# Graph-theoretic estimation of reconfigurability in origami-based
metamaterials

Koshiro Yamaguchi  Hiromi Yasuda  Kosei Tsujikawa  Takahiro Kunimine  Jinkyu
Yang<EMAIL_ADDRESS>William E. Boeing Department of Aeronautics and
Astronautics, University of Washington, Seattle, Washington 98195-2400, USA
Department of Mechanical Engineering and Applied Mechanics, University of
Pennsylvania, Philadelphia, PA 19104, USA  Division of Mechanical Science and
Engineering, Graduate School of Natural Science and Technology, Kanazawa
University, Kanazawa, Ishikawa, 920-1192, Japan  Faculty of Mechanical
Engineering, Institute of Science and Engineering, Kanazawa University,
Kanazawa, Ishikawa, 920-1192, Japan

###### Abstract

Origami-based mechanical metamaterials have recently received significant
scientific interest due to their versatile and reconfigurable architectures.
However, it is often challenging to account for all possible geometrical
configurations of the origami assembly when each origami cell can take
multiple phases. Here, we investigate the reconfigurability of a tessellation
of origami-based cellular structures composed of bellows-like unit cells,
specifically Tachi-Miura Polyhedron (TMP). One of the unique features of the
TMP is that a single cell can take four different phases in a rigid foldable
manner. Therefore, the TMP tessellation can achieve various shapes out of one
originally given assembly. To assess the geometrical validity of the
astronomical number of origami phase combinations, we build a graph-
theoretical framework to describe the connectivity of unit cells and to
analyze the reconfigurability of the tessellations. Our approach can pave the
way to develop a systematic computational tool to design origami-based
mechanical metamaterials with tailored properties.

###### keywords:

Mechanical metamaterials , Graph theory , Reconfigurable systems

††journal: Materials and Design

##  1 Introduction

Mechanical metamaterials can have a wide range of mechanical properties by
leveraging the design freedom in their architectures [1, 2, 3]. Origami has
helped this emerging platform to add the tunability of its shapes and
behaviors after fabrications [4, 5]. Recently, an extensive amount of research
on origami-based metamaterials has been conducted to show their various
characteristics in both mathematical and mechanical aspects [6, 7]. Particular
interest has been placed on the realization of reconfigurable mechanical
metamaterials via the concept of origami, kirigami, or equivalent
architectures [8, 9, 10, 11, 12, 13]. While most research focuses on the
homogeneous arrangements of origami units and their intrinsic properties, the
cross-linkage of heterogeneous origami cells and their effective global
characteristics have been relatively unexplored.

To address this connectivity and reconfigurability problem of heterogeneous
origami architectures, Tachi-Miura Polyhedron (TMP) can serve as an ideal
platform [14, 12]. TMP is one of the origami-based mechanical metamaterials
with bellow-like 3D unit cells constructed from 2D Miura-folding sheets. This
TMP-based metamaterial features several mechanical characteristics, such as
flat and rigid foldability, negative Poisson’s ratio, multistability, and
load-bearing capabilities [15, 12]. Notably, each TMP cell can take four
different states (i.e., phases) while maintaining rigid foldability. Thus,
when we combine them into a tessellation, we can construct a 2D array of
heterogeneous origami cells. However, we cannot arbitrarily assign one of four
states to each unit cell randomly, because it can result in a geometrically
invalid configuration. Therefore, we have to find a method to discover
geometrically valid tessellations efficiently. If we follow the brute-force
method, the number of possible configurations that need to be assessed becomes
$4^{n}$, where $n$ is the total number of the TMP cells. This exponential
increase of the search space is often called the “curse of dimensionality”,
also known as the combinatorial explosion. This leads the search for the
geometrically valid configuration impossible for a large tessellation.

