of a continuous in t family { P_t }_t∈ [0,1]
of polyhedra in ℝ^3 such that, for each t∈ (0,1], 
the corresponding (by continuity) edges of P_t and P_0 have the same length.
If P_t and P_0 are not congruent to each other for each t∈ (0,1] then the family
{P_t }_t∈ [0,1] is called a flex of P_0 and P_0 is called flexible.
P_0 is called rigid if no flex of P_0 does exist. 

For each edge σ of P_0 denote 

∙ by x_σ and y_σ 
the end-points of σ; 

∙ by z_σ' and z_σ” 
the vertices of P_0, such that the triples  
x_σ, y_σ, z_σ' and
x_σ, y_σ, z_σ” 
form faces of P_0;

∙ by ℓ_σ, ℓ'_σ, and ℓ”_σ the lengths of the vectors 
y_σ-x_σ, z_σ'-x_σ,
and z_σ”-x_σ respectively;

∙ by A'_σ and A”_σ the doubled areas 
of the faces of P_0 with the vertices 
x_σ, y_σ, z_σ' and
x_σ, y_σ, z_σ” respectively
(equivalently, the lengths of the vectors 
(y_σ-x_σ)× (z_σ'-x_σ) 
and 
(y_σ-x_σ)× (z_σ”-x_σ)
respectively);

∙ by φ_σ the internal dihedral angle of the polyhedron P_0 at the edge σ.

Moreover, suppose that the notation for the vertices x_σ and y_σ 
is chosen so that the unit vectors 

    n_σ'=(y_σ-x_σ)× 
    (z_σ'-x_σ)/A'_σ  and  n_σ”=(z_σ”-x_σ) × 
    (y_σ-x_σ)/A”_σ

are the vectors of outward normals to the faces with the vertices
x_σ, y_σ, z_σ' 
and x_σ, y_σ, z_σ” 
respectively (see Fig. <ref>).

Here and below a×b stands for the cross product of 
the vectors a and b.

The above agreement that n_σ' and n_σ”, 
defined by the formulas (<ref>), are outward normals to the corresponding faces of
P_0, does not limit the generality of reasoning.
In fact, in the case of violation of this agreement, it is sufficient to swap the notation for
x_σ and y_σ.
After that, the vectors n_σ' and n_σ”
will be replaced by -n_σ' and -n_σ”,
respectively, and the agreement will be fulfilled.

For short, mainly in the proofs, we use the following notation:
w=y_σ-x_σ, 
w'=z_σ'-x_σ, and
w”=z_σ”-x_σ.


Under the above notation, for every edge σ of P_0, the equalities hold true

    A'_σA”_σcosφ_σ    =    ℓ^2_σ(z_σ'-x_σ)·(z_σ”-x_σ)
    
          -[(z_σ'-x_σ)· (y_σ-x_σ)]
    [(z_σ”-x_σ)· (y_σ-x_σ)],
    
    
    A'_σA”_σsinφ_σ    =    
    - ℓ_σ(y_σ-x_σ)·[(z_σ'-x_σ)
    × (z_σ”-x_σ)].

Here and below a·b 
stands for the scalar product of the vectors a and b.



The formula (<ref>) follows from the equality
cosφ_σ = - n_σ'·n_σ” 
and the well-known identity
(a×b)· (c×d) 
= (a·c)(b·d)
-(a·d)(b·c).
Indeed,

    cosφ_σ    =     -w×w'/A'_σ×w”×w/A”_σ
        =     -1/A'_σA”_σ[(w·w”)(w'·w)
    -(w'·w”)(w·w)] 
        =    1/A'_σA”_σ[ℓ^2_σ
    (w'·w”)-(w·w”)(w'·w)].


