Anchored boundary conditions for locally isostatic networks
Aalto Science Institute (AScI) and Department of Computer Science (CS),
Aalto University, PO Box 15500, 00076 Aalto, Finland
Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, England
MESH Consultants Inc., Fields Institute for Research in the Mathematical Sciences,
222 College Street, Toronto, ON, M5T 3J1, Canada
Department of Physics, Arizona State University, Tempe, AZ 85287-1504, USA
Department of Mathematical Sciences, Worcester Polytechnic Institute,
100 Institute Road, Worcester, MA 01609, USA
M. F. Thorpe∗∗
Department of Physics, Arizona State University, Tempe,
AZ 85287-1504, USA and Rudolf Peierls Centre for Theoretical Physics,
University of Oxford, 1 Keble Rd, Oxford OX1 3NP, England
Finite pieces of locally isostatic networks have a large number of ﬂoppy modes because of missing
constraints at the surface. Here we show that by imposing suitable boundary conditions at the
surface, the network can be rendered eﬀectively isostatic. We refer to these as anchored boundary
conditions. An important example is formed by a two-dimensional network of corner sharing trian-
gles, which is the focus of this paper. Another way of rendering such networks isostatic, is by adding
an external wire along which all unpinned vertices can slide (sliding boundary conditions). This ap-
proach also allows for the incorporation of boundaries associated with internal holes and complex
sample geometries, which are illustrated with examples. The recent synthesis of bilayers of vitreous
silica has provided impetus for this work. Experimental results from the imaging of ﬁnite pieces at
the atomic level needs such boundary conditions, if the observed structure is to be computer-reﬁned
so that the interior atoms have the perception of being in an inﬁnite isostatic environment.
arXiv:1508.00666v1 [cond-mat.dis-nn] 4 Aug 2015
FIG. 1. Showing a piece of bilayer of vitreous silica imaged in SPM (Scanning Probe Microscope)5to show the Si atoms as
red discs and the O atoms as black discs. The local covalent bonding leads to the yellow almost-equilateral triangles that are
freely jointed, which we will refer to as pinned. The triangles at the surface have either one or two vertices unpinned.
Boundary conditions are paramount in many areas of computer modeling in science. At the atomic level, ﬁnite
samples require appropriate boundary conditions in order that atoms in the interior behave as if they were part of a
larger or inﬁnite sample, or as closely to this as is possible. One example of this is the calculation of the electronic
properties of covalent materials where the surface is terminated with H atoms so that all the chemical valency is
satisﬁed. In this way the HOMO (highest occupied molecular orbital) and the LUMO (lowest unoccupied molecular
orbital) states inside the sample can be obtained that are not very diﬀerent from those expected in the bulk sample.
In materials science the electronic band structure of a sample of crystalline Si could be obtained by determining the
electronic properties of a ﬁnite cluster terminated with H bonds at the surface. In practice this is rarely done, as it
is more convenient to use periodic boundary conditions and hence use Bloch’s theorem, but this technique has been
used recently in graphene nanoribbons1.
For most samples, the nature of the boundary, ﬁxed, free or periodic only alters the properties of the sample by the
ratio of the number of atoms on the surface to those in the bulk. This ratio is N−1
dwhere Nis the number of atoms
(later referred to as vertices) and dis the dimension. Of course this ratio goes to zero in the thermodynamic limit
as the size of the system N→ ∞ and leads to the important result that properties become independent of boundary
conditions for large enough systems.
Similar statements can be made about the mechanical and vibrational properties of systems except for isostatic
networks that lie on the border of mechanical instability. In this case the boundary conditions are important no
matter how large N, and special care must be taken with devising boundary conditions so that the interior atoms
behave as if they were part of an inﬁnite sample, in as much as this is possible2–4.