To overcome this hurdle, we can consult the graph-theoretical method to
describe and analyze the TMP-based tessellations. Graph theory is a field in
discrete mathematics, which is used to represent pairwise relations between
objects [16]. In this mathematical schematization, the objects are referred to
as vertices, and the connected pairs of the objects are called edges. In
diagrammatic ways, graphs are often represented with dots (or circles) for
vertices and lines for edges. Generally, graphs are classified into two types:
undirected graphs and directed graphs. The former have edges without direction
(two-way relationship), whereas the latter has edges with direction (one-way
relationship). This simple-yet-efficient mathematical structure can be
utilized to model various types of connections in general. Therefore, it has
been applied to numerous systems, such as physical [17], biological [18], and
information [19] ones. However, this versatile feature of graph theory to
model networks has not been fully leveraged to model and analyze tessellations
of mechanical metamaterials composed of unit cells, to the best of the
authors’ knowledge.

In this study, we employ the idea of enumerating the connection of the system
composed of the TMP unit cells, and we search the unique configurations and
count the number of them to assess the reconfigurability of the origami
system. The results show the exact number of unique configurations even for a
large tessellation, which could not have been calculated easily by the
conventional brute-force approach. We also report the growing pattern of the
number of valid configurations as the number of TMP cells increases in
horizontal and vertical directions. Among those results, we find heterogeneous
configurations composed of different states of unit cells. Unlike normal
configurations that are homogenous and have less variety of unit cells, such
heterogeneous configurations are hard to be discovered by human heuristics.
These results suggest that the morphology of the reconfigurable mechanical
metamaterials can meet the engineering requirements that are evolving and
adapting to external environments.

##  2 Methods and materials

###  2.1 TMP tessellation and its transformation

It is well known that TMP can take finite volume or be collapsed to zero
volume during its folding process [15] (see “TMP” and “flat” state in Fig. 1).
In this study, we also consider the other two geometries, so-called origami
tube minus (OT$-$, which is a left-parallelepiped tube) and origami tube plus
(OT$+$, a right-parallelepiped tube), which can be generated from the flat
state without deviating from the rigid-foldable assumption (see Supplementary
Movie 1 for the demonstration of a paper prototype, and Supplementary Movie 2
for the 3D rendered transformation). Thus, in this study, we consider four
possible states of unit cells: TMP, OT$-$, OT$+$, and Defect. Here, defect
mode means the flat state, which does not occupy any volume.

Fig. 1(a) explains the components of TMPs with geometrical parameters
$(l,m,d,\alpha)$. In Fig. 1(b), we see that the folding angle of blue and
green creases decides the reconfigurable states of a unit cell. If either blue
or green creases are folded flat, unit cells become OT$+$ or OT$-$,
respectively. Fig. 1(c) shows the transition of the unit cell from the flat
state to four reconfigurable states of TMP, OT$+$, OT$-$, and Defect. Figs.
1(d) and (e) depict those states and transformations between them for the
3D-rendered model and paper prototype, respectively.

The TMP cells can further be assembled into a tessellation in both horizontal
and vertical manner as shown in Fig. 2(a), and it can have multiple cells with
different states (see Supplementary Movie 3 for a transformation of a
tessellation). For example, Figs. 2(b) and (c) show some configurations of
3D-rendered and paper 3-by-3 stacking with TMP, OT$-$, and OT$+$, which are
represented by gray, blue, and red colors, respectively. Including the defect
mode, we have four different states that can be assigned for each unit.
Therefore, we can consider $4^{n}$ patterns of configurations within a
$n$-cell tessellation. However, there exists a kinematic relationship between
the adjoined TMP cells, and a large portion of the $4^{n}$ patterns are not
valid. Fig. 3 illustrates this problem by using two-cell and three-cell
tessellations. By simulating the stacking of unit cells, we can tell that some
configurations are invalid due to the unmatching of the geometrical boundaries
of the unit cells.