The formula (<ref>) follows from the equality

    n_σ'×n_σ” = sinφ_σ/ℓ_σw

and the well-known identity
(a×b)× (c×d) 
= [(a×b)·d]c 
- [(a×b)·c]d.
Indeed,

    sinφ_σ    =    ℓ_σ/w·ww· (n_σ'×n_σ”) =
    1/ℓ_σw·(
    w×w'/A'_σ×w”×w/A”_σ) 
        =    1/ℓ_σA'_σA”_σw·{[(w×w')·w]w”-
    [(w×w')·w”]w}
        =     -w·w/ℓ_σA'_σA”_σ
    (w'×w”)·w =
    -ℓ_σ/A'_σA”_σw· (w'×w”).



Since {P_t }_t∈ [0,1] is a flex of P_0,
for every t∈ [0,1] and every edge σ of P_0, there is an edge σ (t) of P_t which corresponds to
σ by continuity (in particular, σ (0)=σ).
Similarly, for the dihedral angle φ_σ and vectors
x_σ, y_σ, z_σ', 
z_σ”, w=y_σ-x_σ, 
w'=z_σ'-x_σ, and
w”=z_σ”-x_σ, 
associated with the edge σ of P_0, there are similar quantities which correspond
to the edge σ (t) of P_t.
Denote them by
φ_σ(t), x_σ(t), y_σ(t), z_σ'(t), 
z_σ”(t), w(t)=y_σ(t)-x_σ(t), 
w'(t)=z_σ'(t)-x_σ(t), and
w”(t)=z_σ”(t)-x_σ(t) 
respectively.
Denote the derivatives of these quantities with respect to t calculated at t=0 by
Φ_σ, ξ_σ, η_σ, ζ_σ', 
ζ_σ”, W=η_σ-ξ_σ, 
W'=ζ_σ'-ξ_σ, and 
W”=ζ_σ”-ξ_σ 
respectively, i. e., put by definition

    Φ_σ=d/dt|_t=0φ_σ(t),  ξ_σ=d/dt|_t=0x_σ(t), … ,  W”=d/dt|_t=0w”(t)=ζ_σ”-ξ_σ.


Note that 

∙ the length of the edge σ(t) of P_t is independent of t and is equal to ℓ_σ;

∙ the areas of the faces of P_t with the vertices 
x_σ(t), y_σ(t), z_σ'(t) and
x_σ(t), y_σ(t), z_σ”(t) 
are also indepentent of t and are equal to the areas of the faces of P_0 with the vertices
x_σ, y_σ, z_σ' and
x_σ, y_σ, z_σ” respectively
(equivalently, are equal to 1/2 A'_σ and 1/2 A”_σ respectively).


Under the above notation, for every edge σ of P_0, the equalities hold true

    (A'_σ A”_σsinφ_σ)Φ_σ    =    p_σ· (η_σ-ξ_σ)+ 
    p_σ'· (ζ_σ'-ξ_σ) +
    p_σ”· (ζ_σ”-ξ_σ), 
    
    (A'_σA”_σcosφ_σ)Φ_σ    =    q_σ· (η_σ-ξ_σ)+ 
    q_σ'· (ζ_σ'-ξ_σ) +
    q_σ”· (ζ_σ”-ξ_σ),

where

    p_σ    =    [(y_σ - x_σ)· 
    (z_σ” - x_σ)](z_σ' - x_σ)
    + [(y_σ - x_σ)· 
    (z_σ' - x_σ)](z_σ” - x_σ), 
    p_σ'     =    [(y_σ - x_σ)· 
    (z_σ” - x_σ)](y_σ - x_σ)
    - ℓ^2_σ(z_σ” - x_σ), 
    p_σ”    =    [(y_σ - x_σ)· 
    (z_σ' - x_σ)](y_σ - x_σ)
    - ℓ^2_σ(z_σ' - x_σ), 
    q_σ    =     - ℓ_σ (z_σ' - x_σ)× 
    (z_σ” - x_σ), 
    q_σ'     =    ℓ_σ (y_σ - x_σ)× 
    (z_σ” - x_σ), 
    q_σ”    =     - ℓ_σ (y_σ - x_σ)× 
    (z_σ' - x_σ).



