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Undulations in Tubular Origami Tessellations: a Connection to Area-Preserving Maps
Undulations in Tubular Origami Tessellations: a Connection to
Area-Preserving Maps
Rinki Imada1and Tomohiro Tachi1
Department of General Systems Studies, Graduate School of Arts and Sciences, The University of Tokyo, Meguro-ku, Tokyo,
153-8902, Japan
(*Electronic mail: r-imada@g.ecc.u-tokyo.ac.jp, tachi@idea.c.u-tokyo.ac.jp)
(Dated: 7 August 2023)
Origami tessellations, origami whose crease pattern has translational symmetries, have attracted significant attention in
designing the mechanical properties of objects. Previous origami-based engineering applications have been designed
based on the “uniform-folding” of origami tessellations, where the folding of each unit cell is identical. Although
“nonuniform-folding” allows for nonlinear phenomena that are impossible through uniform-folding, there is no uni-
versal model for nonuniform-folding, and the underlying mathematics for some observed phenomena remains unclear.
Wavy folded states that can be achieved through nonuniform-folding of the tubular origami tessellation called wa-
terbomb tube are an example. Recently, the authors formulated the kinematic coupled motion of unit cells within
waterbomb tube as the discrete dynamical system and identified a correspondence between its quasiperiodic solutions
and wavy folded states. Here, we show that the wavy folded state is a universal phenomenon that can occur in the
family of rotationally symmetric tubular origami tessellations. We represent their dynamical system as the composi-
tion of the two 2D mappings: taking the intersection of three spheres and crease pattern transformation. We show
the universality of the wavy folded state through numerical calculation of phase diagrams and geometric proof of the
system’s conservativeness. Additionally, we present a non-conservative tubular origami tessellation, whose crease pat-
tern includes scaling. The result demonstrates the potential of the dynamical system model as a universal model for
nonuniform-folding or a tool for designing metamaterials.
In recent years, the Japanese art of paper folding, origami,
has gained attention as the tool to program the mechan-
ical properties of an object geometrically. Many previ-
ous engineering applications have been established on “the
uniform-folding of origami tessellations”, i.e., “the periodic
folding of periodic crease patterns”. Though the assump-
tion of the periodicity in folding makes it easy to under-
stand and control the kinematics, it also limits the poten-
tial of origami tessellations. This paper tackles mathe-
matically explaining the nonlinear phenomenon feasible
through the nonuniform-folding of tubular origami tes-
sellations, “the undulation of the folded states”. Using
a dynamical system model representing the coupled fold-
ing motion of the unit cells comprising origami tessella-
tions, we clarify the universality of the phenomenon from
the conservativeness of the dynamical system. The result
not only contributes to understanding the specific phe-
nomenon but also assures the potential of a dynamical sys-
tem model as the universal framework in the analysis of
nonuniform-folding, leading to the novel engineering ap-
plication using nonuniform-folding.
I. INTRODUCTION
The traditional Japanese art of paper folding, known as
origami, has garnered significant attention for its application
in programming the mechanical properties of objects1. By
appropriately adding creases to the objects, it is possible to
achieve a continuous deformation from a flat sheet to an ap-
proximated target surface shape2, as well as discrete deforma-
tions between multiple energy-stable states (multi-stability)3.
Programmed mechanical properties based on geometry have
scalability, so origami structures are expected to have a broad
range of engineering applications. For example, micro de-
vices such as deformable electronic circuits4and meter-scale
deployable structures such as solar panels in space5have al-
ready been proposed.
Origami engineering has developed in parallel with origami
science. In particular, the kinematics of rigid origami has
played a central role in both streams. Rigid origami is a math-
ematical model of origami where we replace its facets and
creases with rigid panels and rotational hinges, respectively.
As a result, we consider origami as a flexible polyhedron that
is not necessarily closed. The first step in considering rigid
origami is to prescribe its crease pattern, which is a graph, the
set of vertices and edges, embedded in a polyhedral surface,
representing a paper. The graph and its embedding provide
the topological and metric properties of the crease pattern, re-
spectively. Note that traditional origami is folded from a flat
sheet of paper, i.e., its crease pattern is embedded in a plane;
however, we do not assume this developability in this paper.
Next, when we fold the given crease pattern, we must con-
sider the geometric constraints, i.e., the preservation of facets’
shapes and their connectivities prescribed by the crease pat-
tern, where we admit the self-intersection of panels. Then,
we define the folded state and folding motion as a folding sat-
isfying these geometric constraints and a continuous family
of folded states, respectively. Because of the geometric con-
straints, the possible folded states or folding motion changes
dramatically depending on the topology or metric of a crease
pattern.
Parameterizing the folding of a crease pattern helps solve
the geometric constraints and clarify the possible folded states
or folding motion. As we will see later, in this paper, we de-
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Undulations in Tubular Origami Tessellations: a Connection to Area-Preserving Maps 2
FIG. 1. Examples of origami tessellations and their uniform-folding motion (Top: Miura-ori, Middle: eggbox pattern, Bottom: waterbomb
tessellation). The regions bordered by black lines indicate the modules in each origami tessellation. These crease patterns have the property
called flat-foldability defined by the existence of a folded state where all fold angles equal ±π. Furthermore, the crease pattern of Miura-ori
and waterbomb tessellations are developable, whereas that of the eggbox pattern is non-developable. Through the uniform-folding, each folded
state of Miura-ori and eggbox pattern approximates a plane, whereas that of waterbomb tessellation approximates a cylinder. Note that each
diagonal edge of the quadrangular faces of Miura-ori and eggbox patterns remains unfolded in the uniform-folding.
fine a specific parameterization for the folding of crease pat-
terns we focus on and represent the geometric constraints as
the preservation of edge lengths which we can solve by tak-
ing the intersection of spheres. Here, we introduce one of the
most common parameters, a fold angle defined as an exterior
dihedral angle in [−π,π]. If a fold angle of an edge is posi-
tive/negative, we call the edge valley/mountain folded. In ad-
dition, we often assign MV (mountain/valley) labels to some
edges to indicate their folding direction and exclude any folds
inconsistent with the assignment. Then, we can formulate the
geometric constraints as the multiple nonlinear equations in
terms of fold angles6,7. The equations are nonlinear, and the
number of variables and equations increase proportional to the
number of creases and inner vertices, respectively; therefore,
solving the geometric constraints for a given crease pattern is
generally challenging.
To relieve difficulties in solving the geometric constraints,
uniform-folding of origami tessellations, i.e., periodic folding
of periodic crease patterns, has been actively researched (see
Figure 1). Origami Tessellations are origami whose crease
pattern has discrete translational symmetry, which we call its
unit cell module. Uniform-folding of a certain origami tessel-
lation is the folding through only uniform folded states where
folded states of its modules are identical. Thus, in uniform-
folding, we can reduce the kinematics of the entire shape into
that of a single module, making it easy to understand and con-
trol. Hence, many conventional engineering applications have
been developed based on uniform-folding of origami tessella-
tions8–13.