While we can achieve various configurations and shapes with this
reconfigurable origami tessellation, it is hard to know the actual number of
total configurations within a given size of tessellation. Considering all
cases (e.g., $4^{9}$ = 262,144 for 3-by-3 TMP tessellations) and sorting out
what are valid configurations one by one is computationally expensive and even
impossible for a large tessellation using conventional computational power. To
overcome this hurdle, we introduce the idea of a graph-theoretical framework
to discover the possible configuration efficiently. With the aid of the graph
representation of a reconfigurable origami tessellation and adjacency matrix
obtained from the graph information, we are able to calculate the total number
of valid configurations systematically. In the following sections, we
introduce the graph representation and adjacency matrices to assess the
combinatorial problem of the TMPs.

###  2.2 Graph representation of TMP tessellation

Recently, several researchers have tackled tessellation-related problems for
materials sciences with the aid of graph-theoretic techniques. Arkus et al.
utilized graph-theoretic enumerations of adjacency matrices and distance
matrices to find the finite sphere packings [20]. Vlassis et al. applied
graphs and graph convolutional deep neural network for anisotropic
hyperelasticity of polycrystal materials [21]. Herein, we further leverage
this technology to investigate the reconfigurability of origami tessellations.

As a first step, we attempt to translate physical TMP tessellations to graph
representations. Fig. 4 shows how we define an undirected graph $G=(V,E)$ from
a single unit cell in Fig. 4(a) and a tessellation in Fig. 4(b), respectively.
Here, vertices $V$ represents the six major faces of a unit cell, and the edge
set $E$ consists of unordered pairs of vertices. Here, the number of vertices
is $|V|=6n$, where $n$ is the number of unit cells in a tessellation. The edge
set counts the connections of faces both within one unit cell and between
other cells (see dashed and solid lines in Fig. 4). Then, we also define a
subgraph $G^{\prime}=(V^{\prime},E^{\prime})$ such that $V^{\prime}\subseteq
V$ and $E^{\prime}\subseteq E$, and $V^{\prime}$ and $E^{\prime}$ represents
the interconnections among the unit cells. For example, in Fig. 4(b) where we
consider a 2 by 2 tessellation, we can see that
$V^{\prime}=\\{1,2,7,11,12,14,15,16,22,23\\}$ and
$E^{\prime}=\\{(1,16),(2,11),(7,22),(12,15),(14,23)\\}$.

Based on this graph representation, we utilize graph-theoretic enumeration of
adjacency matrices. We introduce those definitions in the following. A
tessellation of $n$ unit cells can be described by an $|V|\times|V|$ adjacency
matrix $\bm{A}$. This matrix is defined with the graph $G$ and the subgraph
$G^{\prime}$. Therefore it details which vertices (faces in physical unit
cells) are in contact: $\bm{A}_{ij}=1$ if and only if $(u_{i},u_{j})\in
E^{\prime}$ and otherwise $\bm{A}_{ij}=0$. This means that $\bm{A}_{ij}=1$ if
$i$th and $j$th faces are attached to each other, and $\bm{A}_{ij}=0$ if they
are not. It should be noted that each $x$-by-$y$ tessellation has only one
unique adjacency matrix – regardless of the phases of unit cells – because
once we define the size of the tessellation and its arrangement, we also know
which physical faces of unit cells to be connected to others. The illustration
of the adjacency matrix is also shown in Fig. 4(b). Here, the dots denote the
edge set $E^{\prime}$, which represents the adjoining of TMP faces.

###  2.3 Combinatorial search for valid configurations

To analyze the reconfigurability of the TMP tessellations, we utilize the
aforementioned graph representation and algorithm. The reconfigurability can
be quantified by the number of unique configurations that the tessellation can
achieve. The search of valid configurations consists of three steps: (i)
definition of the size of tessellation, (ii) graph representation and graph-
theoretic enumeration of the tessellation, and (iii) combinatorial search for
valid configurations. First, we define the size and shape of the tessellation.
Our graph-theoretic method is versatile and applicable to various types of
tessellations. However, in order to make comparisons and measure the effect of
the unit cell number on the reconfigurability, we use a rectangular
arrangement of $X$-by-$Y$ tessellation, where $X$ and $Y$ are the number of
unit cells in horizontal and vertical directions, respectively.

