Research

Themed Issue in Origami and Kirigami Mechanics

Yogesh Phalak

Ziyang Zhou

# “Golden Ratio Yoshimura” for Meta-Stable and Massively Reconfigurable
Deployment

Vishrut Deshpande1 Yogesh Phalak1 Ziyang Zhou1 Ian Walker2 and Suyi Li1
1Department of Mechanical Engineering
Virginia Tech, Blacksburg, VA USA
2Department of Electrical and Computer Engineering
Clemson University, Clemson, SC USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Yoshimura origami is a classical folding pattern that has inspired many
deployable structure designs. Its applications span from space exploration,
kinetic architectures, and soft robots to even everyday household items.
However, despite its wide usage, Yoshimura has been fixated on a set of design
constraints to ensure its flat-foldability. Through extensive kinematic
analysis and prototype tests, this study presents a new Yoshimura that
intentionally defies these constraints. Remarkably, one can impart a unique
meta-stability by using the Golden Ratio angle to define the triangular facets
of a generalized Yoshimura. As a result, when its facets are strategically
popped out, a “Golden Ratio Yoshimura” boom with $m$ modules can be
theoretically reconfigured into $8^{m}$ geometrically unique and load-bearing
shapes. This result not only challenges the existing design norms but also
opens up a new avenue to create deployable and versatile structural systems.

###### keywords:

Origami, Meta-Stability, Deployment

##  1 Introduction

Deployable structures have garnered considerable interest over the past
decades due to their unique ability to transform from a compact volume to a
larger-scale, load-bearing shape [1, 2, 3]. Their applications are diverse.
For example, they have been the essential components for satellites, exoplanet
rovers, and space telescopes [4, 5]. They will continue to play a vital role
in the future in in-orbit fabrication [6, 7] and space habit construction [8,
9, 10]. Here on Earth, deployable structures are ideal for making military and
civilian shelters [11, 12] and are even used to develop novel soft robots [13,
14, 15, 16].

A successful deployable structure must meet multiple requirements. It should
be efficient in packing (aka. high packing ratio), lightweight for easy
transportation, and load-bearing at its final stage. To achieve these goals,
researchers have explored various deployment approaches, including pneumatic
inflation [17, 18, 19], origami folding [20, 21, 22, 23, 24], rolling
composite shells [25, 26], responsive materials [27, 28, 29], and multi-stable
mechanics [30, 31, 32]. Furthermore, one can combine some of these approaches
to create innovative solutions. For instance, the combination of pneumatic
actuation, bi-stability, and origami folding can enable rapid actuation, akin
to the Venus flytrap plant [33], and allow for meter-scale construction [34].
Integrating piezo-electric patches onto composite shells can provide versatile
actuation and shape control [35].

However, most deployable structures available today have limited capabilities
in that they can only achieve one targeted state. This constraint highlights
the potential of a more versatile solution. If a deployable structure can
quickly morph into different load-bearing shapes, it can be reconfigured on
demand according to the dynamic working environment and desired
functionalities. This level of reconfigurability can significantly enhance its
appeal. In light of this, we propose and examine a deployable boom structure
with massive reconfigurability by expanding the design space of Yoshimura
Origami.

Figure 1: A big-picture overview of the Golden Ratio Yoshimura as a deployable
and reconfigurable structure. (a) A 3D-printed, meter-scale prototype can be
easily configured into different meta-stable shapes for different purposes,
including a fully folded (compact) configuration, a fully deployed
configuration, a load-bearing arc, or letters “VT” and “CU,” representing the
institute affiliation of the authors. Notice the 3D printed sample weighs
225.6 grams, and a second Yoshimura was used to finish the letter “T.” (b).
The buckling pattern of a cylindrical thin shell reveals the origin of
Yoshimura. Image adapted from [36] with permission.

The Yoshimura pattern is a classical origami design that naturally emerges
from the buckling mechanics of thin-wall cylindrical shells [37] (Fig. 1).
When the cylinder is under axial compression, a uniform tessellation of
rhombus-shaped creases can develop on its surface. Each rhombus consists of
two triangular facets connected by a fold line along the cylinder’s
circumference. As the compression continues, the cylinder buckles along these
crease lines, and the two triangles in each rhombus fold inward, eventually
packing the cylinder into a small volume (Fig. 1). This buckling-induced
crease pattern became the design basis for Yoshimura origami, which has been
made into deployable booms [38, 39, 40, 41] and soft robotic spines [42, 43,
44, 45, 46].

Despite such wide usage, deployable Yoshimura has been fixated on a design
rule in the current literature: The shape of its triangular facets must be
constrained by the number of rhombus creases along the cylinder’s
circumference. This rule can ensure the flat-foldability of Yoshimura.
However, we showed that by intentionally breaking these established design
rules, one can significantly enrich the kinematic configuration space of
Yoshimura and generate a unique “meta-stability.” In this way, one can
selectively pop out pairs of triangular facets to settle the new Yoshimura
into many different elastically stable and load-bearing configurations.
Remarkably, and _for the first time, we mathematically proved that, for a
Yoshimura pattern with three rhombi along its circumference (aka. $n=3$ in
Fig. 2), the required triangular facet shape to attain such meta-stability
follows the Golden Ratio_ (aka. $\cot\beta=\varphi\approx 1.618$ in Fig. 1,
2). We refer to this new origami design as the “Golden Ratio Yoshimura,” and
it can become the foundation for those, as mentioned earlier, massively
reconfigurable and deployable structures.