However, characteristics or phenomena feasible through
uniform-folding are inherently limited compared to
nonuniform-folding. Here, we focus on smooth surfaces
approximated by folded states of origami tessellations while
ignoring the detailed corrugations formed by creases. For
example, an origami approximating the hyperbolic paraboloid
is famous14. Programming a crease pattern so that its folded
state approximates a given surface has been one of the pri-
mary topics in origami science or engineering2,15–17. Hence,
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Undulations in Tubular Origami Tessellations: a Connection to Area-Preserving Maps 3
researchers have tried to characterize and explore the surfaces
approximated by periodic crease patterns, i.e., origami tessel-
lations. One significant result is that uniform folded states of
origami tessellations can approximate only surfaces with zero
Gaussian curvature such as cylinders or planes18–20. Thus,
it needs nonuniform-folding to approximate doubly-curved
surfaces, i.e., surfaces with nonzero Gaussian curvature.
As we mentioned, the complexity of the geometric con-
straints of rigid origami makes it challenging to capture the
kinematics of nonuniform-folding, and researchers have de-
veloped several computational approaches to tackle this prob-
lem. One direct way is to implement the rigid origami model
in computers and numerically solve constraints represented
as equations21. Another way is to use physical simulation
using the bar-and-hinge model, the mass-spring model for
origami, and solve the problem dynamically, not kinemati-
cally22,23. Through the simulation using the bar-and-hinge
model, Schenk and Guest have reported that nonuniform
folded states of certain origami tessellations called Miura-
ori and eggbox patterns approximate surfaces with negative
and positive Gaussian curvature, respectively22. Neverthe-
less, even if discovered a nontrivial phenomenon through such
computational simulations, the underlying geometry or math-
ematical structures behind the phenomenon is not clear and
hard to extract. As for the nonuniform-folding of Miura-
ori and eggbox patterns, Nasser et al. have proposed an-
other mathematical approach applying differential geometry
and confirmed the same result as the previous research24,25;
however, the method is specialized for Miura-ori and eggbox
patterns.
Recently, the authors have proposed a novel mathematical
model of nonuniform-folding, dynamical systems of origami
tessellations26. We focused on an origami tessellation called
waterbomb tube27–30 and formulated the coupling folding mo-
tion of the modules as the recurrence relation, i.e., the discrete
dynamical system, by solving the geometric constraints. We
successfully identified the correspondence between quasiperi-
odic solutions of the derived dynamical system and nontrivial
wavy folded states reported by Mukhopadhyay et al.31 (see
Figure 2).
In this paper, we tackle two problems left in the previous
research. First, it was not clear if the wavy folded states are
unique to the waterbomb tube or a universal phenomenon that
can occur in various tubular origami tessellations. Second,
because the quasiperiodic solutions are the typical type of so-
lution of conservative dynamical systems, we conjectured that
the dynamical system of the waterbomb tube is conservative;
however, this remained an open problem. We show that the
wavy folded states are a universal phenomenon by generaliz-
ing the framework to the family of various tubular origami tes-
sellations including Miura-ori and eggbox patterns. We show
that the underlying dynamical system of the family of tessel-
lation is conservative thereby explaining the wavy shapes as
the undulation near an elliptic fixed point in such a system.
This paper is organized as follows. In Section II, we define
the generalized family of origami tessellations and the dynam-
ical systems representing the folded states. After introducing
the coordinate mapping on the configuration space and param-
eterizing the dynamical system in Section III, we numerically
calculate the phase diagram of the dynamical system for vari-
ous tubular origami tessellations folded from different crease
patterns in Section IV. Using these phase diagrams, we ob-
serve that there exist quasiperiodic solutions and their corre-
sponding wavy folded states in various origami. In Section V,
we prove that the dynamical system is conservative by demon-
strating the area-preservation of the parameterized dynamical
system, explaining the universality of the wavy folded states.
Furthermore, we generalize the dynamical system so that its
scope includes the tubular origami tessellation whose crease
pattern has scaling and show that the dynamical system is
no longer conservative if the crease pattern includes scaling.
Conclusions are given in Section VI.
FIG. 2. This paper model demonstrates the wavy folded state of
waterbomb tube, similar to unduloid, where synclastic and anticlastic
parts appear alternately.
II. SETUP AND METHOD
In this section, we define a family of tubular origami tes-
sellations and their dynamical system. After introducing a
zigzag-based crease pattern generalized from the crease pat-
tern of the waterbomb tube, we discuss its folding. Such a
family of zigzag-based crease patterns allows us to calculate
their tubular folded state by iteratively taking the intersection
of three spheres. Leveraging this property, we define the dy-
namical systems of a family of axisymmetric tubular origami
tessellations. Note that our analysis assumes the rigid origami
model and approaches phenomena from a kinematic point of
view.
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Undulations in Tubular Origami Tessellations: a Connection to Area-Preserving Maps 4
+1-st ring
modules
0-th zigzag 1-st zigzag 2-nd zigzag
⇔
-th ring
FIG. 3. Crease pattern of the waterbomb tube. We identify the left
and right ends of the crease pattern embedded in a plane, which is
equivalent to embedding the crease pattern in a cylinder. The gray
region indicates the ring that consists of Nmodules. We can regard
the crease pattern as the periodic sequence of three types of zigzags
shown in different colors.
A. Crease pattern
Before defining crease patterns included in our scope, we
describe how we see the crease pattern of the waterbomb tube
as an example. As Figure 3 shows, the crease pattern of the
waterbomb tube is a graph embedded in a polyhedral cylinder.
If the crease pattern is developable, embedding in a cylinder
is equivalent to embedding a graph in a plane and identifying
its left and right ends. Then, we can see the crease pattern
as the sequence of rings, each of which consists of Niden-
tical modules (N∈Z>2). Furthermore, we can decompose
each ring into zigzags, where we define zigzag as a closed
polygonal linkage, i.e., a polygonal loop encircling the cylin-
der comprised of 2Nedges with alternating lengths. Two con-
secutive zigzags are connected through Nedges with identical
lengths. From this viewpoint, we can see the crease pattern of
the waterbomb tube as the sequence of zigzags in which the
equivalent zigzag appears in period 3. Similarly, Miura-ori
or eggbox tubes (that are not necessarily developable) (Fig-
ure 12 A, 13 A) can be seen as a sequence of zigzags if each
quadrangular face is triangulated.
In the following, we deal with a crease pattern embedded
in a cylinder that we can regard as a sequence of rings in-
dexed by t=0,1,..., which consists of a sequence of zigzags
indexed by i=0,1,...,I−1 in period I∈Z>0. We let i-th
and i+1-st zigzags share every other vertex. There are two
choices for choosing the shared vertices, which yield differ-
ent graph connectivities as in Figure 4. We distinguish the
two connectivities by assigning left and right to two kinds of
edges and connecting the adjacent zigzag at the left vertex
U1of the left edge and the right vertex U3of the right edge.