In Figure 1, we show a part of a Scanning Probe Microscope (SPM) image5of a bilayer of vitreous silica which
has the chemical formula SiO2. The sample consists of an upper layer of tetrahedra with all the apexes pointing
downwards where they join a mirror image in the lower layer. In the ﬁgure we show the triangular faces of the upper
tetrahedra, which form rigid triangles with a (red) Si atom at the center and the (black) O atoms at the vertices of
the triangles which are freely jointed to a good approximation. We refer to these networks as locally isostatic as the
number of degrees of freedom of the equilateral triangle in two dimensions is exactly balanced by the shared pinning
constraints (2 at each of the 3 vertices, so that 3 −2×3/2 = 0). While the 3D bilayers are locally isostatic, so too
are the 2D projections of corner-sharing triangles which are the focus of this paper. We will use the Berlin A sample
as the example throughout6,7 so that we can focus on this single geometry for pedagogical purposes.
In this paper, we show rigorously that there are various ways to add back the exact number of missing constraints
at the surface, in a way that they are suﬃciently uniformly distributed around the boundary that the network is
guaranteed to be isostatic everywhere. There is some limited freedom in the precise way these boundary condition
are implemented, and the boundary can be general enough to include internal holes. The proof used here involves
FIG. 2. Illustrating sliding boundary conditions, used for a piece of the sample shown in Figure 1. The boundary sites are
shown as blue discs and the 3 purple triangles at the lower left Figure 3 have been removed. The red Si atoms at the centers
of the triangles in Figure 1 have also been removed for clarity. The boundary is formed as a smooth analytic curve by using
a Fourier series 16 sine and 16 cosines terms to match the number of surface vertices, where the center for the radius r(θ) is
placed at the centroid of the 32 boundary vertices13. Note that sliding boundary conditions do not require an even number of
showing that all subgraphs have insuﬃcient edges for redundancy to occur8.
Using the pebble game9,10 , we veriﬁed on a number of samples that anchored boundary conditions in which alter-
nating free vertices are pinned results in a global isostatic state. The pebble game is an integer algorithm, based on
Laman’s theorem8, which for a particular network performs a rigid region decomposition, which involves ﬁnding the
rigid regions, the hinges between them, and the number of ﬂoppy (zero-frequency) modes. We have used it to conﬁrm
that the locally isostatic samples such as that in this paper are isostatic overall with anchored boundary conditions.
The results of this paper imply that, under a relatively mild connectivity hypothesis, this procedure is provably cor-
rect, and thus, relatively robust. Additionally, the necessity of running the pebble game for each individual case is
Figure 2 shows sliding boundary condition11. These make use of a diﬀerent, simpler kind of geometric constraint
at each unpinned surface site. The global eﬀect on the network’s degrees of freedom is like that of the anchored
boundary conditions, and this setup is computationally reasonable. At the same time, the proofs for this case are
simpler, and generalize more easily to handle situations such as holes in the sample.
In Figure 3, we show the anchored boundary conditions. We have trimmed oﬀ the surface triangles in Figure 1 that
are only pinned at one vertex. This makes for a a more compact structure whose properties are more likely to mimic
those of a larger sample, and makes our mathematical statements easier to formulate. In addition we have had to
remove the 3 purple triangles at the lower right hand side in order to get an even number of unpinned surface sites.
When the network is embedded in the plane, this is possible, except for very degenerate samples.12
II. COMBINATORIAL ANCHORING
Intuitively, the internal degrees of freedom of systems like the ones in Figures 1 and 3 correspond to the corners
of trianges that aren’t shared. This is, in essence, the content of Lemma II.1 proved below. Proving Lemma II.1
requires ruling out the appearance of additional degrees of freedom that could arise from sub-structures that contain
more constraints than degrees of freedom.
The essential idea behind combinatorial rigidity14 is that generically all geometric constraints are visible from the
topology of the structure, as typiﬁed by Laman’s8striking result showing the suﬃciency of Maxwell counting15 in
dimension 2. Genericity means, roughly, that there is no special geometry present; in particular, generic instances of
any topology are dense in the set of all instances.
In what follows, we will be assuming genericity, and rely on results similar to Laman’s, in that they are based on an
FIG. 3. Illustrating the anchored boundary conditions used for the sample shown in Figure 1. The alternating anchored sites
on the boundary are shown as blue discs and the 3 purple triangles at the lower left are removed to give an even number of
unpinned surface sites. The red Si atoms at the centers of the triangles in Figure 1 have been suppressed for clarity.