In what follows, section 2 details the design and kinematics of a Golden Ratio
Yoshimura module. Section 3 illustrates the meta-stability and configuration
space of a larger-scale boom structure with multiple modules. Section 4
presents the fabrication and load-bearing test results of 3D-printed Golden
Ratio Yoshimura samples. Finally, Section 5 ends this paper with a conclusion
and discussion.

##  2 Design and Kinematics of the Golden Ratio Yoshimura

Figure 2: The design and kinematics of folded Yoshimura with a generalized
design. (a) The crease pattern design highlights the important parameters.
$n$: number of rhombi along the circumference; $L$: length of the valley
creases; and $\beta$, sector angle between the mountain and valley creases.
(b) The correlations between facets dihedral angle $\theta$, the normalized
folded-height $h/L$, and $\beta$.

The Yoshimura pattern is a tessellation of _mountain_ and _valley_ folds such
that a 2D sheet, upon folding, curls to form a closed shape resembling a
cylinder. Figure 2(a) outlines its layout, where valley creases align with the
circumferential direction of the cylinder, and the mountain creases are
inclined relative to the valley ones. Such sector angle ’$\beta$’ defines the
Yoshimura pattern’s overall shape and will play a central role later in the
kinematic analysis. Besides $\beta$, $L$ is the valley crease’s length and
defines Yoshimura’s size; $n$ is the number of rhombi shapes (pairs of
triangular facets) along the circumference direction. These three parameters
are sufficient to completely define the geometry of the Yoshimura. Moreover,
we designate one layer of rhombi creases as a module.

Traditionally, these three design parameters are not independent. Instead,
they follow a geometric constraint [42]:

$\beta=90\degree-\frac{(n-1)180\degree}{2n}\text{, or }2n\beta=180\degree.$
(1)

For clarity, we call such a design a "traditional Yoshimura." This constraint
ensures that, after folding, the Yoshimura is stress-free in its facets and
has a zero height (aka flat-foldability). For this study, we focused on $n=3$
design, so $\beta=30\degree$.

However, _what if Yoshimura defies this constraint_? To this end, this study
generalizes the Yoshimura design by setting the sector angle $\beta$
independent of $n$. We discovered that when $\beta$ is larger than the
traditional value, the Yoshimura is no longer flat-foldable but has a finite
height after folding, denoted as “folded-height” $h$ in Fig. 2. Moreover, this
height is a function of the dihedral angle $\theta$ between the two triangular
facets in each rhombus so that:

$h=L\tan\beta\sin\theta.$ (2)

This dihedral angle in this folded configuration is also related to $\beta$
and $n$ through the following equation:

$\tan\beta\cos\theta=\tan\frac{\pi}{2n}.$ (3)

The above relationship gives the lower bound for the generalized Yoshimura
design: $\beta\geq\frac{\pi}{2n}$, and any smaller $\beta$ value would not be
geometrically possible, otherwise $h$ and $\theta$ become negative. The
traditional Yoshimura designs precisely follow this lower bound so that $h=0$
and $\theta=0$, giving its flat foldability. On the other hand, when $\beta$
increases, the dihedral angle and folded height increase rapidly (Fig. 2b).
For example, when $\beta=45\degree$, $\theta$ reaches 55°and $h/L=0.8$.

### 2.1 Popping and the Emergence of Golden Ratio Yoshimura

The Yoshimura is often used in folded configurations due to its soft and
conformable nature for robotic manipulations and locomotion tasks. It is
dexterous enough to be manipulated (or actuated) into complex curvatures.
However, a fully folded and soft Yoshimura has minimal load-bearing capacity.
For the first time, this study introduces “pop-out” configurations to the
generalized Yoshimura, allowing it to obtain complex and stable shapes with
significant load-bearing capacity. The idea is to carefully _buckle-out_ the
valley creases (or pushing the valley folds in the mid-section of a module
outwards) so that the rhombus shape inverts its curvature and stabilizes into
a new configuration (Fig. 3a). Due to this transition, the valley fold in the
rhombus “disappears,” and a new vertical fold (or bend) emerges. Remarkably,
popping out/in a rhombus is an isolated event, and it would not induce popping
in the nearby rhombus. Therefore, one can achieve multiple combinations pop-
outs and create a _deployable and reconfigurable_ structure. In this study
focusing on $n=3$, a generalized Yoshimura module has $2^{3}=8$ distinct
configurations that can be categorized into four groups: 0 pop-out, 1 pop-out,
2 pop-out, and 3 pop-out. All of these configurations are elastically stable,
and we refer to this as “meta-stability.” The following sections describe the
kinematic characteristics of 1 and 2 pop-out configurations in detail.