Secondly, we represent the tessellation via graph and enumerate the
information about the connection of the unit cells using an adjacency matrix.
Again, the adjacency matrix is unique for each tessellation and immutable in
the process of tessellation search. After establishing the information about
tessellations, we move to the final step, which is the combinatorial search
for valid configurations. In this step, we utilize the complete list of 3-cell
configurations that are known a priori. It has 32 configurations as shown in
Supplementary Figure S2, and we combine those 3-cell configurations to find
configurations of larger tessellations.

The schematic illustration of the process of this combinatorial search is
shown in Fig. 5. The top row shows the graphical evolution of the TMP
tessellation, the middle row shows their graph representation, and the bottom
panel explains how we use adjacency matrices for this search process. Italic
numbers with red boxes show that unit cells have assignments of
configurations, whereas italic numbers with no boxes show that those unit
cells do not have assignments yet. By using this assignment information and
adjacency matrix, we detect an unassigned unit cell that is adjacent to the
assigned ones. Blue arrows and boxes in the matrix show the process of
searching for an adjacent unit cell to assign. To begin the search, we
allocate configurations to three cells at the lower-left end. Then, based on
the graph structure, we find an adjacent cell and assign a configuration to
the adjacent one by looking up the list of 3-cell configurations (see gray
triangles in Fig. 5b). We repeat this process until all unit cells have
configurations. Also, in this process, we check every one of the
configurations in the list. Therefore, at the end of the process, we can count
all valid configurations out of a given size of tessellation.

##  3 Results and discussion

###  3.1 Estimation and growth of reconfigurability

We start discussing the number and growth of reconfigurability by changing the
size of tessellations. The result of the searching for the number of valid
configurations with 14 different tessellation sizes is summarized in Fig. 6
for selected results and Table. 1 for all results. To verify the result from
our graph-theoretical method, we also implement a brute-force method for
smaller sizes of tessellations in Supplementary Note A. We start to examine
the unique configurations with 2-by-2 tessellations with the total number of
four unit cells. We then enlarge the size of the tessellation by adding a row
or column of unit cells to 49 unit cells in total. If we follow the brute-
force method for this number of unit cells, we have to investigate
$4^{49}\approx 10^{29}$ candidates of configurations that are almost
impossible to examine even with the aid of high-performance computers.
Therefore, by utilizing our method, we achieve a substantial amount of
efficiency in estimating the reconfigurability of the origami-based
metamaterials.

As we increase the size of the tessellation, we see that the number of unique
configurations increases exponentially for $X$-by-2 and 2-by-$Y$ cases where
$X$ and $Y$ are the number of horizontal and vertical stacking, respectively.
For $N$-by-$N$ tessellation where $N$ is for both horizontal and vertical
numbers, we see that the rate of increase decays gradually. Furthermore, we
observe that the rate of increase in the number differs by the direction of
adding a new column or row (horizontally or vertically). Adding a column of
unit cells to the tessellation doubles the unique configurations to the
original one, whereas adding a row quadruples them. In Fig. 7, we show an
example of adding a row or a column to a 2-by-2 tessellation with four TMPs.
While we obtain two different configurations in Fig. 7(b) for 3-by-2
tessellation, we can achieve four configurations for 2-by-3 tessellations in
Fig. 7(c). We observe this difference is caused by the direction of the phase
transition of unit cells. Since the transition among TMP, OT$+$, and OT$-$ is
lateral movement, the column-wise attachment of unit cells is more bounded to
the existing tessellations than the row-wise attachment. Actually, a column-
wise attachment involves only TMP and OT$+$ in Fig. 7(b). However, a row-wise
addition includes TMP, OT$+$, OT$-$, and defect in Fig. 7(c).