FIG. 4. Showing a typical subgraph from Figure 3 used in the proof that there are no rigid subgraphs larger than a single
triangle. (See Lemma II.3.)
appropriate variation of Maxwell counting. Our proofs have a graph-theoretic ﬂavor, which relate certain hypothesis
about connectivity16 to hereditary Maxwell-type counts.
A. Triangle ring networks
We will model the ﬂexibility in the upper layer of vitrious silica bilayers as systems of 2D triangles, pinned togehter
at the corners. The joints at the corners are allowed to rotate freely. A triangle ring network is rigid if the only
available motions preserving triangle shapes and the network’s connectivity are rigid body motions; it is isostatic if
it is rigid, but ceases to be so once any joint is removed. These are an examples of body-pin networks17 from rigidity
The combinatorial model is a graph Gthat has one vertex for each triangle and an edge between two triangles if
they share a corner (Figure 6). Since we are assuming genericity, we will identify a geometric realization with the
graph Gfrom now on. In the sequel, we are interested in a particular class of graphs G, which we call triangle ring
networks. The deﬁnition of a triangle ring is as follows: (a) Ghas only vertices of degree 2 and 3; Gis 2-connected18 ;
(b) there is a simple cycle Cin Gthat contains all the degree 2 vertices, and there are at least 3 degree 2 vertices;
(c) any edge cut set19 in Gthat disconnects a subgraph containing only degree 3 vertices has size at least 3.
To set up some terminology, we call the degree 2 vertices boundary vertices and the degrees 3 vertices interior
vertices. A subgraph spanning only interior vertices is an interior subgraph.
The reader will want to keep in mind the speciﬁc case in which Gis planar with a given topological embedding and
Cis the outer face, as is the case in our ﬁgures. This means that subgraphs strictly interior to the outer face have
FIG. 5. Illustrating two, at ﬁrst sight, more complex anchored boundary conditions that by our results can be used for the
sample shown in Figure 2, with the 3 purple triangles at the lower left are removed to give an even number of unpinned surface
sites. The anchored sites are shown as blue discs, with an even number of surface sites in both graphs. The graph at the right
has an even number of surface sites in both the outer and inner boundary. The red Si atoms at the centers of the triangles have
been suppressed for clarity. The green line goes through the boundary triangles.
FIG. 6. The triangle ring network, complementary to that in Figure 3, where the Si atoms, shown as red discs, at the center
of each triangle are emphasized in this three-coordinated network. Dashed edges are shown connecting to the anchored sites.
only interior vertices, which explains our terminology. However, as we will discuss in detail later, the setup is very
general. If the sample has holes, Ccan leave the outer boundary and return to it: provided that it is simple, all the
results here still apply.
A theorem of Tay–Whiteley20,21 gives the degree of freedom counts for networks of 2-dimensional bodies pinned
together. Generically, there are no stressed subgraphs in such a network, with graph G, of vbodies and epins if and
2e0≤3v0−3 for all subgraphs G0⊂G. (1)
where v0and e0are the number of vertices and edges of the subgraph. If (1) holds for all subgraphs, the rigid subgraphs
are all isostatic, and they are the subgraphs where (1) holds with equality.
Lemma II.1. Any triangle ring network Gsatisﬁes (1).
Proof. Suppose the contrary. Then there is a vertex-induced subgraph Ton v0vertices that violates (1). If Tcontains
a vertex vof degree 1 then T−valso violates (1) so we may assume that Thas minimum degree 2. In this case, T
has at most 2 vertices of degree 2, since it has maximum degree 3. In particular, Tmay be disconnected from Gby
removing at most 2 edges. If Tis an interior subgraph, we get a contradiction right away. Alternatively, at least one
of the degree 2 vertices in Tis degree 2 in G, and so on C. If exactly one is, then Gis not 2-connected. If both are,
then T=Gand there are only 2 boundary vertices. Either case is a contradiction.
Corollary 1. The rigid subgraphs of Gare the subgraphs containing exactly 3vertices of degree 2and every other
vertex has degree 3. Moreover, any proper rigid subgraph contains at most one boundary vertex of G.
Proof. The ﬁrst statement is straightforward. The second follows from observing that if a rigid subgraph Thas two
vertices on the boundary of G, then Gcannot be 2-connected, since all the edges detaching Tfrom Gare incident on
a single vertex.