#### 1 Pop-out Kinematics

A 1 pop-out configuration emerges when a single valley crease is pushed
outwards from the mid-section of a module. Figure 3(b) details the geometry of
a 1 pop-out module from three different viewing angles. The isometric view
highlights three essential features to explain its kinematics. A top interface
triangle ABC, a mid-surface OPQR, and a base interface triangle A’B’C’. Here,
_we made a critical assumption that the shape of the top and base interface
triangle remain fixed and equilateral, regardless of the popping
configurations_. In this way, one can seamlessly assemble multiple modules at
different configurations into a more complex structure (which we will discuss
in the later sections.)

The top view in Figure 3(b) clearly shows the kite-shaped mid-surface OPQR.
Here, the valley crease PR is popped out, and we needed two angles to define
this kite shape: $\alpha$ and $\eta$, which follows the relationship:

$\sin\eta=\frac{\sin\alpha}{2}.$ (4)

To further correlate these two angles to the dihedral facet angle ($\theta$)
and mountain crease incline angle $\beta$, we started by calculating the
height of the original triangular facet ($w$ as in Fig. 2), in that:

$w=\frac{L}{2}\tan\beta.$ (5)

Then, using vertex O as the reference point, we can calculate the positions of
vertices A, B, and R relative to O:

$\begin{split}O&\equiv[0,0,0],\quad(\text{Reference})\\\
A&\equiv[0,L\cos\eta+\frac{L}{2}\cos\alpha,w],\\\
B&\equiv[\frac{L}{2},w\cos\theta,w\sin\theta],\\\
R&\equiv[L\sin\eta,L\cos\eta,0].\end{split}$ (6)

Figure 3: The kinematics of 1 and 2 pop-out Yoshimura modules. (a) Paper
models illustrating the sequence of popping out the horizontal valley fold
lines to create new configurations. (b) The detailed view of a 1 pop-out
module, highlighting several important geometric features used in the
kinematic analysis, such as the top and base interface triangles, mid-surface,
tilt angle $\gamma$, and dihedral angle between facets $\theta$. (c, d) The
correlation between $\gamma$, $\theta$, and $\beta$, highlighting the Golden
Ratio Yoshimura design. (e-g) A similar kinematics analysis on a 2 pop-out
module.

Based on the above-mentioned assumption on fixed interface triangles, the
distance between vertices A and B does not change after pop-out; therefore

$|A-B|^{2}=L^{2}=\Big{[}\frac{L}{2}-0\Big{]}^{2}+\Big{[}w\cos\theta-(L\cos\eta+\frac{L}{2}\cos\alpha)\Big{]}^{2}+\Big{[}w\sin\theta-w\Big{]}^{2}.$
(7)

Simplifying this equation with the help of Eq. (5) yields:

$2\tan^{2}\beta(1-\sin\theta)+4\cos^{2}\eta+\cos^{2}\alpha-2\tan\beta\cos\theta(2\cos\eta+\cos\alpha)=3.$
(8)

Similarly, the distance between the vertices R and B does not change so that,

$|R-B|^{2}=\left[\frac{L}{2\cos\beta}\right]^{2}=\Big{[}L\sin\eta-\frac{L}{2}\Big{]}^{2}+\Big{[}L\cos\eta-w\cos\theta\Big{]}^{2}+\Big{[}0-w\sin\theta\Big{]}^{2}.$
(9)

Simplifying again with the help from Eq. (5) yields,

$\tan\beta\cos\theta=\frac{1-\sin\eta}{\cos\eta}.$ (10)

Using Eq. (4-10), we could fully define the geometry of a 1 pop-out Yoshimura
module. These equations are highly non-linear and transcendental, so they were
solved numerically. Figure 3(c) summarizes the solution showing the
correlation between the dihedral facet angle $\theta$ and the Yoshimura design
parameter $\beta$. Surprisingly, we observe no solution when
$\beta<31.72\degree$, which is directly related to the Golden Ratio. This
discovery guided us to define the “Golden Ratio Yoshimura” pattern for which
the dihedral angle ($\theta$) reduces to zero. As a result, the vertices B and
C on the top interface triangle will coincide with their counterpart B’ and C’
on the base triangle (i.e., the top surface and base triangle share a common
edge). For any $\beta<31.72\degree$, the two adjacent facets PCO and PC’O
would interfere and create geometric frustration, resulting in unreliable
mechanics.

The tilt angle ($\gamma$) is defined as the angle between the top and base
interface triangle. It can be calculated with the help of a vector joining the
mid-point of BC and vertex A. The relative coordinates of the mid-point of BC
(let’s say E) as,

$E\equiv[0,w\cos\theta,w\;\sin\theta],$ (11)

Now, the angle between this vector EA and the mid-section plane is half of the
tilt angle, which can be found as,

$\tan\frac{\gamma}{2}=\frac{w-w\sin\theta}{L\cos\eta+\frac{L}{2}\cos\alpha-w\cos\theta}.$
(12)

The above equation explains the relation of the tilt angle for a generalized
Yoshimura that depends on the pattern angle, $\beta$, dihedral angle,
$\theta$, and the angle $\eta$. This tilt angle for our Golden-ratio Yoshimura
is $(\gamma_{\text{gold}})_{1-p/o}\approx 41.8\degree$, and it is
interestingly the maximum for the $n=3$ design.