 
Differentiating the formula (<ref>) with respect to t when t=0 yields

    -   (A'_σA”_σsinφ_σ)Φ_σ
           = ℓ^2_σ (W'·w” + w”·W”) 
    - (W'·w + w'·W)(w”·w) 
          
    - (w'·w)(W”·w + w”·W) 
           = -[(w·w”)w' + (w·w')w”]·W - [(w·w”)w 
    - ℓ^2_σw”]·W'
          
    - [(w·w')w - ℓ^2_σw']·W”.

This formula differs from (<ref>) by notation only.

Differentiating the formula (<ref>) with respect to t when t=0 yields

    (A'_σA”_σ/ℓ_σ)(cosφ_σ)Φ_σ    =    
    - W· (w'×w”) 
    - w· (W'×w”) - w· (w'×W”) 
        =     - (w'×w”)·W 
    - (w”×w)·W' - (w×w')·W”.

This formula differs from (<ref>) by notation only.



Under the above notation, for every edge σ of P_0, the equalities hold true

    Φ_σ=
    r_σ· (η_σ-ξ_σ)
    + r_σ'· (ζ_σ'-ξ_σ) 
    + r_σ”· (ζ_σ”-ξ_σ), 
       if  sinφ_σ≠ 0, 
    s_σ· (η_σ-ξ_σ)
    + s_σ'· (ζ_σ'-ξ_σ) 
    + s_σ”· (ζ_σ”-ξ_σ), 
       if  cosφ_σ≠ 0,

where

    r_σ    =     -[(y_σ - x_σ)· 
    (z_σ” - x_σ)](z_σ' - x_σ)
    + [(y_σ - x_σ)· (z_σ' - x_σ)]
    (z_σ” - x_σ)/ℓ_σ(y_σ-
    x_σ)·[(z_σ'-x_σ)× 
    (z_σ”- x_σ)], 
    r_σ'     =     -[(y_σ - x_σ)· 
    (z_σ” - x_σ)](y_σ - x_σ)
    - ℓ^2_σ(z_σ” - x_σ)/ℓ_σ(y_σ-
    x_σ)·[(z_σ'-x_σ)× 
    (z_σ”-x_σ)], 
    r_σ”    =     -[(y_σ - x_σ)· 
    (z_σ' - x_σ)](y_σ - x_σ)
    - ℓ^2_σ(z_σ' - x_σ)/ℓ_σ(y_σ-
    x_σ)·[(z_σ'-x_σ)× 
    (z_σ”-x_σ)], 
    s_σ    =     - ℓ_σ/θ (z_σ' - x_σ)× 
    (z_σ” - x_σ), 
    s_σ'     =    ℓ_σ/θ
    (y_σ - x_σ)× (z_σ” - x_σ), 
    s_σ”    =     - ℓ_σ/θ (y_σ - x_σ)× 
    (z_σ' - x_σ),

and

    θ = ℓ^2_σ(z_σ'-x_σ)
    ·(z_σ”-x_σ)-[(z_σ'-x_σ)
    · (y_σ-x_σ)]
    [(z_σ”-x_σ)· (y_σ-x_σ)].




Lemma <ref> follows immediately from Lemmas <ref> and <ref>.


In <cit.>, among other results, A.A. Gaifullin and L.S. Ignashchenko proved that the Dehn invariants of 
any flexible oriented polyhedron with non-degene­rate triangular faces in ℝ^3 do not alter during a flex.
In <cit.>, this fact was used to prove the following theorem

<cit.>
Let P_0  be an oriented compact boundary-free flexible polyhedron in ℝ^3
with non-degenerate triangular faces and let {P_t}_t∈ [0,1] be a flex of P_0.
Let ℓ_σ be the length of an edge σ of P_0 and let
L be the ℚ-linear span of the set ∪_σ⊂ P_0{ℓ_σ} in ℝ.
Let {λ_1,…,λ_m}  be a ℚ-basis in L,  i. e., the reals
λ_1,…,λ_m are ℚ-linearly independent and, for every σ,
there are α_σ j∈ℚ such that

    ℓ_σ=∑_j=1^m α_σ jλ_j.

