Supplementary material
Paul Plucinsky
a
, Benjamin A. Kowalski
b
, Timothy J. White
b
, and Kaushik Bhattacharya
c
a
Aerospace and Engineering Mechanics, University of Minnesota
b
Materials and Manufacturing Directorate, Air Force Research Lab
c
Engineering and Applied Sciences, California Institute of Technology
March 6, 2018
Supplementary videos
The video nobias.mov shows the degeneracy of a pyramid designed with no bias (i.e., no twists near
the ridges). The pyramid can easily be manipulated to be stable in either the up or the down states.
The video bias.mov shows the robustness of the pyramid designed with a bias that is shown in
Figure 5. The video shows that the pyramid promptly returns to its programmed direction even
after being manipulated by hand into the “wrong” configuration.
1
Electronic
Supplementary
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(ESI)
for
Soft
Matter.
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journal
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©
The
Royal
Society
of
Chemistry
2018
Supplementary material on nonisometric origami
Formulation of compatibility for general building blocks
Generic nonisometric origami designs are made up of building blocks where straight interfaces, which
separate regions of distinct constant director, merge to a point as sketched in Figure 1. (These
interfaces, if designed appropriately, become the ridges of the origami structure upon actuation.)
Each of these building blocks must independently satisfy the metric constraint for the whole origami
structure to be compatible upon actuation.
To formulate this notion of compatibility of building blocks, we consider an initially flat
k
-faced
building block
ω
̃
p
=
∪
α
=1
,...,k
S
α
⊂
R
2
in which
k
sectors
S
α
of piecewise-constant director merge to
a point
̃
p
∈
R
2
, which is in the interior of
ω
̃
p
. We let
̃
t
α
∈
S
1
denote the outward tangent vector at
the point
̃
p
describing the interface between the sectors
S
α
and
S
α
+1
of director
n
0
α
and
n
0(
α
+1)
,
respectively (where we define
n
0(
k
+1)
:=
n
01
and likewise for
S
k
+1
). The notation is also provided
in Figure 1.
Given all this, there exists a design and deformation at the building block
ω
̃
p
which satisfies the
metric constraint if and only if the collection of interfaces
{
̃
t
1
,...,
̃
t
k
}
and directors
{
n
01
,...,
n
0
k
}
satisfies
R
α
(
`
1
/
2
n
0
α
)
3
×
2
̃
t
α
=
R
α
+1
(
`
1
/
2
n
0(
α
+1)
)
3
×
2
̃
t
α
,
α
∈{
1
,...,k
}
(1)
for some
{
R
1
,...,
R
k
} ∈
SO
(3)
, where
(
`
1
/
2
n
0
α
)
3
×
2
:=
r
−
1
/
6
(
I
3
×
2
+ (
r
1
/
2
−
1)
n
0
α
⊗
̃
n
0
α
)
for each
α
∈ {
1
,...,k
}
. (Again, we define
R
k
+1
:=
R
1
.) We have addressed the intimate connection
between this compatibility and the metric constraint (1) elsewhere [1, 2], as well as the justification
of these configurations as designable actuation. Thus here, we focus on examples which highlight
the richness of the design landscape for building blocks.
In this direction, we note that if the equations (1) are solved for some collection
{
R
α
}∈
SO
(3)
,
then any deformation
y
:
ω
̃
p
→
R
3
of the form
y
(
̃
x
) =
QR
α
(
`
1
/
2
n
0
α
)
3
×
2
(
̃
x
−
̃
p
) +
y
(
̃
p
)
,
̃
x
∈
S
α
,
α
∈{
1
,...,k
}
,
Q
∈
SO
(3)
and
y
(
̃
p
)
∈
R
3
(2)
realizes the metric on
ω
̃
p
and is, consequently, a candidate for the deformation of the building
block upon actuation. Furthermore, we can use the freedom afforded us in the rigid rotation
Q
and
translation
y
(
̃
p
)
of the building block to attempt to link multiple such building blocks together to
actuate complex shapes. This framework permits a systematic investigation: (i) characterize the
{
t
α
}
,
{
n
0
α
}
and
{
R
α
}
which satisfy (1), (ii) use this characterization to build nonisometric origami
building blocks described by deformations of the form in (2), and (iii) use these building blocks as
building blocks to form more complex shapes.
Some efforts on characterization
For the characterization herein, we focus on the case that each of the directors
n
0
α
∈
S
2
is planar
(i.e.,
n
0
α
·
e
3
= 0
for each
α
). This encapsulates all such nonisometric origami building blocks which
can be realized using current experimental techniques (e.g., the voxelation technique of Ware et al.
[4]). We also assume that
n
0
α
and
n
0(
α
+1)
are linearly independent (otherwise, the interface
̃
t
α
would be superfluous). Finally, we assume that the sheet is under actuation (i.e.,
r
6
= 1
).