#### 2 Pop-out Kinematics

A 2 pop-out configuration emerges when two valley creases in the mid-section
of a module are pushed outwards. One can reach this configuration either by
popping out an additional valley fold from a 1 pop-out state or popping in a
rhombus from the 3 pop-out configuration. Figure 3(e) details the geometry of
this configuration with three different views. Again, we assume the top and
base interface triangles remained unchanged.

From the top view, one can see that mid-surface OPQSR—where the valley creases
PR and RO are popped out — has a square shape, which is relatively more
straightforward to calculate than the 1 pop-out case. To correlate the
dihedral facet angle $\theta$ with the facet sector angle $\beta$, we start by
calculating the positions of vertices B and C relative to the mid-point of PO
crease so that:

$\begin{split}C&\equiv[0,-w\cos\theta,w\sin\theta],\\\
B&\equiv[\frac{L}{2},\frac{L}{2},w].\end{split}$ (13)

Here, the distance between the vertices B and C is conserved according to the
fixed interface triangle assumption,

$|B-C|^{2}=L^{2}=\Big{[}\frac{L}{2}-0\Big{]}^{2}+\Big{[}\frac{L}{2}-(-w\cos\theta)\Big{]}^{2}+\Big{[}w-w\sin\theta\Big{]}^{2}.$
(14)

Simplifying this equation yields the following nonlinear relationship,

$\tan^{2}\beta(1-\sin\theta)+\tan\beta\cos\theta=1.$ (15)

Figure 3(f) summarizes the solution of Eq. (15). Again, we observe that the
dihedral angle between facet PCO and PC’O drops to zero when the sector angle
$\beta$ equals the golden ratio angle of 31.72°. Any smaller $\beta$ angle
will give a negative $\theta$, meaning that the two facets intersect and are
not physically possible.

_Therefore, we mathematically proved that the Golden Ratio angle is the lower
bound of the $\beta$ value for kinematically admissible pop-out operations_
($n=3$).

To calculate tilt angle $\gamma$, we consider a vector from the mid-point of
crease AB to vertex C. Its angle with the mid-surface is half of the $\gamma$
angle. The coordinates of this mid-point (let’s say D) relative to the mid-
point of crease OP are,

$D\equiv[0,\frac{L}{2},w]$ (16)

Hence, the tilt angle is defined by,

$\tan\frac{\gamma}{2}=\frac{w-w\sin\theta}{\frac{L}{2}+w\cos\theta}$ (17)

The correlation between $\gamma$ and $\beta$ is presented in Fig. 3(f).
Interestingly, the maximum tilt angle also occurs at the Golden Ratio design,
and it is the same as that for the 1 pop-out configuration, i.e.,
$(\gamma_{\text{gold}})_{2-p/o}\approx 41.8\degree$. Although the orientation
of this tilt angle is different from the 1 pop-out configurations (more on
this in the following section).

##  3 Configuration Space of A Golden Ratio Yoshimura Boom

### 3.1. Forward Kinematics Formulation

The previous section’s kinematic analysis on the Golden Ratio Yoshimura shows
its modules’ state depends on the number of popped-out facets and the relative
orientation of the adjacent modules. Here, the equilateral and fixed
triangular interface shared by two adjacent modules is an essential feature of
this arrangement, allowing for seamless and independent assembly of all
possible states. We used a binary “pop state” representation, where “1”
corresponds to a pop-out (deployed) facet, and “0” represents a pop-in
(folded) facet. For the $n=3$, there are a total of $2^{3}=8$ distinct states,
as shown in Fig. 4(a).

Naturally, the following fundamental question arises: How can we derive a
comprehensive model to analyze the kinematic configuration space of a large-
scale Golden Ratio Yoshimura boom with multiple modules? To this end, we
formulated a modified Denavit-Hartenberg (mDH) convention that is typically
used in robot kinematics [47], essentially treating each module like a robotic
link. First, we defined a coordinate frame $\mathbf{r}_{i}$ attached to module
#$i$’s base interface triangle, where the vertex A’ of the base triangle lies
on the $y-$axis, and the side formed by the remaining vertices is parallel to
the $x-$axis (highlighted in Fig. 3g). All dimensions are re-scaled so that
the side length of the interface triangle equals 1 unit.

To calculate the position and orientation of the next coordinate frame
$\mathbf{r}_{i+1}$ — which defines the top interface triangle of module #$i$
or the base interface triangle of the next module #$i+1$ (Fig. 3g) — one must
apply two consecutive transformations based on Yoshimura’s kinematics model:
One from the base triangle A’B’C’ to the mid-surface
(${}^{\text{base}}\mathbf{T}_{\text{mid}}$), and the other from the mid-
surface to the top interface triangle ABC
(${}^{\text{mid}}\mathbf{T}_{\text{top}}$).