Then, for every j=1, …, m, the expression 

    ∑_σ⊂ P_0α_σ jφ_σ (t)

is independent of t, i. e., it remains unaltered during the flex {P_t}_t∈ [0,1].
In (<ref>), the summation is taken over all edges σ of P_0 and 
φ_σ(t) stands for the internal dihedral angle of the polyhedron P_t at the edge σ(t).


Observe that Theorem <ref> implies the following statement:


Under the hypothesis of Theorem <ref>, for every j=1, …, m, the equality

    ∑_σ⊂ P_0α_σ j[g_σ· (η_σ-ξ_σ)
    + g_σ'· (ζ_σ'-ξ_σ)
    + g_σ”· (ζ_σ”-ξ_σ)] = 0

holds true, where

    g_σ=
    r_σ,    if  cosφ_σ= 0; 
    s_σ,    if  sinφ_σ= 0; 
    either r_σ or s_σ,    if  (cosφ_σ)(sinφ_σ)≠ 0,


    g_σ'=
    r_σ',    if  cosφ_σ= 0; 
    s_σ',    if  sinφ_σ= 0; 
    either r_σ'  or s_σ',    if  (cosφ_σ)(sinφ_σ)≠ 0,


    g_σ”= 
    r_σ”,    if  cosφ_σ= 0; 
    s_σ”,    if  sinφ_σ= 0; 
    either r_σ” or s_σ”,    if  (cosφ_σ)(sinφ_σ)≠ 0

and the vectors r_σ, r_σ', r_σ”, 
s_σ, s_σ', and s_σ”
are defined in Lemma <ref>.



Differentiate the expression (<ref>) with respect to t when t=0 and replace
d/dt|_t=0φ_σ (t)=Φ_σ by 
a suitable expression obtained in Lemma <ref>.


Generally speaking, for a given polyhedron P_0 we can write several equations (<ref>), not a single one.
There are several reasons for that.
Firstly, for each j=1,…, m, we can write a specific equation (<ref>).
Secondly, for each edge σ, such that (cosφ_σ)(sinφ_σ)≠ 0, 
there are two ways to write each of the coefficients 
g_σ, g_σ', and g_σ”.
That is, each edge of this type generates eight equations (<ref>).
In general, these eight equations are independent from each other, but the velocity vectors of
the vertices of a flexible polyhedron P_0 must satisfy each of them.



§ NECESSARY CONDITIONS FOR THE EXTENDIBILITY OF A FIRST-ORDER FLEX OF A POLYHEDRON TO ITS FLEX,
GENERATED BY THE DEHN INVARIANTS


Let P be an arbitrary polyhedron in ℝ^3, let σ be its edge,  
and let x_σ, y_σ, z_σ', z_σ” 
be the vertices of P, associated with σ as described in Section <ref>.
Let us assign arbitrary vectors a_σ, b_σ, c_σ', and 
c_σ” 
to the vertices x_σ, y_σ, z_σ', and 
z_σ”, respectively. 
Note that this notation is overdetermined in the following sense:
a vector, denoted as a_σ, b_σ, c_σ', or 
c_σ” for the edge σ, can get any of the notation
a_τ, b_τ, c_τ', or 
c_τ” for another edge τ.
Suppose that P is deformed in such a way that the verticies 
x_σ, y_σ, z_σ', z_σ” 
move according to the formulas

    x_σ+ta_σ,  y_σ+tb_σ,  z_σ'+tc_σ',  z_σ”+tc_σ”,
 
where t∈ℝ.
In Section <ref>, we called this deformation a first-order flex of P
if it changes the length of every edge of P by a value which is o(t) as t→ 0.
From now we say that, the set of vectors 
∪_σ⊂ P{a_σ, b_σ, 
c_σ', c_σ”} is a first-order flex of P 
if the deformation (<ref>) changes the length of every edge of P by a value which is o(t) as t→ 0.
Such a switch from deformations to collections of vectors is common in rigidity theory, see, e. g.,
<cit.> or <cit.>. 