When Gis planar, these rigid subgraphs are regions cut out by cycles of length 3 in the Poincar´e dual. More
generally in the planar case, subgraphs corresponding to regions that are smaller triangle ring networks with tdegree
2 vertices have tdegrees of freedom.
B. Anchoring with sliders
Now we can consider our ﬁrst anchoring model, which uses slider pinning11. A slider constrains the motion of a
point to remain on a ﬁxed line, rigidly attached to the plane. When we talk about attaching sliders to a vertex of the
graph, we choose a point on the corresponding triangle, and constrain its motion by the slider. In the results used
below, this point should be chosen generically; for example the theory does not apply if the slider is attached at a
pinned corner shared by two of the triangles. Since we are only attaching sliders to triangles corresponding to degree
2 vertices in N, we may always attach sliders at an unpinned triangle corner.
The notion of rigidity for networks of bodies with sliders is that of being pinned: the system is completely
immobilized.22 A network with sliders is pinned-isostatic if it is pinned, but ceases to be so if any pin or slider
The equivalent of the White-Whiteley counts in the presence of sliders is a result of Katoh and Tanigawa23, which
says that a generic slider-pinned body-pin network Gis independent if and only if the body-pin graph satisﬁes (1)
2e0+s0≤3v0for all subgraphs G0⊂G, (2)
where s0is the number of sliders on vertices of G0. Here is our ﬁrst anchoring procedure.
Theorem 2. Adding one slider to each degree 2boundary vertex of a triangle ring network Ggives a pinned-isostatic
Proof. Let Tbe an arbitrary subgraph with v0vertices and v00 vertices of degree at most 2. That (1) holds is Lemma
II.1. The fact that the only vertices of Twhich get a slider are vertices with degree 2 in Gimplies that (2) is also
satisﬁed, and, by construction 2e+s= 3v.
We may think of this anchoring as rigidly attaching a rigid wire to the plane then constraining the boundary
vertices to move on it. Provided that the wire’s path is smooth and suﬃciently non-degenerate, this is equivalent, for
analyzing inﬁnitesimal motions, to putting the sliders in the direction of the tangent vector at each boundary vertex.
See also Figure 2
C. Anchoring with immobilized triangle corners
Next, we consider anchoring Gby immobilizing (pinning) some points completely. Combinatorially, we model
pinning a triangle’s corner by adding two sliders through it. Since we are still using sliders, the deﬁnitions of pinned
and pinned-isostatic are the same as in the previous section.
The analogue for (2) when we add sliders in groups of 2 is:
2e0+ 2s0≤3v0for all subgraphs G0⊂G, (3)
where sis the number of immobilized corners.
FIG. 7. Illustrating even more complex boundaries, developed from the sample shown in Figure 3 by removing triangles to
form two internal holes. The boundary sites are shown as blue discs and the 3 purple triangles at the lower left Figure 3 have
been removed. The red Si atoms at the centers of the triangles in Figure 1 have also been removed for clarity. The green line
forms a continuous boundary which goes through all the surface sites which must be an even numbers. The anchored (blue)
sites then alternate with the unpinned sites on the green boundary curve which has to cross the bulk sample in two places to
reach the two internal holes. Here there are 32 boundary sites, 5 boundary sites in the upper hole and 7 in the lower hole,
giving a total even number of 44 boundary sites. Where these crossings take place is arbitrary, but it is important that the
anchored and unpinned surface sites alternate along whatever (green) boundary line is drawn.
Theorem 3. Let Gbe a triangle ring network with an even number tof degree 2vertices on C. Then, following C
in cyclic order, pinning every other boundary vertex that is encountered results in a pinned-isostatic network.
Proof. Let Tbe an arbitrary subgraph of G. If at most one of the vertices of Tare pinned, there is nothing to do.
For the moment, suppose that no vertex of degree 1 in Tis pinned. Let tbe the number of pinned vertices in T.
We will show that for each of the tpinned vertices, there is a distinct unpinned vertex of degree 1 or 2 in t. This
implies that 2e0≤3v0−2tin T, at which point we know (3) holds for T.