First, assuming the Yoshimura module’s configuration is symmetric about the
$y-z$ reference plane, which applies to states 000, 100, 011, and 111 in Fig.
4(a), one can write this transformation matrices as:

$^{i}\mathbf{T}_{i+1}=^{\text{base}}\mathbf{T}_{\text{mid}}\cdot^{\text{mid}}\mathbf{T}_{\text{top}},$
(18)

where

${}^{\text{base}}\mathbf{T}_{\text{mid}}$
$\displaystyle=\mathbf{Rot}_{x}\left(\gamma/2\right)\mathbf{Tr}_{x}(0)\mathbf{Rot}_{z}(0)\mathbf{Tr}_{z}\left(d/2\right),$
(19) ${}^{\text{mid}}\mathbf{T}_{\text{top}}$
$\displaystyle=\left[\mathbf{Rot}_{x}\left(-\gamma/2\right)\mathbf{Tr}_{x}(0)\mathbf{Rot}_{z}(0)\mathbf{Tr}_{z}\left(-d/2\right)\right]^{-1}.$
(20)

Figure 4: Kinematics and configuration space of the multi-modular Golden Ratio
Yoshimura boom: (a) Illustration of the eight distinct meta-stable states in
the binary ‘pop state’ representation. (b) Plots comparing the achievable
configuration from Yoshimura boom with 1, 2, and 3 modules, showing a fractal
nature. (c) Representation of the state transition scheme using a 3-bit Gray
code. (d) Examples of length scaling (left), curvature scaling (center), and a
complex curve (right) that are achievable with Yoshimura booms.

Here, $\mathbf{Rot}_{k}$ and $\mathbf{Tr}_{k}$ are the rotation and
translation matrices along the $k-$coordinate axes ($k=x,y$ or $z$). $\gamma$
is the tilt angle between the top and base interface triangles, as defined in
Fig. 3, and $d$ is the distance between the center points of these two
triangles (also referred to as the “slant height”). For example,

$\mathbf{Tr}_{x}(s_{x})=\begin{bmatrix}1&0&0&s_{x}\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\\\ \end{bmatrix},$ (21)

$\mathbf{Tr}_{z}(s_{z})=\begin{bmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&s_{z}\\\
0&0&0&1\\\ \end{bmatrix},$ (22)

$\mathbf{Rot}_{x}(\phi_{x})=\begin{bmatrix}1&0&0&0\\\
0&\cos{\phi_{x}}&-\sin{\phi_{x}}&0\\\ 0&\sin{\phi_{x}}&\cos{\phi_{x}}&0\\\
0&0&0&1\\\ \end{bmatrix},$ (23)

$\mathbf{Rot}_{z}(\phi_{z})=\begin{bmatrix}\cos{\phi_{z}}&-\sin{\phi_{z}}&0&0\\\
\sin{\phi_{z}}&\cos{\phi_{z}}&0&0\\\ 0&0&1&0\\\ 0&0&0&1\\\ \end{bmatrix}.$
(24)

Here, $s_{x}$, $s_{z}$, $\phi_{x}$, and $\phi_{z}$ are generic length and
angle variables. If the Yoshimura module’s configuration is not symmetric
about the $y-z$ reference plane (which applies to meta-stable states 001, 010,
101, and 110 in Fig. 4a), one can first rotate the module about its $z-$axis
at the base by a “phase angle” $\psi$ until becoming symmetric, apply the
transformation defined in Eq. (18), and then rotate back. Therefore, a more
generic transformation matrix becomes:

${}^{i+1}\mathbf{T}_{i}=\mathbf{Rot}_{z}(\psi)\mathbf{Rot}_{x}\left(\gamma/2\right)\mathbf{Tr}_{z}(d)\mathbf{Rot}_{x}\left(\gamma/2\right)\mathbf{Rot}_{z}(-\psi).$
(25)

More specifically,

${}^{i+1}\mathbf{T}_{i}=\begin{bmatrix}\cos^{2}{\psi}+\cos{\gamma}\sin^{2}\psi&\frac{1}{2}(1-\cos{\gamma})\sin{2\psi}&\sin{\gamma}\sin{\psi}&d\sin{\frac{\gamma}{2}}\sin{\psi}\\\
\frac{1}{2}(1-\cos{\gamma})\sin{2\psi}&\cos{\gamma}\cos^{2}{\psi}+\sin^{2}{\psi}&-\sin{\gamma}\cos{\psi}&-d\sin{\frac{\gamma}{2}}\cos{\psi}\\\
-\sin{\gamma}\sin{\psi}&\sin{\gamma}\cos{\psi}&\cos{\gamma}&d\cos{\frac{\gamma}{2}}\\\
0&0&0&1\end{bmatrix}.$ (26)

The corresponding transformation parameters for each module are listed in
Table 1. Finally, by treating each module as an independent building block, we
could determine the kinematic state of the entire structure, by applying a
chain of transformations from the $i^{th}$ and $j^{th}$ modules ($i<j$):

$\mathbf{x}_{j}=^{j}\mathbf{T}_{j-1}{}^{j-1}\mathbf{T}_{j-2}\dots^{i+1}\mathbf{T}_{i}\cdot\mathbf{x}_{i},$
(27)

whereas $\mathbf{x}_{j}$ and $\mathbf{x}_{i}$ are the vertices coordinates of
$j^{th}$ and $i^{th}$ module.