The condition that the length of σ is stationary at t=0 for the  first-order flex
∪_σ⊂ P{a_σ, b_σ, 
c_σ', c_σ”} yields

    0     =    d/dt|_t=0[(y_σ+tb_σ
    -x_σ-ta_σ)·
    (y_σ+tb_σ-x_σ-ta_σ)]
        =    [2(y_σ-x_σ)· (b_σ-a_σ) + 
    2t(b_σ-a_σ)· (b_σ-a_σ)
    ]|_t=0
        =     2(y_σ-x_σ)· (b_σ-a_σ),

i. e.,

    (y_σ-x_σ)· (b_σ-a_σ)=0.

The equation (<ref>), corresponding to an edge σ of P,
is called an equation of the first-order flexes of P.

Denote by ⟨ FOF⟩ the set of all finite ℝ-linear combinations of the equations (<ref>). 
This notation reflects the fact that this set is generated by the equations of the first-order flexes.

We say that a first-order flex 
∪_σ⊂ P{a_σ, b_σ, 
c_σ', c_σ”} of P 
can be extended to a flex of P if there is a flex { P_t}_t∈ [0,1] of 
P=P_0, such that if the functions
x_σ(t), y_σ(t), z_σ'(t), z_σ”(t)
represent the motions of the vertices of P_t corresponding to the vertices 
x_σ, y_σ, z_σ', z_σ” 
of P_0 then

    a_σ=d/dt|_t=0x_σ(t), .   b_σ=d/dt|_t=0y_σ(t), .   c_σ'=d/dt|_t=0z_σ'(t) . ,   c_σ”=d/dt|_t=0z_σ”(t). .


Let us observe that Theorem <ref> implies the following statement:


Let P  be an oriented compact boundary-free flexible polyhedron in ℝ^3
with non-degenerate triangular faces.
Let ℓ_σ be the length of an edge σ of P and let
L be the ℚ-linear span of the set ∪_σ⊂ P{ℓ_σ} in ℝ.
Let {λ_1,…,λ_m}  be a ℚ-basis in L,  i. e., the reals
λ_1,…,λ_m are ℚ-linearly independent and, for every σ,
there are α_σ j∈ℚ such that

    ℓ_σ=∑_j=1^m α_σ jλ_j.

Let ∪_σ⊂ P{a_σ, b_σ, 
c_σ', c_σ”} be a first-order flex of P
which can be extended to a flex of P. 
Then, for every j=1, …, m, the equality

    ∑_σ⊂ Pα_σ j[g_σ· (b_σ-a_σ)
    + g_σ'· (c_σ'-a_σ)
    + g_σ”· (c_σ”-a_σ)] = 0

holds true, where the vectors
g_σ, g_σ', g_σ”
were defined in Theorem <ref>.



Immediately follows from Theorem <ref> and from the definition of a first-order flex 
which can be extended to a flex.


Denote by ⟨ DI⟩ the set of all finite ℝ-linear combinations of the equations (<ref>). 
This notation reflects the fact that this set is generated by the Dehn invariants.

In our opinion, the set ⟨ DI⟩ deserves deep study.
The following problems are especially interesting:

(i) How many new (compared to (<ref>)) equations are contained among the equations (<ref>), 
i. e., how much the dimension of the linear span of the set ⟨ DI⟩∪⟨ FOF⟩ 
is greater than  the dimension of the linear space ⟨ FOF⟩?

(ii) Characterize all n⩾ 2 such that the equations (<ref>) are not a consequence of the 
equations of the nth order flex.

In <cit.>, Hellmuth Stachel constructed  explicit examples of octahedra in ℝ^3 which
admit the first- and second-order flexes as well as explicit necessary and
sufficient conditions for an octahedron to be nth-order infinitesimally flexible for n<8,
provided the octahedron under consideration is not totally flat.
These results may be useful in the study of the problems (i) and (ii). 
But we left such a study for the future.