The intermediate vertex U2is connected to the next zigzag via
an edge. In the following, we denote the length of the left
and right bars in i-th zigzag by li,Land li,R, respectively, and
the length of the edges connecting i-th and i+1-st zigzags
li+1,M. Each time a zigzag is appended, we have two choices
+1-st zigzag
-th zigzag
+2-nd zigzag
Left Right
Case 0 Case 1
FIG. 4. Construction of the crease pattern through connecting
zigzags. Edges in different types of zigzags are shown in different
colors, and thick/thin edges represent their left/right edges. From
bottom to top, first, we append i+1-st zigzag to i-th one, following
the left-right assignment of i-th one. Next, give i+1-st one to its left-
right assignment. Finally, append i+2-nd one based on the left-right
assignment of i+1-st one. When giving the left-right assignment to
i+1-st zigzag, there exist two possible ways. Let U0
1,U0
2,U0
3denote
the three representative vertices in i+1-st zigzag. We call the left-
right assignment of i+1-st zigzag Case 0 if U0
2is shared with i-th
zigzag. If i+1-st zigzag and i-th zigzag share U0
1and U0
3, then we
call the left-right assignment of i+1-st zigzag Case 1. After con-
necting i+2-nd zigzag, each Case results in a different topology of
the crease pattern.
for the left-right assignment with respect to the previous left-
right assignment, which we call Case 0 and Case 1 indicated
by ki∈ {0,1}. Note that all the faces of such a zigzag-based
crease pattern are triangles, implying that the metric of each
face can be specified by the lengths of each edge.
B. Folded state and dynamical system
Second, we consider the rigid folding of a given crease
pattern. We assume that the folded state is N-fold symmet-
ric, i.e., there exists an axis, the axis of N-fold symmetry,
around which the rotation of the folded state by 2π/Nresults
in the same shape. This assumption enables us to construct
the folded state of the whole crease pattern from that of the
1/Npart of the crease pattern. Under the assumption, the
folded state of each zigzag comprising the tube forms the N-
fold symmetric zigzag with respect to the same axis. Then,
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Undulations in Tubular Origami Tessellations: a Connection to Area-Preserving Maps 5
Axis of N-fold symmetryAxis of N-fold symmetry
FIG. 5. Two independent configuration changes of the
zigzag through the N-fold symmetric folded state. Top: expan-
sion/shrinkage. The distance between the white-circled vertices and
the axis of N-fold symmetry changes, while the positions of other
vertices relative to the white-circled vertices remain fixed so that all
vertices are coplanar. It is worth noting that the left-most and right-
most states in the top are degenerate, meaning that some of the ad-
jacent edges are colinear. Bottom: rotational motion. The distance
between the white-circled vertices and the axis is fixed, but the po-
sitions of other vertices relative to the white-circled vertices change.
Each of the dashed lines shows the axis of rotation of a movable ver-
tex.
the folded state of each zigzag can be represented by two pa-
rameters, one representing the expansion and one representing
the rotation, as illustrated in Figure 5.
Significantly, when the folded state of i-th zigzag is de-
termined, that of i+1-st is uniquely determined by the con-
straint from the rigid origami model, i.e., the preservation of
the facets’ shapes and their connectivities prescribed by the
crease pattern (i=0,1,...,I−1). For triangulated crease
patterns like zigzag-based ones that we focus on, these con-
straints are equivalent to preserving the lengths of edges. Con-
sidering this edge length preservation, we can calculate the
folded state of i+1-st zigzag by taking the intersection of
three spheres as below (see Figure 6). Let U1,U2,U3de-
note vertices in i-th zigzag taken in the same manner as Fig-
ure 4. Then, there is the vertex Vin i+1-st zigzag that is
connected with three consecutive vertices U1,U2,U3by edges
with length ri,1:=|U1V|,ri,2:=|U2V|,ri,3:=|U3V|. Thus,
we can identify the position of Vby taking the intersection
of three spheres centering at Ujwith radius ri,j(j=1,2,3).
The radii are the edge lengths defined from the crease pattern,
i.e., when Case ki=0, (ri,1,ri,2,ri,3)=(li+1,R,li+1,M,li+1,L),
and when Case ki=1, (ri,1,ri,2,ri,3)=(li+1,L,li+1,M,li+1,R).
Let us assume that the spheres meet at two points, which are
mirror-symmetric through the plane spanned by U1,U2,U3.
Selecting the point above/below the mirror as Vresults in
the positive/negative fold angle of U2V, i.e., the choice de-
termines whether U2Vis valley or mountain folded. Here,
we prescribe MV-assignment σi∈ {M,V}to U2V, enforces
us to select the one consistent with the given value of σias
the vertex V. Then, by rotating Varound the axis of N-fold
symmetry by 2 jπ/N(j=1,2,...,N−1), we can identify the
positions of all vertices in i+1-st zigzag and obtain its folded
state. As a remark, if the spheres have no intersection point,
there is no folded state of i+1-st zigzag compatible with that
of i-th one. In the following, we assume the existence of in-
tersection points, meaning that we focus on the case where we
can infinitely increase the number of zigzags.
Here, the I-times composition of the above procedure forms
the recurrence relation between the 0-th zigzag of t-th ring and
the 0-th zigzag of t+1-st rings (t=0,1,. . . ). Since the 0-th
zigzag determines the entire folded state of the ring, we rep-
resent the folded state of t-th ring by that of 0-th zigzag of the
ring. We can see this I-times composition as the recurrence
relation between the folded states of t-th and t+1-st rings,
representing their coupling folding motion. Thus, we can for-
mulate this recurrence relation as the following discrete dy-
namical system in which “time” is represented by the indices
of rings:
F:=FI−1◦···◦F0:M(l0,L,l0,R,N)→M(l0,L,l0,R,N).(1)
Fiis the mapping from the folded state of i-th zigzag to that of
i+1-st representing the one step of three-sphere-intersection
(i=0,1,...,I−1):
Fi:M(li,L,li,R,N)→M(li+1,L,li+1,R,N),(2)
where M(lL,lR,N)denotes the configuration space, i.e., the
set of all possible N-fold symmetric folded states of a zigzag
with its left and right edge lengths lL∈R>0and lR∈R>0,
respectively. In the following, let xdenotes as an element of
the configuration space. Furthermore, we decompose Fiinto
Fi=gki
i◦fidepending on the value of ki. We define firepre-
senting the i-th step for Case 0 as follows:
fi(x):=f(x;li,L,li,R,N,ri,1,ri,2,ri,3,σi),(3)
where frepresents the three-sphere-intersection with given
parameters for Case 0 assignment:
f:M(lL,lR,N)→M(r3,r1,N)
x7→ f(x;lL,lR,N,r1,r2,r3,σ).(4)
We define gki
ias follows:
gki
i(x):=(xif ki=0
g(x;ri,3,ri,1,N)if ki=1,(5)
Where gis the connectivity-transformation that swaps the left-
right assignments:
g:M(lL,lR,N)→M(lR,lL,N)
x7→ g(x;lL,lR,N).(6)
As we mentioned, an N-fold symmetric zigzag can be ex-
pressed by a 2-degrees-of-freedom (2-DOF) system: expan-
sion/shrinkage within a finite interval and cyclic motion. This
implies that M(lL,lR,N)is isomorphic to a cylinder. Hence,
Fis a 2-dimensional system, resulting in the N-fold sym-
metric tubular origami tessellation as a 2-DOFs mechanism
whose folded state is completely determined by that of the ini-
tial ring. In the following sections, we clarify their kinematics
by analyzing F.