Table 1: Forward kinematics model parameters for each meta-stable state. Note that $\varphi$ represents the golden ratio, i.e. $\varphi=\frac{1+\sqrt{5}}{2}\approx 1.618$. All angles are given in radians. Pop in/out state |  $\psi$ (phase angle) |  $\gamma$ (tilt angle) |  $d$ (slant height)
---|---|---|---
$000$ | $0$ | $0$ | $\frac{1}{\varphi}\sqrt{1-\frac{\varphi^{2}}{3}}$
$001$ | $-\frac{2\pi}{3}$ | $2\sin^{-1}\left(\frac{1}{\sqrt{3}\varphi}\right)$ | $\frac{1}{3\varphi}$
$010$ | $\frac{2\pi}{3}$ | $2\sin^{-1}\left(\frac{1}{\sqrt{3}\varphi}\right)$ | $\frac{1}{3\varphi}$
$011$ | $0$ | $-2\sin^{-1}\left(\frac{1}{\sqrt{3}\varphi}\right)$ | $\frac{2}{3\varphi}$
$100$ | $0$ | $2\sin^{-1}\left(\frac{1}{\sqrt{3}\varphi}\right)$ | $\frac{1}{3\varphi}$
$101$ | $\frac{2\pi}{3}$ | $-2\sin^{-1}\left(\frac{1}{\sqrt{3}\varphi}\right)$ | $\frac{2}{3\varphi}$
$110$ | $-\frac{2\pi}{3}$ | $-2\sin^{-1}\left(\frac{1}{\sqrt{3}\varphi}\right)$ | $\frac{2}{3\varphi}$
$111$ | $0$ | $0$ | $\frac{1}{\varphi}$

### 3.2. Exploring and Accessing the Configuration Space of Yoshimura

Since the modules are independent of each other as they switch between meta-
stable states, the achievable configuration space of a deployable Yoshimura
can grow exponentially as its number of modules increases. Using the
transformation Eq. (27), we can calculate the reachable workspace of a single
Yoshimura module by calculating the positions of its top interface triangle’s
center at the eight meta-stable states (the first plot of Fig. 3b). With each
additional module, the workspace ’replicates’ and ’grows,’ similar to a
fractal tree canopy. This self-similarity allows for diverse shapes and
configurations, as illustrated in the second and third plots in Fig. 3(b). As
a result, even with a few modules, a substantial portion of space becomes
accessible, emphasizing the Golden Ratio Yoshimura’s massive deployability and
reconfigurability.

Adding more modules to the Yoshimura boom presents a challenge in actuation;
attaching one actuator to every facet might create an overly complicated
mechatronic setup. To solve this problem, we suggest using a single actuator
that can travel freely through a hollow cavity inside the Yoshimura and along
its spline. This way, the actuator can selectively pop out or in individual
facets, one at a time. While developing such an actuator is out of the scope
of this paper, we formulated a 3-bit Gray code to establish an efficient state
transition scheme for a single module, as demonstrated in Fig. 4(c). This code
ensures that adjacent states differ by only one switch, providing a single
face-popping action per step. As a result, following this Gray code minimizes
the actuation requirements. One could repeat the same scheme iteratively when
dealing with structures containing multiple modules (e.g., using a 6-bit Gray
code for two modules and a 3$n$-bit Gray code length for $n$ modules).

The Golden Ratio Yoshimura boom offers a versatile way to construct complex
shapes by combining different modules. We can adjust the length scaling and
curvature to achieve intricate shapes overall. For example, we could reach the
maximum increment in length ($l^{\prime}$) by stacking either the 111 module
or a pair of 1 and 2 pop-out modules with the same phase $\psi$ (e.g., 001 and
110 according to Table 1). The corresponding increment in length is
$1/\varphi$. On the other hand, the minimum increment in length ($l$) is
achieved with 000 modules, and the value is given by
$1/\varphi\sqrt{1-\varphi^{2}/3}$.

Curvature also varies across modules. Yoshimura’s curvature can be calculated
as the ratio of tilt angle $\gamma$ to slant height $d$. Based on the
kinematics model and its results summarized in Table 1), Modules 000 and 111
exhibit zero curvature. 2 pop-out modules have the intermediate curvature, and
1 pop-out module has maximum curvature. Stacking different models with
curvatures in the same plane could achieve varying $\kappa$ in that,

$\kappa=\frac{\text{total angle swept}}{\text{average arc-
length}}=\frac{\sum_{i}{\gamma_{i}}}{\sum_{i}{d_{i}}}.$ (28)

Figure 4(d) demonstrates the possibilities of these combinations.

##  4 Experimental Assessments

It’s worth noting that besides the rich configuration space via meta-
stability, another significant advantage of the Golden Ratio Yoshimura boom is
that each meta-stable state should be load-carrying (except for the fully
folded 0 pop-out one). This is because the popped-out facets take up the shape
of a cylindrical shell. Evidently, a more robust fabrication method is
necessary to assess such load-carrying performance. Although the paper-folded
samples used in the previous sections exhibit kinematic properties similar to
those predicted by the theoretical model, they are too fragile to carry any
meaningful payloads. To this end, we developed a customized fabrication
technique using dual-material 3D printing.