In Example 1 below, we answer the simplest problem similar to the problems (i) and (ii).
Namely, we show that, there are a polyhedron and its edge σ such that, the equation (<ref>),
written for σ, is linearly independent of all equations (<ref>).

Example 1. Let Q be a polyhedron in ℝ^3 shown in Fig. <ref>.

In other words, let Q be a triangular bipyramid with vertices p_i, i=1,…, 5.
Suppose that p_1, p_2, p_3, p_4
form a regular tetrahedron  with edge length 8 and p_5 lies inside the face 
p_2, p_3, p_4 in such a manner that the length of each of the 
two edges with the endpoints p_3, p_5 and p_4, p_5 
is equal to 5.
For convenience, we assume that 

    p_1 =  (4/3√(3)-3, 0, -8/3√(6)),  p_2 =  (4√(3)-3, 0, 0),
    p_3 =  (-3, 4, 0),  p_4 =  (-3,-4, 0),  p_5 =  (0, 0, 0).


Let us associate the velocity vector v_i to each of the vectors  p_i, i=1,…,5,
and put by definition v_5 = (0, 0, 1) and v_i=(0,0,0) for every  i=1,2,3,4.

Since v_5 is orthogonal to the plane containing the vertices p_i, i=2,3,4,5,
then (p_i-p_5)·v_5=0 for every i=2,3,4. 
These formulas differ from (<ref>) by notation only.
Hence, the vectors v_i, i=1,2,3,4,5, constitute the first-order flex of Q.

Now let us verify that the vectors v_i, i=1,2,3,4,5, do not satisfy the equation (<ref>), 
written for the edge with the end-points p_2, p_5.
Denote this egde by σ. Then
x_σ=p_2, y_σ=p_5, 
z_σ'=p_4, z_σ”=p_3, 
n_σ'=n_σ”=(0,0,1), 
ξ_σ=ζ_σ'=ζ_σ”=(0,0,0),
η_σ=v_5=(0,0,1), ℓ_σ=4√(3)-3, ℓ'_σ=ℓ”_σ=8.

By definition, put λ_1=1 and λ_2=4√(3)-3.
Since the length of any edge of Q is equal to 5, 8, or 4√(3)-3, we conclude that the set
{λ_1, λ_2} is a ℚ-basis in the ℚ-linear span of the set of all edge-lengths of Q.
Hence, α_σ 1=0, α_σ 2=1 and the equation  (<ref>) takes the form

    g_σ·v_5 =0,

where g_σ = s_σ since sinφ_σ=0.
Calculating s_σ according to a suitable formula from Lemma <ref>, we get

    g_σ= s_σ=(0,0,128-6√(3)/313).


Since v_5=(0,0,1), we conclude that the equality (<ref>) is not satisfied.
Thus, the equation (<ref>) corresponding to the above chosen edge σ is not satisfied.

Hence, for Q, one of the equations (<ref>) is not a linear combination of the equations 
(<ref>). This completes Example 1.



§ NECESSARY CONDITIONS FOR THE EXTENDIBILITY OF A FIRST-ORDER FLEX OF A POLYHEDRON TO ITS FLEX,
GENERATED BY THE RIGIDITY MATRIX


In Sections <ref> and <ref> we have shown that the Dehn invariants imply the equations (<ref>) 
which are necessary conditions for a given first-order flex of a polyhedron to be extendable to a flex.
In this Section we show that the rigidity matrix also implies a finite set of linearly independent equations
which are necessary conditions for a given first-order flex of a polyhedron to be extendable to a flex.
 
First we fix the notation which will differ slightly from the notation used in Sections <ref> and <ref>.

Let P_0 be a flexible polyhedron in ℝ^3 with triangular faces. 
(Note that in this Section we do not exclude the cases when P_0 is non-orientable or 
some of its faces are degenerated.) Suppose P_0 has V vertices.
Let us enumerate them in an arbitrary order and denote by 
p_i=(x_i, y_i, z_i)∈ℝ^3, i=1,…,V.
Let { P_t }_t∈ [0,1] be a flex of P_0.
Generally speaking, the vertices of the polyhedron P_t are functions of t.
Denote them by p_i(t)=(x_i(t), y_i(t), z_i(t)), i=1,…,V.
Moreover, we will assume that p_i(0)=p_i.