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Undulations in Tubular Origami Tessellations: a Connection to Area-Preserving Maps 6
Mountain fold
Valley fold
Plane spanned by
Case
-th zigzag
Left Right +1-st zigzag
FIG. 6. Calculation of the folded state of i+1-st zigzag from that of i-th one by taking the intersection of three spheres. Given the folded states
of i-th zigzag and positions of U1,U2,U3,Vmust be the intersecting point of three spheres centered at Ujwith radius rjbecause edge-length
must be consistent with the crease pattern ( j=1,2,3). If spheres intersect at two points, which are mirror symmetric through the plane spanned
by U1,U2,U3. Thus, the MV assignments of the crease connecting each of the two and U2are different, where we denote one of the two VM
if VMU2is mountain folded, and the other VV. We select Vσias V, where σi∈ {M,V}is the prescribed MV assignment of U2V, and obtain
the whole i+1-st zigzag by rotating Vσiaround the axis of N-fold symmetry. Finally, we give the prescribed left-right assignment of i+1-st
zigzag specified by ki, where the figure shows the case of ki=0.
III. PARAMETERIZATION OF DYNAMICAL SYSTEMS
Now, let’s turn to introduce the coordinates on M(lL,lR,N)
to parameterize F. After defining the coordinates on
M(lL,lR,N), we reformulate the dynamical system as the
composition of fiand gi, which is the parameterization of fi
and gi.
A. Parameterization of Configuration Space
A coordinate must have a one-to-one correspondence with
a point on M(lL,lR,N). Note that we must distinguish two
identical folded states with different left-right assignments
representing the connectivity with the next zigzag because the
way to place the three spheres depends on the connectivity.
Hence, we define the coordinate mapping on M(lL,lR,N)that
explicitly depends on the left-right assignment representing
the connectivity.
Consider parameterizing x∈M(lL,lR,N). We define the
coordinate mapping x7→ [d,ρ]Tas follows: first, take three
consecutive vertices U1,U2,U3in the zigzag represented by x
in the same manner as Figure 4. Then, measure half the dis-
tance d∈(|lL−lR|/2,(lL+lR)/2)between U1and U3and the
signed angle ρ∈(−π,π]between the axis of N-fold symme-
try and the vector from the foot of the perpendicular dropped
from point U2to the line U1U3to U2, which correspond to the
expansion/shrinkage, and the rotational motion, respectively.
This parameterization depends on the left-right assignment.
Figure 7 shows the parameterizations of the identical zigzag
with different left-right assignments.
In the following, we denote the coordinate mapping by
φlL,lR,N:M(lL,lR,N)→P(lL,lR,N), where P(lL,lR,N):=
(|lL−lR|/2,(lL+lR)/2)×(−π,π]denotes the parameter
space, i.e., the set of possible values of coordinates. For the
derivation of the explicit form of φlL,lR,Nand its inverse, see
Appendix A.
Axis of N-fold symmetryAxis of N-fold symmetry
𝑖-th zigzag
Left Right
FIG. 7. Parameterizations of the identical folded state with different
left-right assignments. In the left and right case, the folded state is
regarded as the point on M(a,b,N)and M(b,a,N), then, parame-
terized as [d,ρ]Tand [d∗,ρ∗]T, respectively.
B. Dynamical system as the composition of two kinds of
mappings
Using the coordinate mappings, we parameterize fas f:
P(lL,lR,N)→P(r3,r1,N);[d,ρ]T7→ [d0,ρ0]T:
f:=φr3,r1,N◦f◦φ−1
lL,lR,N
[d0,ρ0]T=f([d,ρ]T;lL,lR,N,r1,r2,r3,σ),(7)
and gas g:P(lL,lR,N)→P(lR,lL,N);[d,ρ]T7→ [d∗,ρ∗]T:
g:=φlR,lL,N◦f◦φ−1
lL,lR,N
[d∗,ρ∗]T=g([d,ρ]T;lL,lR,N).(8)
For instance, gmaps [d,ρ]Tin Figure 7 left to [d∗,ρ∗]in the
right. Then, we can parameterize Fas F:P(l0,L,l0,R,N)→
P(l0,L,l0,R,N);[dt,ρt]T7→ [dt+1,ρt+1]Tas follows:
F:=FI−1◦···◦F0,(9)
Fi:=gki
i◦fi(i=0,...,I−1),(10)
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where fiand gki
iare the parameterization of fiand gki
i:
fi([d,ρ]T):=f([d,ρ]T;li,L,li,R,N,ri,1,ri,2,ri,3,σ),(11)
gki
i([d,ρ]T):=([d,ρ]Tif ki=0
gi:=g([d,ρ]T;ri,3,ri,1,N)if ki=1.
(12)
Figure 8 shows the image of [d,ρ]Tin the left in Figure 7 by
fiand gi◦fi. See Appendix B for the derivation of fand g.
Case 0 Case 1
𝑖-th zigzag
Left Right 𝑖+1-st zigzag
FIG. 8. The left and right shows the image of [d,ρ]Tin Figure 7 by
fand g◦f, which are denoted by [d0,ρ0]Tand [(d0)∗,(ρ0)∗]Tcorre-
sponding to Case 0 and Case 1, respectively.
IV. EXAMPLES
In this section, we confirm the universality of the wavy
configuration in the N-fold symmetric tubular origami tessel-
lations through the numerical computation of the phase dia-
grams of Funder some values of ki,I,N,li,L,li,M,li,R,σi(i=
0,...,I−1). In the visualization, we also show some folded
states corresponding to certain orbits, including wavy and
cylindrical ones.
A. Two topology of crease patterns
Here, we consider the crease patterns having one of two
types of topology: the one consists of 6-valent vertices (Fig-
ure 9), and the other consists of 4 or 8-valent vertices (Fig-
ure 10). As for 6-valent tessellations, the connectivity trans-
formation is no longer needed at any steps. Hence, their dy-
namical system can be written as fI−1◦··· ◦f0. We consider
the two cases: I=3 and I=4 which results in the differ-
ent shapes of modules shown in left and right in Figure 9,
respectively. Ignoring the MV assignment and focusing on
the topology and the module-shape, we can see the 6-valent
tessellation with I=3 and I=4 as the generalization of wa-
terbomb tessellation (Figure 11 A and A’) and triangulated
Miura-ori (Figure 12 A and A’), respectively. On the other
hand, the dynamical system of 4 or 8-valent tessellations in
Figure 10 can be represented as f3◦g2◦f2◦f1◦g0◦f0. In the
same way, we can regard this type of crease patterns as the
generalization of the triangulated eggbox pattern (Figure 13
A).
1-st zigzag
0-th zigzag
2-nd zigzag 3-rd zigzag
Left Right
FIG. 9. Crease pattern consists of 6-valent vertices. The left and
right pattern is the case for I=3 and I=4, respectively. Edges in
different types of zigzags are shown in different colors, and thin/thick
edges represent their left/right edges. The gray regions indicate the
module for each case.
1-st zigzag
0-th zigzag
2-nd zigzag 3-rd zigzag
Left Right
FIG. 10. Crease pattern consists of 4 or 8-valent vertices. Edges in
different types of zigzags are shown in different colors, and thin/thick
edges represent their left/right edges. The gray region indicates the
module.
B. Observation of phase diagrams
Varying the values of parameters N,li,L,li,M,li,R,σi(i=
0,...,I−1), we draw phase diagrams of F. In the computation
of the phase diagram, we try some parameter values to find the
case where Fhas the elliptic fixed point or p-periodic point
[¯
d,¯
ρ]Tsatisfying Fp([ ¯
d,¯
ρ]T)=[¯
d,¯
ρ]T(p∈Z>0), where the
Jacobian matrix DFp([ ¯
d,¯
ρ]T)has complex conjugate eigen-
values with magnitude 1. The existence of such a point de-
pends on the crease patterns’ parameters but is not rare; these
fixed points stay persist if the crease patterns are perturbed.