In Fig. 6(a), the red line connects the results from 2-by-2, 2-by-3, 2-by-4,
and 2-by-8 tessellations. Likewise, the blue line connects the results from
2-by-2, 3-by-2, 4-by-2, and 8-by-2 tessellations. Both lines show linear
growth in $\text{log}_{2}$ scale along the y-axis because there is a pattern
for the growth of the unique configurations, as we state previously. The
yellow line shows the results up to 7-by-7 tessellations. Since these results
are from the combinations of the horizontal and vertical increments, the
increase rate is not constant.

Table. 1 also includes the number of unique configurations only with three
states (OT$+$/OT$-$/Flat, TMP/OT$+$/Flat, TMP/OT$-$/Flat, and TMP/OT$+$/OT$-$)
and the rate of increase of the number of configurations from using three
states to four states. Here, each combination of three states
(OT$+$/OT$-$/Flat, TMP/OT$+$/Flat, TMP/OT$-$/Flat, and TMP/OT$+$/OT$-$) has
the same number of valid configurations. The comparison between the four
states indicates that each TMP state equally contributes to the excessive
growth of reconfigurability. However, one of the advantages of inducing defect
states is that we can achieve an enormous variety of reconfigurable shapes
without incurring volume increases. Based on the graph-theoretic evaluations,
we confirm that the increase of the phase number from three to four
contributes directly to the surge of the overall variety of the tessellations.
Further results and discussions about the effect of defect modes are included
in Supplementary Note B.

###  3.2 Highly heterogeneous configurations

Now we discuss some examples of configurations we discover in the search
process. Fig. 8 shows the four examples of anomalous configurations in 4-by-4
tessellations. They consist of four different states (TMP, OT$+$, OT$-$, and
Defect). Also, the composition of the states is highly mixed and distributed
within the tessellation. For example, a tessellation in Fig. 8(a) consists of
six TMPs, two OT$+$, six OT$-$, and two Defects. Besides, 3-cell groups of
TMPs are separated by OT$+$, OT$-$, and defect cells. Unlike normal
configurations that have a homogeneous occurrence of deformation states, such
heterogeneous configurations composed of a variety of the different states are
hard to be discovered by human heuristics.

We further extend this result to see if they can simulate specific geometries
because such reconfigurability is one of the essential aspects when we utilize
these mechanical metamaterials for engineering applications. To assess how
much the tessellation’s reconfigurability contributes to the ability to
simulate the real-life objects, we examine cross-sections of 4-by-4
tessellations and compare them with some images in Supplementary Note C. The
result of simulating the various type of geometries (chair, buffalo, boot, and
boat) suggests that this platform of TMP tessellations has a great potential
to satisfy the geometrical requirements for mechanical, aerospace, and medical
applications potentially.

##  4 Conclusion

In this study, we have demonstrated a graph-theoretical approach to discover
the valid configurations of TMP tessellations. Given the challenges of
combinatorial problems of origami-based mechanical metamaterials, we have
built a computationally efficient framework to account for a complete set of
reconfigurable shapes of the TMP tessellations. To this end, we have
enumerated the information about the connections of the TMP unit cells using
adjacency matrices. One of the unique features of this versatile graph-
theoretical approach is that it has the possibility to be applied for
designing various types of mechanical metamaterials. This simple approach of
using adjacency matrices has proposed an efficient method to describe and
analyze the tessellation of the mechanical metamaterials. The result of the
aforementioned analysis indicates that the TMP tessellations have an abundance
of reconfigurability owing to the heterogeneity of the TMP cells and their
versatile connectivity. Such reconfigurability can be exponentially improved
by adopting a larger number of TMP cells in the tessellation, and the exact
number of possible configurations has been calculated by the proposed graph-
theoretic method in an accurate and efficient manner.