Let v_i(t)=(v_i,1(t), v_i,2(t), v_i,3(t))∈ℝ^3 
be the velocity vector of the vertex v_i(t), i. e., put by definition

    v_i(t)=d/dtp_i(t),   
    v_i,1(t)=d/dtx_i(t),   
    v_i,2(t)=d/dty_i(t),   
    v_i,3(t)=d/dtz_i(t).


If the vertices v_i and v_j of P_0 are joint with each other by an edge
then the length of the corresponding (by continuity) edge of P_t is independent of t, namely,

    (x_i(t)-x_j(t))^2  +  (y_i(t)-y_j(t))^2     +    (z_i(t)-z_j(t))^2 
        =     (x_i-x_j)^2 + (y_i-y_j)^2 + (z_i-z_j)^2.

Differentiating the last relation in t we get

    (x_i(t)-x_j(t))(v_i,1(t)-v_j,1(t)) 
    + (y_i(t)-y_j(t))(v_i,2(t)-v_j,2(t)) 
    
    + (z_i(t)-z_j(t))(v_i,3(t)-v_j,3(t)) = 0.


For a fixed t, the equations (<ref>) are linear with respect to 3V variables v_i,k(t), i=1,…,V, k=1,2,3.
Jointly, the equations (<ref>) form a homogeneous system of linear algebraic equations.
The matrix of this system is called the rigidity matrix of the polyhedron P_t.
It has 3V columns and E rows, where E stands for the number of edges of P_0.


Let P_0 be a flexible polyhedron in ℝ^3 with triangular faces and V vertices
and let { P_t }_t∈ [0,1] be a flex of P_0.
Then, for every t∈ [0,1], every (3V-7)× (3V-7) minor of the rigidity matrix of 
the polyhedron P_t is equal to zero.



Denote by F the number of faces of P_t. According to Euler's formula, 

    V-E+F=χ
    =
    2-2g,    if  P_t is orientable;
    
     2-k,    if  P_t is non-orientable.
 
Here χ is the Euler characteristic of P_t, g⩾ 0 is genus of P_t (i. e.,
the number of tori in a connected sum decomposition of P_t),  and k⩾ 0 is non-orientable genus of P_t
(i. e., the number of real projective planes in a connected sum decomposition of P_t).
Since P_t has triangular faces only,  3F=2E.
Substituting the latter relation to (<ref>), we obtain E=3V-3χ⩾ 3V-6. 
This inequality implies that the set of (3V-6)× (3V-6) minors of the rigidity matrix of P_t is non-empty.

Since the isometry group of ℝ^3 has dimension 6, the kernel of the rigidity matrix of 
any polyhedron with triangular faces has dimension greater than or equal to 6.
Therefore, every (3V-6)× (3V-6) minor of the rigidity matrix of P_t is equal to zero.

Recall that 

(a) the first-order flex of a polyhedron is called nontrivial if it is not
generated by a continuous family of isometries of ℝ^3 or, equivalently,
if there are two vertices of the polyhedron (which are not connected by an edge) 
such that the Euclidean distance between these vertices is not stationary;

(b) a polyhedron is called infinitesimally flexible, if it admits a nontrivial first-order flex;

(c) a polyhedron in ℝ^3 with triangular faces and V vertices is infinitesimally flexible 
if and only if every (3V-7)× (3V-7) minor of its rigidity matrix is equal to zero;

(d) every flexible polyhedron is infinitesimally flexible.

These definitions and statements are standard in rigidity theory.
The reader can find them, for example, in <cit.> (for the simplest case of sphere-homeomorphic polyhedra) 
or in <cit.> and <cit.> (for a more general case of tensegrity frameworks).

Now we a ready to finalize the proof of Lemma <ref>.
Obviously, P_t is flexible for every t∈ [0,1]. According to (d), it is infinitesimally flexible.
Then, (c) implies that every (3V-7)× (3V-7) minor of the rigidity matrix of P_t is equal to zero.