We have manually chosen 13 sets of parameters that have such
points. To find fixed points or p-periodic points for a given
crease pattern, we used Newton’s method and linear stability
analysis. If we get parameter values where elliptic point ex-
ists, we compute orbits {[dt,ρt]T=Ft([d0,ρ0]T)|t=0,1, . . . }
in which we take [d0,ρ0]Tnear [¯
d,¯
ρ]T. Then, we plot the or-
bits and get the phase diagram of F.
Figure 11, Figure 12, and Figure 10 show the phase dia-
grams and the folded states corresponding to some solutions
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0.2 0.3 0.4 0.5 0.6
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
dt
ρt
Phase Diagram
0.2 0.3 0.4 0.5 0.6
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
dt
ρt
Phase Diagram
(𝑑0, 𝜌0) = (0.460135, 0.111611)
(𝑑0, 𝜌0) = (0.35, 0.111611)
(𝑑0, 𝜌0) = (0.2, 0.111611)
(𝑑0, 𝜌0) = (0.441827, 0.0632163)
(𝑑0, 𝜌0) = (0.4, 0.0632163)
(𝑑0, 𝜌0) = (0.35, 0.0632163)
0.670 0.675 0.680
1.62
1.64
1.66
1.68
1.70
1.72
1.74
1.76
dt
ρt
Phase Diagram
(𝑑0, 𝜌0) = (0.675736, 1.70178)
(𝑑0, 𝜌0) = (0.682688, 1.70946)
(𝑑0, 𝜌0) = (0.61657, 1.57903)
(𝑑0, 𝜌0) = (0.65,1.57903)
0.55 0.60 0.65
1.0
1.2
1.4
1.6
1.8
2.0
dt
ρt
Phase Diagram
A
A′
B
B′
FIG. 11. The plots of solutions of f2◦f1◦f0, i.e., phase diagrams, where the solutions initiated from different (d0,ρ0)are shown in
different colors. Figures A and A0represent two cases under (σ0,σ1,σ2) = (M,M,M), with (l0,L,l0,M,l0,R,l1,L,l1,M,l1,Rl2,L,l2,M,l2,R,N) =
(1,0.620284,1,1,√2,1,0.712417,0.620284,0.712417,12)and (1,0.620284,1,1.1,1.4,0.90.72,0.7,0.72,12), respectively. Additionally, the
configurations below each diagram correspond to the initial 19 terms in specific solutions marked as ◦,4, and in the phase diagram, where
tincreases from left to right in the configuration. In particular, the solution marked as ◦is the fixed point whose corresponding configuration
forms the cylindrical shape. The difference between the A and A0is that the configurations on A have the symmetry described by the
point symmetry CNV because li,L=li,Rfor i=0,1,2, while there is no mirror plane in the ones on A0. Figures B and B0show the case of
(σ0,σ1,σ2) = (M,M,V)in the same format as A and A0. B and B0represent two cases, with (l0,L,l0,M,l0,R,l1,L,l1,M,l1,R,l2,L,l2,M,l2,R, , N) =
(0.815,0.95,0.815,1.24,1.223,1.24,1.33,.525,1.33,10)and (0.7,0.95,0.851.1,1.2,1.2,1,0.5,1.33,10), respectively.
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(𝑑0, 𝜌0) = (0.781493, 2.94836)
(𝑑0, 𝜌0) = (0.67, 2.94836)
(𝑑0, 𝜌0) = (0.752223, 2.97614)
(𝑑0, 𝜌0) = (0.67, 2.97614)
(𝑑0, 𝜌0) = (0.497844, 2.49896)
(𝑑0, 𝜌0) = (0.54, 2.49896)
(𝑑0, 𝜌0) = (0.504408, 2.46821)
(𝑑0, 𝜌0) = (0.535, 2.46821)
(𝑑0, 𝜌0) = (0.426646, -2.94949)
(𝑑0, 𝜌0) = (0.5, -2.94949)
0.46 0.48 0.50 0.52
2.30
2.35
2.40
2.45
2.50
2.55
2.60
2.65
dt
ρt
Phase Diagram
0.40 0.45 0.50
2.3
2.4
2.5
2.6
2.7
dt
ρt
Phase Diagram
2.7
2.8
2.9
3.
3.1
dt
ρt
Phase Diagram
0.70 0.75 0.80
-3.1
2.8
2.9
3.
3.1
dt
ρt
Phase Diagram
0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80
-3.1
(𝑑0, 𝜌0) = (0.407726, -2.92555)
(𝑑0, 𝜌0) = (0.5, -2.92555)
0.20 0.25 0.30 0.35 0.40 0.45 0.50
-3.
-2.8
-2.6
-2.4
2.8
3.
dt
ρt
Phase Diagram
2.8
3.
0.20 0.25 0.30 0.35 0.40 0.45 0.50
-3.
-2.8
-2.6
-2.4
dt
ρt
Phase Diagram
A
A′
B
B′
C
C′
FIG. 12. The phase diagrams of f3◦f2◦f1◦f0in the same setting as Figure 11. A and A0, B and B0, and C and
C0are the case where the values of (σ0,σ1,σ2,σ3)are: (V,M,M,V),(M,V,M,V), and (V,V,V,M), respectively. The values of
(l0,L,l0,M,l0,R,l1,L,l1,M,l1,R,l2,L,l2,M,l2,R,l3,L,l3,M,l3,R,N)in A, A0, B, B0, C, and C0are: (0.86,0.855,0.86,1.175,0.95,1.175,1.255,0.525,
1.255,1.31,0.925,1.31,15),(0.8,0.86,0.85,1.15,1.,1.175,1.2,0.525,1.255,1.2,0.925,1.31,15),(0.72,0.81,0.72,0.88,0.75,0.88,0.63,
0.55,0.63,1.03,0.6675,1.03,25),(0.7,0.8,0.75,0.85,0.75,0.9,0.62,0.55,0.65,1,0.68,1.1,25),(0.81,0.45,0.81,1.345,0.615,1.345,0.91,
0.745,0.91,0.725,0.965,0.725,18), and (0.8,0.45,0.81,1.3,0.615,1.345,0.9,0.745,0.91,0.7,0.965,0.725,18), respectively.
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0.2 0.3 0.4 0.5 0.6 0.7 0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
dt
ρt
Phase Diagram
0.40 0.45 0.50 0.55 0.60 0.65 0.70
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
dt
ρt
Phase Diagram
(𝑑0, 𝜌0) = (0.595209, 2.6609)
(𝑑0, 𝜌0) = (0.475, 2.6609)
(𝑑0, 𝜌0) = (0.702775, 2.88329)
(𝑑0, 𝜌0) = (0.657060, 2.851836)
(𝑑0, 𝜌0) = (0.633417, -0.0714159)
(𝑑0, 𝜌0) = (0.3, -0.0714159)
2.8
2.9
3.
3.1
dt
ρt
Phase Diagram
0.60 0.65 0.70 0.75
-3.1
-3.