Given the simplicity of this framework, the graph-based approach can be
applied to the other types of the tessellations of mechanical metamaterials
(e.g., metal-organic hinged cube tessellation [22], voxelated mechanical
metamaterials [8, 23], and other origami lattices in 2D or 3D settings [24,
13, 25, 26]) by building graph representations for each architecture of
mechanical metamaterials and by understanding the connections within the
tessellations. Also, the ability to simulate the various shapes and
morphologies suggests the possibility of answering the following question;
given the target geometry and properties we want to achieve, can we dial-in
the local phases in unit cells and find the optimal metamaterial
configurations globally? This can be assessed by utilizing the graph-
theoretical framework developed herein and combining with combinatorial
optimization techniques.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## CRediT authorship contribution statement

Koshiro Yamaguchi: Conceptualization, Methodology, Investigation, Software,
Visualization, Writing- Original draft preparation. Hiromi Yasuda:
Conceptualization, Methodology, Investigation, Software, Visualization,
Writing - Review & Editing. Kosei Tsujikawa: Investigation, Software. Takahiro
Kunimine: Supervision, Writing - Review & Editing. Jinkyu Yang:
Conceptualization, Supervision, Writing - Review & Editing, Funding
acquisition.

## Acknowledgments

K.Y. and J.Y. are grateful for the support from the U.S. National Science
Foundation (1553202 and 1933729) and the Washington Research Foundation. K.Y.
is supported by the Funai Foundation for Information Technology. We thank Dr.
Dillon Foight and Prof. Mehran Mesbahi at the University of Washington, and
Dr. Jesse Silverberg at Multiscale Systems for helpful discussions.