To prove the claim, let vbe a pinned vertex of T. Traverse the boundary cycle Cfrom v. Let wbe the next
pinned vertex of Tthat is encountered. If the chain from vto walong Cis in T, the alternating pattern provides
an unpinned degree 2 vertex that is degree 2 in G. Otherwise, this path leaves T, which can only happen at a vertex
with degree 1 or 2 in T. Continuing the process until we return to v, produces at least tdistinct unpinned degree 2
vertices, since each step considers a disjoint set of vertices of C.
Now assume that Tdoes have a pinned vertex vof degree 1. The theorem will follow if (3) holds strictly for T−v.
Let wand xbe the pinned vertices in Timmediately preceding and following v. The argument above shows that
there are at least 2 unpinned degree 1 or 2 vertices in Ton the path in Cbetween wand xon C. Since these are in
T−v, we are done.
When there are an odd number of boundary vertices in G, Theorem 3 does not apply. This next lemma gives a
simple reduction in many cases of interest.
Lemma II.2. Let Gbe a planar triangle ring network, with Cthe outer face. Suppose that there are an odd number
tof boundary vertices. If Gis not a single cycle, then it is possible to obtain a network with an even number of
boundary vertices by removing the intersection of a facial cycle of Gwith C, unless G=C.
Sketch. The connectivity requirements for a triangle ring network, combined with planarity of Gimply that the
intersection of Cand any facial cycle Dof Gis a single chain. Every boundary vertex is in the interior of such a
chain, so some facial cycle Dcontributes an odd number of boundary vertices. Removing the edges in D∩Cchanges
the parity of the number of boundary vertices.
FIG. 8. Illustrating two additional boundary conditions used for the sample shown in Figure 3, with the 3 purple triangles at
the lower left removed to give an even number of unpinned surface sites. On the left, alternating surface sites are connected to
one another through triangulation of ﬁrst and second neighbors, with the last three connections not needed (these would lead
to redundancy). Hence there are three additional macroscopic motions when compared to Figure 3 which can be considered
as being pinned to the page rather than to the internal frame shown by green straight lines here. On the right we illustrate
anchoring with additional bars which connect all unpinned surface sites, except again three are absent, to avoid redundancy,
and to give the three additional macroscopic motions when compared to Figure 3
D. Anchoring with additional bars
So far, we have worked with networks of triangles pinned together. Now we augment the model to also include bars
between pairs of the triangles. We will always take the endpoints of the bars to be free corners of triangles that are
boundary vertices in the underlying network G. Combinatorially we model this by a graph Hon the same vertex set
as G, with an edge for each bar between a pair of bodies. In this case, the Tay–Whiteley count becomes:
2e0+b0≤3v0−3 for all subgraphs G0⊂G. (4)
where e0is the number of edges in G0and b0is the number of edges in Hspanned by the vertices of G0. The anchoring
procedures with sliders or immobilized vertices have analogues in terms of adding bars to create an isostatic network.
These boundary conditions are illustrated on the right hand side of Figure 8. Also shown in Figure 8 in the left panel
is a triangular scheme involving alternating unpinned surface sites, that is equivalent to anchoring. In both cases
shown here the sample is free to rotate with respect to the page.
Theorem 4. If Nhas boundary vertices v1, . . . , vt, we obtain an isostatic framework by taking the edges of Hto be
v1v2, v2v3, . . . , vt−3vt−2.
Proof. Consider the t−3 new bars. By construction and Lemma II.1 we have 2e+t−3 = 3v−t+t−3 = 3v−3.
Corollary 1 and the connectivity hypotheses imply that no rigid subgraph of Nhas more than 1 of its 3 degree 2
vertices on the boundary of N. This shows that no rigid subgraph of Nhas a bar added to it.
Theorem 5. If Ghas tboundary vertices and tis even, then taking Hto be any isostatic bar-joint network with vertex
set consisting of t/2boundary vertices chosen in an alternating pattern around Cresults in an isostatic network.
A triangulated t/2-gon is a simple choice for H.
Sketch. By Lemma II.1, we are adding enough bars to remove all the internal degrees of freedom. The desired statement
then follows from Theorem 3 by observing that pinning down the boundary vertices is equivalent, geometrically, to
pinning down Hand then identifying the boundary vertices of Gto the vertices of H.