-2.9
A CB
FIG. 13. The phase diagrams of f3◦g2◦f2◦f1◦g0◦f0in the same setting as Figure 11. A, B, and
C are the cases where the values of (σ0,σ1,σ2,σ3)are (M,M,V,V),(M,M,M,M)and (M,V,V,M), respec-
tively. The values of (l0,L,l0,M,l0,R,l1,L,l1,M,l1,R,l2,L,l2,M,l2,R,l3,L,l3,M,l3,R,N)in A, B and C are (0.9,0.7,0.9,
0.9,0.7,0.9,0.9,0.7,0.9,0.9,0.85,0.9,10),(0.835,0.67,0.835,1.055,0.77,1.055,0.95,0.815,0.95,0.89,0.86,0.89,10), and
(0.975,0.3,0.975,1.486,0.75,1.486,1.215,0.548,1.215,0.94,0.562,0.94,12), respectively.
of f2◦f1◦f0,f3◦f2◦f1◦f0, and f3◦g2◦f2◦f1◦g0◦f0, re-
spectively. In the figures, we can observe the same structure
as the waterbomb tube described in the previous research26,
that is, there are nested closed cyclic plots, namely, quasiperi-
odic solutions around the elliptic fixed point, which neither
converge/diverge to/from the fixed point. Each solution of F
corresponds to the certain folded state of the crease pattern.
In particular, the folded state corresponding to the fixed point
[¯
d,¯
ρ]Tforms the cylinder-like shape in which the folded stats
of rings are identical, i.e., uniform folded state. The orbit
in the empty regions outside of the quasiperiodic solutions
results in the “finite solution” observed in the previous re-
search26, where the three-sphere-intersection has no solution
at some time steps.
Here, in each quasiperiodic solution around [¯
d,¯
ρ]T, the
value of dt, representing the diameter of t-th ring, oscillates
between values smaller/larger than ¯
d. As a result, the diameter
of the corresponding tubular folded state also oscillates, giv-
ing rise to the wavy folded state, where the peaks and valleys
exhibit larger or smaller diameters compared to the cylinder-
like state. Although the wavy folded state seems to approxi-
mate a periodic surface like an unduloid, the quasiperiodicity
of the solution implies that the folded states of a ring and the
one after the “one-period” of the wavy shape do not coincide.
In contrast, periodic solutions around [¯
d,¯
ρ]Tyield the folded
states of a ring identical to the one after one period of wavy
shape (see the 3-periodic solution in Figure 11 B and the 2-
periodic solution in 13 C).
The folding motion from one wavy folded state to another
can be achieved through a continuous change in an initial
value from a point in the orbit corresponding to the initial state
to that of the terminal state. In particular, we can tune the “am-
plitude” and the “phase” of the wavy folded state, where the
changes in “radius” of the plot and the selection of an initial
value within the same orbit correspond to the change in am-
plitude and phase, respectively.
We can still observe the same phenomenon even if we
slightly change the parameter values; however, note that the
existence of quasiperiodic solutions depends on the parame-
ter values. Specifically, the existence of fixed/periodic points
and their stability, i.e., whether they are elliptic or not, de-
pends on crease pattern parameters. The detailed analysis of
the dependence on crease pattern parameters for the water-
bomb tube is performed in the previous work26. The results
include some generalizations of the previous research. For ex-
ample, the scope of the previous research26 was limited to the
case of the waterbomb tube having point symmetry denoted
by CNV , i.e., N-fold rotational symmetry with a vertical mir-
ror plane. The upper figures in Figure 11,12, and 13 show the
folded states having the symmetry of CNV , which is equivalent
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to li,L=li,Rholds for every i. As shown in the figures in the
lower row in Figure 11,12, we observed that quasiperiodic so-
lutions and their corresponding wavy folded states could exist
even if li,L6=li,Rat some i. As another generalization, in the
previous research, the center of the quasiperiodic solutions is
a fixed point in every case; however, we found a case in which
a periodic point locates at the center of quasiperiodic orbits.
For instance, the 2-periodic points center at quasiperiodic so-
lutions in Figure 13 B.
More specifically, Figure 12 A and A0show that the gen-
eralized Miura-ori can approximate wavy surfaces. Similarly,
Figure 13 A shows the generalized eggbox pattern approx-
imating the wavy surfaces. The one reason why the original
Miura-ori/eggbox pattern, whose module consists of four con-
gruent parallelograms, can only approximate the surface with
negative/positive Gaussian curvature22,24,25 is, their dynami-
cal systems have no fixed point that is elliptic.
V. CONSERVATIVENESS
Quasiperiodic solutions, which neither converge to some
points nor diverge under time evolution, are typically ob-
served in conservative dynamical systems that preserve some
kinds of measures. Here, we demonstrate that Fis the con-
servative dynamical system regardless of the values of pa-
rameters, which explains the universality of the wavy con-
figurations in N-fold symmetric tubular origami tessellations.
Specifically, we give proof that Fpreserves the area of param-
eter spaces, i.e., Fis the area-preserving system one possible
parameterization of a conservative system. Furthermore, we
extend the scope of Finto the crease pattern with the scaling
and show that the dynamical system is no longer conserva-
tive if the crease pattern includes the scaling. For a general
discussion of conservative systems and area-preserving map-
pings, see, for example, Arnold (1989)32 or Mackay (1994)33.
A. Proof of Conservation
To prove that Fis area-preserving, we show that g◦f
and gare area-preserving. Actually, if g◦fand gare area-
preserving, fis also area-preserving because the inverse of
gunder (lL,lR)=(a,b)is given as gunder (lL,lR)=(b,a).
Then, we conclude that Fis area-preserving because Fis the
composition of area-preserving mappings fand g.
First, the area-preservation of g◦fcan be proven geo-
metrically. The area-preservation of g◦fis equivalent to
detD(g◦f)≡1. Let we denote [(d0)∗,(ρ0)∗]T=g◦f([d,ρ]T).
Here, the definition of fand gprovides (d0)∗≡d, which im-
plies ∂(d0)∗/∂d≡1 and ∂(d0)∗/∂ ρ ≡0 (see Figure 7 and Fig-
ure 8 right). Therefore, it is enough to prove ∂(ρ0)∗/∂ ρ ≡1.
To prove this, let us use Figure 8 and suppose that we incre-
ment ρby ∆ρ. This incrementation of ρcauses the rotation of
the tetrahedron U1U2U3Varound −−−→
U1U3by ∆ρ. After all, the
increment of ρby ∆ρsimply results in the increment of (ρ0)∗
by ∆ρ. Hence, g◦fis area-preserving.
Second, we must take the algebraic proof for the area-
preservation of g. Based on the explicit form of gshown
in B 2, we have proved det Dg≡1 regardless of the param-
eter values using Mathematica; however, the expression was
highly complicated. Hence, in the following, we deal with the
case lL=lR=lwhere the length of the edges is uniform. The
form of gunder this assumption is reduced as follows:
d∗=sin π
Nrpl2−d2sinρ+dcot π
N2,(13)
ρ∗=arctan2 −1
4pd2l2−d4cosρ
r4d2cot π
N+4pd2l2−d4sinρ2,
sin π
Nd2cot π
N+pd2l2−d4sinρ
d2−pd2l2−d4cot π
Nsinρ.