2
e
1
e
2
̃
t
↵
̃
n
0
↵
̃
n
0(
↵
+1)
̃
p
!
!
̃
p
S
↵
S
↵
+1
Figure 1: At each vertex, the actuation induced shape change of the sectors surrounding the vertex
must be comapatible for the pattern to be nonisometric origami.
Under these assumptions, we can satisfy (1) only if the set of tangent vectors
{
̃
t
α
}∈
S
1
satisfies
̃
t
α
∈
span
{
̃
n
0
α
+
̃
n
0(
α
+1)
,
̃
n
0
α
−
̃
n
0(
α
+1)
}
,
α
∈{
1
,...,k
}
.
(3)
In words, compatibility requires that the tangent vector at each interface bisect the two correspond-
ing planar directors (up to a reflection of one or both of the directors). This follows from taking the
squared norm of both sides of (1), and manipulating this quantity using the stated assumptions.
This condition on the interface tangents is necessary for a compatible nonisometric origami
building block, but it is not sufficient. For sufficiency, we need (additionally) to find a set of rotations
{
R
α
} ∈
SO
(3)
which satisfies (1). In this direction, we note that all solutions to three-faced
building blocks of nonisometric origami have been worked out explicitly in the thesis of Plucinsky
[3]. For instance, if we are given any three distinct planar director
{
̃
n
01
,
̃
n
02
,
̃
n
03
}∈
S
1
, then there
are generically 16 nonisometric origami building block designs associated to these directors that
are compatible for heating. These design are bi-stable, (i.e., the pattern can actuate either up or
design). The formulas associated to the designs and actuations are cumbersome, but explicitly stated
in Appendix A of the thesis and easily amenable to calculation (in mathematica, for instance). We
showcase the rich design landscape for three faced building blocks by considering one such example
below.
Now in trying to generalize these results to (
k >
3)
-faced building blocks, there are important
distinctions to be made. Most notably, such building blocks, when compatible, are generically
and non-trivially degenerate. Specifically, if we can find a (
k >
3
)-faced building block which has
a solution to (1) upon actuation, then there are, in fact, continuous families of solutions. The
basic idea is that additional interfaces, beyond three, add degrees of freedom to the systems (i.e.,
we pick up an extra folding angle rotation for every additional interface). These extra degrees of
freedom allow for continuous families of solutions. Indeed, recall that for the symmetric four-faced
3
building block design, we showed that the interfaces can be continuously deformed from the (initially
symmetric) pyramid to generate an entire family of “metric invariant” actuations arising from this
design (Figure 3(c) in the main text). We emphasize again that this is a generic fact of (
k >
3
)-faced
building blocks (though we will refrain from supplying the formal argument here).
From a design point of view, these result have important implications: due to the degeneracies
of the nonisometric building blocks, we need a strategy that breaks the up-down symmetry and
distinguishes between various metric-invariant actuations associated to building blocks with more
that three interfaces. We believe that in tailoring the twist angles
τ
and the width of the wedges
w
, our strategy to bias the interfaces can enable this simultaneous functionality.
Examples: General three-faced building blocks
To highlight the richness of the design landscape for nonisometric origami building blocks, we
consider a single set of directors,
{
n
01
,
n
02
,
n
03
}
=
{
e
1
,
cos(
5
π
36
)
e
1
+ sin(
5
π
36
)
e
2
,
cos(
5
π
18
)
e
1
−
sin(
5
π
18
)
e
2
}
,
(4)
and provide all the compatible nonisometric origami building blocks and corresponding actuations
associated to this set. There are 16 such compatible designs, and each design and actuation is
provided in Figures 2, 3, 4 and 5 (explicit formulas for the actuation are a direct result of Theorem
A.3.6 in [3]). The actuation parameter in all the examples shown is
r
= 0
.
5
, but the solutions exist
and are continuous for all
r
∈
(0
,
1]
. The bi-stable solutions are reflections of each other. In other
words, the second solution is obtained by replacing the folding angles of the first with folding angles
of the same magnitude but opposite sign. The colors are associated to a particular director: red is
n
01
=
e
1
, green is
n
02
= cos(
5
π
36
)
e
1
+ sin(
5
π
36
)
e
2
, and blue is
n
03
= cos(
5
π
18
)
e
1
−
sin(
5
π
18
)
e
2
.
We emphasize again that all these examples correspond to a
single set
of directors (notice the
arrows in Figure 2(a), 3(a), 4(a) and 5(a)). If we change the set, then we would obtain a new
collection of nonisometric origami building blocks, likely 16 but sometimes less if their is some
symmetry in associated to the set of directors. Thus, there are an infinite number of three-faced
building blocks, and this is the simplest possible case. There is still much to be explored in the
direction of
(
k >
3)
-faced building blocks.
4
(b)
(a)
(c)
Figure 2: (a) Director design for the building block. (b),(c) The bi-stable actuations for
r
= 0
.
5
.
5