### 4.1. Sample Fabrication via 3D Printing

Figure 5: 3D printing and fabrication process of a Golden Ratio Yoshimura
sample. (a) Diagram of a flat Yoshimura sample being printed using the layer-
by-layer approach. (b) Detailed design parameter of the 3D-printed sample. (c)
Fabrication steps after printing.

In this study, we employed the widely-used Fused Deposition Modeling (FDM)
3D-printing technique to fabricate the Golden Ratio Yoshimura samples (dual-
material Ultimaker S5 printer). As outlined in Fig. 5, the Yoshimura structure
is printed flat first and then folded, reflecting the origami paper folding
philosophy. However, unlike paper, the 3D-printed model utilizes two
materials: stiff nylon and soft thermoplastic polyurethane (TPU) 95A. Here,
nylon is used for the triangular facets, significantly enhancing Yoshimura’s
meta-stability and overall stiffness. TPU 95A primarily functions as the
crease lines, ensuring sufficient flexibility for folding and reconfiguration.
It is important to note that creating a strong bond between these two
materials during the printing process poses significant challenges, and our
repeated trials and errors showed that nylon has the best compatibility with
TPU 95A. To further strengthen the bonding, we took inspiration from prior
works [48] and developed a layer-by-layer printing method. First, we reduced
the 3D printing’s layer thickness to 0.1mm. Then, we printed the flat
Yoshimura pattern in three parts: a TPU base, nylon facets, and a TPU cover
(Fig. 5a,b). The TPU base — with a thickness ($t_{3}$) 0.4mm and a zig-zag
printing path — is a durable foundation for the entire structure. Thin TPU
walls ($d_{3}$) built on top of this base create space for the nylon facets
and provide lateral connections between nylon and TPU. The nylon facets are
0.6mm thick ($t_{1}$). Finally, a 0.2mm thin TPU cover ($t_{2}$) securely
seals the nylon facets within the structure, ensuring proper bonding even
under significant load and deformation.

Overall, the 3D-printed sample used thicker and stiffer nylon to define the
facets and relied on the thinner and softer TPU 95 to define the fold lines.
Its design aims to closely replicate the kinematics model (e.g., the printed
facet sector angle matched the Golden Ratio $\beta=31.7\degree$). Still, it
differs from the paper-folded sample, which has a negligible thickness and
sharp folds. So, we made a few design accommodations. For example, we designed
a gap, labeled as ($d_{1}$ and $d_{2}$), between the thick facets to
facilitate proper folding. We also strategically placed holes of radius
$r_{1}$ or $r_{2}$ at the vertices to reduce stress concentration. Table 2
summarizes the design parameters selected for 3D printing.

Once printing was complete, the Yoshimura sample was wrapped around a plastic
tube and covered with a plastic film for annealing. Note that the tube’s
radius was large enough so that the Yoshimura sample would not overlap itself,
and the plastic film cover was tight. Then, the sample was annealed in a
temperature-controlled oven at 120°C for 5 minutes (EasyComposites OV301).
This step created a curvature in the Yoshimura to facilitate the subsequent
heat welding (Fig. 5c). We used a manual soldering gun set at 265°C to weld
the curved Yoshimura sample into a closed cylinder. It is critical to point
out that the triangular nylon facets on one side of the Yoshimura sample
protrude out of the base TPU layer, and this protrusion directly matches the
exposed TPU base layers on the other side (highlighted in Fig. 5b,c). This
design maximized the surface area for heat welding while ensuring the finished
specimen’s overall thickness and mechanical properties are consistent between
welded and unwelded portions.

Once the welding was complete, we folded the Yoshimura sample along the crease
lines into the fully folded 0 pop-out state. We custom-built a frame to
compress and hold the sample and then put everything into the oven for another
annealing at 140°C for 15 minutes. This was a crucial step in reducing tension
along the crease lines to facilitate folding. After cooling to room
temperature, the sample remained fixed to the frame for an additional 30
minutes to eliminate any residual stress. Following these procedures, the
sample was removed from the frame, and as shown in Fig. 5(c), the Yoshimura
sample exhibited meta-stability and retained its configuration once the
desired facets were popped out.

Table 2: Design Parameters for 3D printed Yoshimura sample Parameter |  Value
---|---
Triangular facet angle $\beta$ |  31.72∘
Gap width for folding $d_{1}$ |  1.0mm
Gap width for folding $d_{2}$ |  1.5mm
Vertical TPU wall thickness $d_{3}$ |  0.5mm
Stress relieving hole radius $r_{1}$ |  1.5mm
Stress relieving hole radius $r_{2}$ |  1.0mm
Nylon facet thickness $t_{1}$ |  0.6mm
Top TPU layer thickness $t_{2}$ |  0.2mm
Base TPU layer thickness $t_{3}$ |  0.4mm

### 4.2 Load Bearing Capacity Tests

We conducted longitudinal compression and 3-point bending tests on the
3D-printed Yoshimura samples to assess their mechanical load-bearing capacity.