(14)
Then, the components of Dgiis calculated as follows:
∂d∗
∂d=−dsin π
Nsinρ−√l2−d2cos π
N
√l2−d2,(15)
∂d∗
∂ ρ =pl2−d2sin π
Ncosρ,(16)
∂ ρ ∗
∂d=l2sin π
Ncosρ/pl2−d2
−dsin π
Nsinρ−pl2−d2cos π
N2
−lsin π
Ncosρ2,
(17)
∂ ρ ∗
∂ ρ =
√l2−d2dsin π
Nsinρ−√l2−d2cos π
N
−dsin π
Nsinρ−√l2−d2cos π
N2−lsin π
Ncosρ2.
(18)
Thus, detDg≡1 holds. From the above, Fis the area-
preserving dynamical system.
The area-preservation of Fimmediately explains the struc-
tural stability of elliptic fixed points against the perturbation
of a crease pattern mentioned in Section IV. This is because
the multipliers of a fixed point must be reciprocal, which im-
plies that its stability can change only through a fixed point
with its multipliers equal to 1 or −1.
B. Non-conservative example: crease pattern including
scaling
Now, to reveal what causes the conservativeness of the sys-
tem, we introduce the scaling to the zigzag-based crease pat-
tern, where the pattern of i+1-st ring is that of i-th one
scaled by a factor s∈R>0(see Figure 14). To extend Fto
˜
F:P(l0,L,l0,R,N)→P(l0,L,l0,R,N);[dt,ρt]T7→[dt+1,ρt+1]T
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so that its scope includes the crease pattern with scaling, we
modify Fas follows:
[dt+1,ρt+1]T=˜
s◦˜
gkI−1
I−1◦˜
fI−1◦···◦˜
gk0
0◦˜
f0[dt,ρt]T,
˜
fi([d,ρ]T):=
fi([d,ρ]T)if i=0,...,I−2
f([d,ρ]T;li,L,li,R,N,sri,1,ri,2,sri,3,σi)
if i=I−1
,
˜
gi([d,ρ]T):=(gi([d,ρ]T)if i=0,...,I−2
g([d,ρ]T;sri,3,sri,1,N)if i=I−1,
˜
s([d,ρ]T):= [d/s,ρ]T.
The difference between ˜
Fand Fis that rI−1,1and rI−1,3in fI−1
and gI−1are scaled by a factor sbecause they are the lengths
of edges in the next ring. Additionally, the scaling mapping
˜
sis introduced, which makes the initial space and the termi-
nal space of ˜
Fidentical. Without ˜
s, the initial space of ˜
Fis
P(l0,L,l0,R,N)but the terminal space is P(sl0,L,sl0,R,N).˜
s
resolves this discrepancy by dividing dby s, which is equiva-
lent to scaling the zigzag by a factor of 1/s.
Let us consider the characteristics of ˜
Ffocusing on its con-
servativeness. Because fiand giare area-preserving map-
pings,
detD˜
F=detD˜
s≡1/s; (19)
therefore, if s6=1, ˜
Fis non-conservative. Specifically, if
s>1 (s<1), ˜
Fexhibits the contracting (expanding) behavior,
where there can be a sink (source), which is not possible in
conservative systems. Note that ˜
Fwith s=s0is the inverse of
˜
Fwith s=1/s0.
For example, Figure 15 shows the phase diagrams of ˜
F=
˜
f2◦˜
f1◦˜
f0and ˜
F=˜
f3◦˜
f2◦˜
f1◦˜
f0with s>1. We can ob-
serve that the orbits spirally converge to the point, which is
the sink corresponding to the cone-like folded state in which
rings are similar. Hence, the folded states corresponding to
the orbits near the fixed point are still wavy; however, their
amplitude relative to the edge lengths attenuates when we ig-
nore the scale. In other words, if we perturb the folded state
of the 0-th ring of the cone-like state, the change in the folded
state of the t-th ring decreases as tincreases.
VI. CONCLUSION
This paper presented the generalization and the extension
of the dynamical system model of origami tessellations sug-
gested by the authors26. We focused on the crease pat-
tern constructed by connecting zigzag patterns and their N-
fold symmetric tubular folded states. Based on the fact we
can calculate their folded state by repeating the three-sphere-
intersection operation, we formulated the coupled motion of
rings as the dynamical system defined on the configuration
space of the N-fold symmetric zigzag. Then, by introduc-
ing the coordinate mapping reflecting the connectivity be-
tween adjacent zigzags, we parameterized the dynamical sys-
tem as the composition of two mappings defined on the pa-
1-st zigzag
0-th zigzag
2-nd zigzag
Left Right
1-st zigzag
0-th zigzag
2-nd zigzag 3-rd zigzag
FIG. 14. Waterbomb pattern includes the scaling, in which the
length of edges in t+1-st ring is that in t-th one scaled by a factor s.
rameter spaces: three-sphere-intersection and connectivity-
transformation. Using the parameterized dynamical system,
we visualized the phase diagram for various origami tessella-
tions and observed the correspondence of quasiperiodic solu-
tions and wavy folded states, the same phenomenon as the wa-
terbomb tube. We finally proved that the dynamical system is
conservative by showing the area-preservation of the parame-
terized dynamical system regardless of crease patterns, which
explains the universality of the phenomenon. Furthermore,
we demonstrated that the conservativeness of the dynamical
system vanishes when the crease pattern includes the scaling.
In general, an area-preserving map can have a solution
densely fill some regions of a phase space having a posi-
tive measure, called chaotic sea. The phase space structure
in Figure 11, 12, and 13 implies the existence of such a
chaotic sea. For example, we can see 9-periodic point that
is elliptic or saddle appears alternately in Figure 11 A0re-
flecting Poincaré–Birkhoff fixed point theorem. These points
form a chaotic sea bounded by stable/unstable manifolds of 9-
periodic saddle; however, the chaotic sea is very small. Thus,
the change in a chaotic solution is small, and the correspond-
ing tubular folded state results in the wavy folded state not
distinguishable from that of quasiperiodic solutions. If we
straightforwardly try to create a chaotic solution filling a large
region, there can be a case where three spheres have no in-
tersections at some time steps. In such a situation, the folded
state of i+1-st zigzag compatible with a given folded state
of i-th zigzag does not exist, which results in a finite solution.
Assuring the existence of an infinite chaotic solution to dis-
cover and create a novel chaotic origami is our future research
direction.
In this study, we omitted the detailed exploration of the
phase space structure and its dependence on the parameters
because there are too many parameters. Hence, detailed anal-
ysis such as the bifurcation analysis and the computation of
the stable/unstable manifolds is the future task, which en-
riches the understanding of the nonuniform-folding of tubular
origami tessellations. We can make this direction of analysis
easier by imposing constraints such as developability or flat-
foldability, which can decrease the number of parameters. In
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Undulations in Tubular Origami Tessellations: a Connection to Area-Preserving Maps 13
0.30 0.35 0.40 0.45
0.0
0.1
0.2
0.3
0.4
0.5
dt
ρt
Phase Diagram
(𝑑0, 𝜌0) = (0.399226, 0.211836)
(𝑑0, 𝜌0) = (0.35, 0.211836)
(𝑑0, 𝜌0) = (0.3, 0.211836)
2.8
2.9
3.
3.1
Phase Diagram
dt
ρt
0.50 0.55 0.60 0.65 0.70 0.75 0.80
-2.8
-2.9
-3.