A result of White and Whiteley24 on “tie downs”, then gives:
Corollary 6. In the situation of Theorems 4 and 5, adding any 3 sliders results in a pinned-isostatic network.
E. Stressed regions
The results so far have shown how to render a ﬂoppy triangle ring network isostatic or pinned-isostatic. It is
interesting to know when adding a single extra bar or slider results in a network that is stressed over all its members.
This is a somewhat subtle question when adding bars or immobilizing vertices, but it has a simple answer for the
sliding boundary conditions.
We say that a triangle ring network is irreducible if: (a) every minimal 2 edge cut set either detaches a single vertex
from Gor both remaining components contain more than one boundary vertex of G; (b) every minimal 3 edge cut
set disconnects one vertex from G.
Lemma II.3. A triangle ring network Ghas no proper rigid subgraphs if and only if Gis irreducible.
Proof. Recall, from Corollary 1, that a proper rigid subgraph Tof Ghas exactly 3 vertices of degree 2 and the rest
degree 3. Thus, Tcan be disconnected from Gby a cut set of size 2 or 3.
In the former case, Corollary 1 implies that exactly one of the degree 2 vertices in Tis a boundary vertex of G.
This means that Twitnesses the failure of (a), and Nis not irreducible. Conversely, (a) implies that, for a 2 edge cut
set not disconnecting one vertex, either side is either a chain of boundary vertices or has at least 4 vertices of degree
Finally, observe that cut sets of size 3 are minimal if and only if they disconnect an interior subgraph on one side.
Corollary 1 then implies that there is a proper rigid component that is an interior subgraph of Gif and only if (b)
Theorem 7. Let Gbe a triangle ring network anchored using the procedure of Theorem 2. Adding any bar or slider
to Gresults in a network with all its members stressed if and only if Gis irreducible.
Proof. First consider adding a slider. Because Gis pinned-isostatic, the slider creates a unique stressed subgraph T.
A result of Streinu-Theran11 implies that Tmust have been fully pinned in G. Since any proper subgraph has an
unpinned vertex of degree 1 or 2, (2) holds strictly. Thus, the stressed graph is all of N.25
If we add a bar, there is also a unique stressed subgraph. This will be all of Gby11, unless both endpoints of the
bar are in a common rigid subgraph. That was ruled out by assuming that Gis irreducible.
In this paper we have demonstrated boundary conditions for locally isostatic networks that incorporate the right
number of constraints at the surface so that the whole network is isostatic. These boundary conditions should be
useful in numerical simulations which involve ﬁnite pieces of locally isostatic networks. The boundary can be quite
complex and involve both an external boundary with internal holes.
Although our deﬁnition of a triangle ring network is most easily visualized when Gis planar and Cis the outer
face, the combinatorial setup is quite a bit more general. The natural setting for networks with holes is to assume
planarity, and then that all the degree 2 vertices are on disjoint facial cycles in G. The key thing to note is that the
cycle Cin our deﬁnition does not need to be facial for Theorem 3. For example, in Figure 7, Cgoes around the
boundary of an interior face that contains degree 2 vertices. In general, the existence of an appropriate cycle Cis a
non-trivial question, as indicated by Figure 7 (See also Figure 5 for other examples of complex anchored boundary
What is perhaps more striking is that Theorem 2 still applies whether or not such a Cexists, provided faces in N
deﬁning the holes in the sample are disjoint from the boundary and each other.
In applying anchored boundary conditions, it is important that the complete boundary has an even number of
unpinned sites, which can include internal holes, which must then be connected using the green lines shown in the
various ﬁgures. This gives a practical way of setting up calculations with anchored boundary conditions in samples
with complex geometries and missing areas.
Support by the Finnish Academy (AKA) Project COALESCE is acknowledged by LT. We thank Mark Wilson and
Bryan Chen for many useful discussions and comments. This work was initiated at the AIM workshop on conﬁguration
spaces, and we thank AIM for its hospitality.