Figure 6: Experiment Testing on 3D-printed Golden Ratio Yoshimura’s load-
bearing capacity. (a) Longitudinal compression test results, highlighting the
differences between a dual-material sample and a pure TPU sample. (b) Details
of the experimental setups for the 3-point bending tests. (c) 3-point bending
test results of the 3D printed Yoshimura sample, which was bent either via the
valley crease or via the vertex. Here, the solid lines are averaged results,
and the accompanying shaded regions are standard deviations of 7 consecutive
loading cycles.

#### Longitudinal Compression Test

Two different Yoshimura samples were fabricated for the compression tests: The
first sample had a dual-material setup with nylon facets, while the second was
entirely printed from TPU 95. The two samples shared identical geometry with
three modules. Comparing the load-bearing capabilities of these two samples
could help us investigate how the material composition influences the overall
structural performance.

Figure 6(a) summarizes the longitudinal compression test results (Instron 6500
with 100N load cell). Overall, the 3D-printed Yoshimura samples exhibited
remarkable durability, giving repeatable mechanical behaviors even after
extensive usage. The dual-material Yoshimura sample withstood over 50 Newtons
of compression force before beginning to buckle and transitioning from the
deployed 3 pop-out state to the folded 0 pop-out state eventually (the sample
itself weighs about 23 grams). Such buckling and folding occurred in sequence,
so each small peak on the force-deformation curve indicates an inward buckling
(or pop-in) of one triangular facet pair. The Yoshimura sample fully folded
after being compressed by 62mm, and once it transitioned into the folded
state, the Yoshimura sample behaved like a soft linear spring. In contrast,
the pure TPU Yoshimura sample could carry only about 25 Newtons of compression
force. These compression tests demonstrated that the nylon facets can
significantly enhance the overall load-bearing, so we only used the dual
material sample for the subsequent 3-point bending tests.

#### 3-Point Bending Test

For the 3-point bending test, we designed a Yoshimura sample with five
modules. The central module was the focus of the test: It was switched between
0 pop-out, 2 pop-out, and 3 pop-out states, while the four modules at both
ends remained in the fully pop-out configuration. This setup ensured
consistent boundary conditions when the input force was applied to the central
module. Moreover, we bent the Yoshimura sample in two opposite directions: one
involving loading on a valley crease and the other on a vertex. A 3D-printed
ring fixture distributed the transverse load more evenly over Yoshimura’s
surface (Fig. 6b).

Figure 6(c) summarizes the tested external moment-bending angle relationships,
elucidating the bending stiffness of the Yoshimura structure under various
configurations. Overall, the Yoshimura could carry significantly more bending
load in their 2 pop-out and 3 pop-out configurations than the 0 pop-out state,
validating its advantage as a load-carrying reconfigurable structure. The
bending response is generally linear, except for the 2 pop-out results. When
loaded on the valley crease, the 2 pop-out Yoshimura exhibits a close-to-
linear bending response. However, when loaded on the vertex, the 2 pop-out
Yoshimura showed a significantly nonlinear behavior with two distant “snap-
through” events. These snap-through events occurred when the popped-out
facets, under compression in this loading condition, eventually buckled and
folded inwards. This behavior is similar to what was observed in the
longitudinal compression test.

##  5 Conlusion and Discusion

By intentionally breaking the traditional design rules of Yoshimura origami,
this study presents a new “Golden Ratio Yoshimura” as the backbone for new
deployable and massively re-configurable structures. The conventional
Yoshimura design requires the sector angle of its triangular facets ($\beta$)
to be directly related to the number of rhombus patterns ($n$) along its
circumference. However, we generalized the design in this study by setting
$\beta$ as an independent variable. For the first time, we proved
mathematically that when $n=3$, the more general Yoshimura pattern would
obtain meta-stability when the facet sector angle is larger than the Golden
Ratio: $\beta\geq 31.7\degree=\cot^{-1}1.618$. With such meta-stability, one
can selectively pop out pairs of triangular facets of the Yoshimura and
transition it between many elastically stable states. On the other hand, if
the $\beta<31.7\degree$, these meta-stable states become kinematically
inadmissible.

Each pop-out/in operation is independent, meaning that an elementary module of
the $n=3$ Golden Ratio Yoshimura has $2^{3}=8$ meta-stable states. As a
result, a deployable Yoshimura Boom structure with $m$ modules can
theoretically achieve $8^{m}$ geometrically unique and load-bearing shapes.
Furthermore, we demonstrated the practical use of Golden Ratio Yoshimura by
fabricating meter-scale prototypes using multi-material FDM 3D printing. These
prototypes, with their soft TPU 95A base and stiff Nylon facets, validated the
potential of the Golden Ratio Yoshimura boom to be re-configured into a wide
variety of shapes, each suited for different tasks.

Philosophically speaking, this study’s results highlight that even a seemingly
minor tweak to the decade-old origami design rules could unveil significant
new potential, potentially fostering a new generation of deployable and re-
configurable structures.

V. Deshpande., Y. Phalak. Z. Zhou. and S. Li acknowledge the gracious support
of the National Science Foundation (CMMI-2240211 and 2312422) and Virginia
Tech (via startup funds and graduate assistantship). I. Walker acknowledges
the gracious support of the National Science Foundation (CMMI-2312423 and
HCC-2221126).

The authors declare no conflict of interests.

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