-3.1
(𝑑0, 𝜌0) = (0.698938, -3.09219)
(𝑑0, 𝜌0) = (0.75, -3.09219)
(𝑑0, 𝜌0) = (0.8, -3.09219)
A B
FIG. 15. Figure A and B shows the phase diagram of ˜
f2◦˜
f1◦˜
f0and ˜
f3◦˜
f2◦˜
f1◦˜
f0, respectively. The parameter values of ˜
f2◦˜
f1◦˜
f0and
˜
f3◦˜
f2◦˜
f1◦˜
f0are the same as A0in Figure 11 and A0in Figure 12, but scale factor is set as s=1.08 and s=1.05, respectively. In the phase
diagrams, the time evolution is represented by the gray arrows.
addition, there are some origami tessellations we cannot con-
struct through the zigzag-based procedure. The famous Ron
Resch pattern34 is an example. Thus, generalizing the present
result to other tubular origami tessellation is also a possible fu-
ture work. Furthermore, though we assumed the axisymmetry
in the folding, which makes tubular origami tessellations the
2-DOFs mechanisms, we can consider the folding with higher
DOFs without the symmetry assumption, which results in the
dynamical system on a higher dimensional space. How we can
interpret the 2-dimensional conservative system developed in
this paper from the viewpoint of a higher-dimensional system
is the question we should examine.
The dynamical system model is grounded in the approach
based on the coupled folding motion of modules which is uni-
versal in origami tessellations other than ones with tubular
forms. Thus, the model has the potential to be the univer-
sal mathematical model for understanding the nonuniform-
folding of origami tessellations with planer or cellular forms.
For future engineering applications, connecting the mathe-
matical structure of the dynamical system with the physical
property of the object would be essential.
ACKNOWLEDGMENTS
This work was supported by JST PRESTO Grant Num-
ber JPMJPR1927 and JSPS KAKENHI Grant Number
JP23KJ0682.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0160803
Undulations in Tubular Origami Tessellations: a Connection to Area-Preserving Maps 14
Author Contributions
R. Imada: Conceptualization (equal); Funding acquisition
(equal); Formal Analysis; Methodology (lead); Software; Vi-
sualisation; Writing–Original Draft (lead); Writing–review &
editing (equal). T. Tachi: Conceptualization (equal); Funding
acquisition (equal); Methodology (supporting); Supervision;
Writing–Original Draft (supporting); Writing–review & edit-
ing (equal).
DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
Appendix A: Coordinate Mapping
1. Coordinate mapping
Consider parameterizing the folded state x∈M(lL,lR,N),
where vertices U1,U2,U3are taken in the manner depicted
in Figure 4. First, we define the coordinate system in 3-
dimensional Euclidean space so that X-axis coincides with the
axis of N-fold symmetry and Y-axis is parallel with −−−→
U3U1and
U1and U3are on Y Z plane. Hence, letting rUjdenotes the
position vector of Uj(j=1,2,3),rU3can be represented as
rU3=RX(2π/N)rU1, where RX(2π/N)denotes the rotation
matrix of angle 2π/Naround X-axis. Then, we can obtain
[d,ρ]T=φlL,lR,N(x)∈P(lL,lR,N)as follows:
d=||rU3−rU1||
2,(A1)
ρ=arctan2v
||v|| ·eX,v
||v|| ×eX·rU3−rU1
2d,(A2)
where eX= [1,0,0]Tand
v=rU2−rU1+lLl2
L+ (2d)2−l2
R
2lL(2d)rU3−rU1
2d.
(A3)
2. Inverse of coordinate mapping
Next, we consider the inverse of coordinate mapping, i.e.,
construct the folded state of zigzag from given [d,ρ]T∈
P(lL,lR,N). Using the coordinate system defined above, the
position vector of U1and U2can be represented as follows:
rU1=h0,d,dcot π
NiT,(A4)
rU2=0,d−lLl2
L+ (2d)2−l2
R
2lL(2d),dcot π
NT
+lLs1−l2
L+ (2d)2−l2
R
2lL(2d)2
[cosρ,0,sin ρ]T.
(A5)
Then, the coordinates of other vertices can be obtained as
(RX(2π/N))jrU1or (RX(2π/N))jrU2(j=1,...,N−1).
Appendix B: Two Mappings on Parameter Spaces
1. Explicit form of three-sphere-intersection
Consider [d,ρ]T∈P(lL,lR,N)is given and calculating
f[d,ρ]T;lL,lR,r1,r2,r3,N,σ. First, obtain the folded state
corresponding to [d,ρ]Tusing the inverse of coordinate map-
ping (see A 2). Then, take the intersection of three spheres
centered at rU1,rU2,rU3with radius r1,r2,r3. Let us assume
that these spheres intersect at two points VMand VV, which
correspond to σ=Mand σ=V, respectively. rVMand rVV
can be calculated as follows: 1) consider the local coordinate
system defined by e1= (rU2−rU1)/|rU2−rU1|,e2= ((rU3−
rU1)−((rU3−rU1)·e1)e1)/p(2d)2−((rU3−rU1)·e1)2, and
e3=e1×e2, which origin is set at rU1. 2) let e1,e2,e3denote
the e1and e2component of rVσ−rU1, which can be repre-
sented as follows:
e1=r2
1+|rU2−rU1|2−r2
2
2|rU2−rU1|,(B1)
e2=1
2rU3−rU1·e2r2
1−r2
3+(rU3−rU1)·e12
+rU3−rU1·e22−2e1(rU3−rU1)·e1,
(B2)
e3=±qr2
1−e2
1−e2
2,(B3)
where ±sign in e3correspond to σ=M,V, respectively. 3)
Then, we get rVσ=rU1+e1e1+e2e2+e3e3. Finally, we ob-
tain f[d,ρ]T;lL,lR,N,r1,r2,r3,σusing the coordinate map-
ping where vertices positioned at rVσ,rU2,RX(2π/N)rVσare
set as U1,U2,U3.
2. Explicit form of connectivity-transformation
Consider calculating [d∗,ρ∗]T∈P(r1,r3,N)from the
given [d,ρ]T∈P(r3,r1,N). First, obtain the folded
state corresponding to [d,ρ]Tusing the inverse of coor-
dinate mapping (see A 2). Then, we get [d∗,ρ∗]Tby
reparameterize the obtained folded state using the co-
ordinate mapping (see A 1) where we take vertices at
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PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0160803
Undulations in Tubular Origami Tessellations: a Connection to Area-Preserving Maps 15
rU2,rU3,RX(2π/N)rU2as U1,U2,U3. The explicit form of
[d∗,ρ∗]T=g[d,ρ]T;r3,r1,Nobtained through the above
procedure is as follows:
d∗=sin π
N
4d
v
u
u
u
u
u
tr2
3−r2
12+4d2cot π
N
+sinρq16d2r2
3−4d2+r2
3−r2
122
,
(B4)
ρ∗=arctan2 −cos ρq16d2r2
3−4d2+r2
3−r2
12
v
u
u
u
u
u
tr2
1−r2
32+4d2cot π
N
+sinρq16d2r2
3−4d2+r2
3−r2
122,
−sin π
Nsinρq16d2r2
3−4d2+r2
3−r2
12
+4d2cot π
Ncot π
Nsinρq16d2r2
3−4d2+r2
3−r2
12
−4d2+cot π
Nr2
1−r2
32.
(B5)
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