As before, suppose that P_0 is a flexible polyhedron in ℝ^3 with triangular faces and V vertices,
{ P_t }_t∈ [0,1] is a flex of P_0, and x_1(t), y_1(t), z_1(t), …, x_V(t), y_V(t), z_V(t)
stand for the coordinates of the vertices of P_t.
Then, for every t∈ [0,1], every  (3V-7)× (3V-7) minor Δ(t) of the rigidity matrix of P_t is 
a homogeneous polynomail in the variables
x_1(t), y_1(t), z_1(t), …, x_V(t), y_V(t), z_V(t).
Moreover, according to Lemma <ref>, this polynomial is equal to zero if the values of these variables 
correspond to P_t.
Differentiating Δ(t) with respect to t when t=0 and replacing the derivatives of the variables
x_1(t), y_1(t), z_1(t), …, x_V(t), y_V(t), z_V(t) 
according to the formulas (<ref>), yields that some finite linear combination of the vectors
v_1(0), v_2(0), …, v_V(0)
is equal to zero.
The coefficients of this linear combination are homogeneous polynomials in the variables
x_1(0), y_1(0), z_1(0), …, x_V(0), y_V(0), z_V(0).
Each such linear combination is a new (compared with the stationarity condition of the edge lengths)
relation that should be satisfied by every first-order flex
v_1(0), v_2(0), …, v_V(0) of P_0 
which can be extended to a flex { P_t }_t∈ [0,1].

Let us list a few problems related to the above constructed new linear relations for the first-order flexes:

(A) Is it true that the stationarity of all (3V-7)× (3V-7) minors of the rigidity matrix is a 
consequence of the stationarity of the edge lengths?

(B) How many new linearly independent relations are generated by (3V-7)× (3V-7) minors of 
the rigidity matrix of a given infinitesimally flexible polyhedron?

(C) How these new relations are related to the possibility of extension of a given first-order flex
to an nth order flex for n⩾ 2?

(D) Do these new relations imply that the volume bounded by a flexible polyhedorn is stationary?
(It is known that the stationarity of the edge lengths only does not imply the stationarity of the
volume, see, e. g., <cit.>).

In Example 2, we make the first step in the study of the problems (A)–(D).
Namely, we give a negative answer to the problem (A).

Example 2. Let Q be the polyhedron with V=5 vertices constructed in Example 1.
Let us make sure that at least one linear relation for the first-order flexes obtained by differentiating 
the (3V-7)× (3V-7) minors of the rigidity matrix of Q is not a consequence of
the equations expressing the stationarity of the edge lengths of Q.

To do this, we consider a first-order flex ∪_i=1^5{v_i} of Q
such that  v_i=0 for every i=1,…, 4 and describe all vectors
v_5 such that all (3V-7)× (3V-7) minors of the rigidity matrix of Q are stationary 
at t=0 for the deformation 

    t↦p_i+tv_i,    t∈ℝ,    i=1,…, 5.


Obviously, the vector v_5 must lie in the plane passing through the vertices
p_2, p_3, p_4 of Q.
This follows from the fact that every polyhedron Q' with the vertices p_i', i=1,…,5, 
which has the same combinatorial structure as Q and is sufficiently close to Q
(i. e., such that the length of each of the vectors p_i'-p_i, i=1,…,5,
does not exceed a sufficiently small number ε>0) 
is infinitezimally rigid if and only if the vertex p_5' lies in the plane passing through the vertices
p_2', p_3', p_4'.

Since ∪_i=1^5{v_i} is a first-order flex of Q, 
v_5 is orthogonal to the plane passing through the vertices p_2, p_3, 
p_4 of Q. 
For v_5≠ 0, this implies that v_5 does not lie in this plane
and, thus, some of the (3V-7)× (3V-7) minors of the rigidity matrix of Q is not stationary under the deformation
(<ref>) at t=0.
  
plain