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## List of figures

Figure 1: Folding behavior of a single Tachi-Miura Polyhedron (TMP) unit cell.
a, crease patterns and geometrical parameters of two flat sheets composing the
TMP. b, the folding angles of blue and green creases decide the reconfigurable
states of a unit cell. When blue (green) creases are folded flat, the unit
cell takes the OT$+$ (OT$-$) phase. c, 3D rendered images of the transition
from the flat state four reconfigurable states. d, 3D rendered images of a TMP
unit cell with the initial flat state and the three reconfigurable states of
TMP, OT$+$, and OT$-$. e, images of paper prototypes with corresponding
configurations to d. Figure 2: Examples of various configurations within a
3-by-3 tessellation. a, 3D rendered images showing how a tessellation is built
with the nine unit cells. Certain faces of unit cells are adjoined altogether
to form one tessellation. b, 3D rendered images of the 3-by-3 tessellation
with four different configurations. c, digital images of the paper prototypes
corresponding to four tessellations shown in b. Figure 3: Examples of how each
tessellation becomes valid or invalid. Cases of a, two-cell tessellations and
b, three-cell tessellations. In a, one unit cell (TMP, OT$+$, or OT$-$) is
attached to one TMP cell. The attachment of TMP and OT$+$ results in the valid
configuration. However, we can see that OT$-$ tube does not fit with the
upper-right side of the TMP cell. Likewise, cases of three-cell stacking in b
shows that the attachment of TMP and OT$-$ to a two-TMP stacking brings the
valid configuration, whereas the attachment of OT$+$ tube makes the
tessellation invalid.  Figure 4: Examples of the graph representation for a, a
single unit cell and b, a 2-by-2 tessellation. Red dots in a represent the
midpoints of the major side edges of TMPs. In a graph representation, those
red dots are considered as nodes. Dashed lines in a and b mean the connection
of the nodes within a unit cell. Solid lines in b means the connection of the
nodes between different unit cells. Italic numbers in blue circles denote the
TMP unit cells, and upright numbers denote the middle points in physical TMPs
and corresponding nodes in graph. In the illustration of the adjacency matrix,
left and bottom labels shows the node numbers as represented in the graph.
Likewise, right and top labels in italic numbers denote the TMP unit cells.
Gray lines in adjacency matrix show which part of the matrix corresponds to
the TMP unit cell numbers. Dots in this adjacency matrix indicate the
connected nodes, i.e., adjoined faces in the TMP tessellation. Figure 5: An
example of discovering a valid configuration for the 3-by-3 tessellation with
logistic illustrations of the computation processes using adjacency matrices.
In a and b, cyan, red, blue, and yellow colors in the graph representation
correspond to the TMP, OT$+$, OT$-$, and defect states of the unit cell,
respectively. In a, 3d-rendered images represent the graphical process of
building tessellations. Likewise, in b, the pictures of graphs with colored
parts show the schematic process of the tessellation search. Three unit cells
enclosed by gray lines in a and b indicates where the 3-cell configurations
are referenced and adopted. In c, adjacency matrices are illustrated in the
same way as Fig.4(b). Italic numbers with red boxes show that unit cells have
assignments of configurations, whereas italic numbers with no boxes show that
those unit cells do not have assignments yet. Blue arrows and boxes in the
matrix show the process of searching an adjacent unit cell to assign the
proper unit cell configuration. Figure 6: Number of valid configuration for
each tessellation size. a, 14 results of counting valid configurations with
four configurations (TMP, OT$+$, OT$-$, and Defect) based on the
categorization of N-by-N, X-by-2, and 2-by-Y tessellations where $X$, $Y$, and
$N$ are the number of horizontal and vertical stackings of the tessellation.
b, 14 results of counting valid configurations with three configurations
(OT$+$/OT$-$/Flat, TMP/OT$+$/Flat, TMP/OT$-$/Flat, and TMP/OT$+$/OT$-$). Each
three-state configuration has the same number of valid configurations. Other
results and exact numbers are shown in Table 1. Figure 7: a, an image of a
2-by-2 tessellation with four TMPs. Red box shows the position where the unit
cells are attached to build 2-by-3 tessellations. Likewise, blue box shows the
position for 3-by-2 tessellations. b, 3-by-2 valid tessellations that are
emerged from a 2-by-2 tessellation with four TMPs shown in a. c, 2-by-3 valid
tessellations that are emerged from a 2-by-2 tessellation with four TMPs shown
in a. In both b and c, these tessellations are the only ones that can be
emerged from the 2-by-2 tessellation in a. Cyan, red, blue, and yellow colors
in the 3D rendered images correspond to the TMP, OT$+$, OT$-$, and defect
states of the unit cell, respectively. Figure 8: Examples of heterogeneous
configurations in 4-by-4 tessellations. a, a configuration consisting of 6
TMPs, 2 OT$+$, 6 OT$-$, and 2 defects. b, a configuration consisting of 2
TMPs, 2 OT$+$, 5 OT$-$, and 7 defects. c, a configuration consisting of 5
TMPs, 3 OT$+$, 5 OT$-$, and 3 defects. d, a configuration consisting of 1 TMP,
3 OT$+$, 5 OT$-$, and 7 defects.

Table 1: Number of unique configurations for various tessellation sizes. Size | 4 states | 3 states | Increase | Size | 4 states | 3 states | Increase
---|---|---|---|---|---|---|---
2-by-2 | 32 | 13 | 146% | 3-by-4 | 1024 | 120 | 753%
3-by-2 | 64 | 21 | 205% | 4-by-4 | 2048 | 184 | 1013%
2-by-3 | 128 | 32 | 300% | 8-by-2 | 2048 | 517 | 296%
4-by-2 | 128 | 37 | 246% | 2-by-8 | 131072 | 4181 | 3035%
2-by-4 | 512 | 89 | 475% | 5-by-5 | 16383 | 672 | 2338%
3-by-3 | 256 | 50 | 412% | 6-by-6 | 131072 | 2480 | 5185%
4-by-3 | 512 | 82 | 524% | 7-by-7 | 1048576 | 9312 | 11160%


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