∗Email: louis.theran@aalto.ﬁ; Web: http://theran.lt
†Email: email@example.com; Web: http://www.lancaster.ac.uk/maths/about-us/people/anthony-nixon
‡Email: firstname.lastname@example.org; Web: http://www.elissaross.ca
¶Email: email@example.com; Web: http://users.wpi.edu/˜bservat
∗∗ Email: firstname.lastname@example.org; Web: http://thorpe2.la.asu.edu/thorpe
1O. Hod, J. E. Peralta, and G. E. Scuseria, Phys. Rev. B 76, 233401 (2007).
2M. Thorpe, Journal of Non-Crystalline Solids 182, 135 (1995).
3T. C. Lubensky, C. L. Kane, X. Mao, A. Souslov, and K. Sun, Reports on Progress in Physics 78, 073901 (2015).
4W. G. Ellenbroek, V. F. Hagh, A. Kumar, M. F. Thorpe, and M. van Hecke, Phys. Rev. Lett. 114, 135501 (2015).
5L. Lichtenstein, C. B¨uchner, B. Yang, S. Shaikhutdinov, M. Heyde, M. Sierka, R. W lodarczyk, J. Sauer, and H.-J. Freund,
Angewandte Chemie International Edition 51, 404 (2012).
6M. Wilson, A. Kumar, D. Sherrington, and M. F. Thorpe, Phys. Rev. B 87, 214108 (2013).
7A. Kumar, D. Sherrington, M. Wilson, and M. F. Thorpe, Journal of Physics: Condensed Matter 26, 395401 (2014).
8G. Laman, J. Engrg. Math. , 331.
9D. J. Jacobs and M. F. Thorpe, Phys. Rev. Lett. 75, 4051 (1995).
10 D. J. Jacobs and M. F. Thorpe, Phys. Rev. E 53, 3682 (1996).
11 I. Streinu and L. Theran, Discrete Comput. Geom. 44, 812 (2010).
12 If all rings intersect the sample boundary in a contiguous chain of triangles, and there are at least 2 such rings, some ring
intersects the surface in a chain containing an odd number of unpinned sites. We can remove this chain to get an even
number of unpinned surface sites.
13 M. Sadjadi, M. Wilson, and M. F. Thorpe, “Computer reﬁnement of experimentally determined structures at the atomic
level,” Preprint, in preparation (2015).
14 See, e.g., the monograph by Graver, et al.26 for an introduction.
15 J. C. Maxwell, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 27, 294 (1864).
16 To make this paper somewhat self-contained, we will brieﬂy explain the concepts we use. Our terminology is standard, and
can be found in, e.g., the textbook by Bondy and Murty27.
17 Since only two triangles are pinned together at any point, we are dealing with the 2-dimensional specialization of body-hinge
frameworks ﬁrst studied by Tay21 and Whiteley20 in general dimensions. In 2D, there is a richer combinatorial theory of
“body-multipin” structures, introduced by Whiteley28. See Jackson and Jord´an29 and the references therein for an overview
of the area.
18 This means that to disconnect G, we need to remove at least 2 vertices.
19 This is a inclusion-wise minimal set of edges that, when removed from G, results in a graph 2 connected components.
20 W. Whiteley, SIAM J. Discrete Math. 1, 237 (1988).
21 T.-S. Tay, Graphs Combin. 5, 245 (1989).
22 Rigid body motions are not “trivial”, because slider constraints are not preserved by them.
23 N. Katoh and S.-i. Tanigawa, SIAM J. Discrete Math. 27, 155 (2013).
24 N. L. White and W. Whiteley, SIAM J. Algebraic Discrete Methods 4, 481 (1983).
25 It is worth noting that, so far, irreducibility of Nwas not required. It is needed only for adding bars.
26 J. Graver, B. Servatius, and H. Servatius, Combinatorial rigidity, Graduate Studies in Mathematics, Vol. 2 (American
Mathematical Society, Providence, RI, 1993) pp. x+172.
27 J. A. Bondy and U. S. R. Murty, Graph theory, Graduate Texts in Mathematics, Vol. 244 (Springer, New York, 2008) pp.
28 W. Whiteley, Discrete Comput. Geom. 4, 75 (1989).
29 B. Jackson and T. Jord´an, Discrete & Computational Geometry 40, 258 (2008).