=Paper=
{{Paper
|id=Vol-3754/paper07
|storemode=property
|title=An e-origami artwork of a big wing crane
|pdfUrl=https://ceur-ws.org/Vol-3754/paper07.pdf
|volume=Vol-3754
|authors=Tetsuo Ida
|dblpUrl=https://dblp.org/rec/conf/sycss/Ida24
}}
==An e-origami artwork of a big wing crane==
https://ceur-ws.org/Vol-3754/paper07.pdf
An e-origami artwork of a big wing crane
Tetsuo Ida†
University of Tsukuba, 1-1-1 Tennoudai, Tsukuba, 305-8573, Japan
Abstract
We present the construction of a big wing crane as an application of folding with virtual cutting and gluing edges
of origami faces in the e-origami environment. With the new methodology, we can reason about the construction
algorithm for classical origami in finer steps rather than relying on the skill of the human hand. We employ a
new origami model, as described in our earlier work, and we have developed software that implements classical
origami folds.
Keywords
e-origami, classical fold, paper folding rule, cut and glue edges
1. Introduction
In our previous paper [1], we introduced a new technique of cut-and-glue of a shared face edge to
e-origami1 . We observed that an origami artwork is a complex arrangement of bounded two-sided flat
planes, or faces, intricately connected and superposed by repeated folds of a single (virtual) sheet of
paper. This collection of faces culminates in an object with a remarkable shape.
Our research demonstrates that cutting an edge shared by two faces unveils a class of classical
folds. By gluing the faces divided by the cut, we restore the connection of the separated faces. This
cut-and-glue technique opens up vast possibilities, enabling the discovery of new folds that were
previously deemed impossible by Huzita-Justin folds, which, when applied to practical constructions,
have certain limitations that our approach overcomes [2] and [3]. The inside reverse fold, one of the
most straightforward classical folds, is not included in the Huzita-Justin folds. When we apply the
cut-and-glue technique, we can realize the inside reverse fold by combining Huzita-Justin folds. We
demonstrate the practical application of our method by constructing a big wing crane, a well-versed
sophisticated origami structure [4] that demands a deep research investigation.
2. Modeling for e-origami
Origami is a term meaning “folding paper.” It also refers to a sheet of paper used for origami. Folding
an origami along a fold line and unfolding the fold to the previous shape leaves a line segment, called
a crease, on the origami. We can construct various interesting geometric objects when we freely
choose fold lines and allow overlaps of faces without breaking the original sheet. When we impose
mathematically plausible fold line construction rules, we can define an origami geometry that deserves
deep mathematical investigation.
Euclidean (plane) geometry constructs geometrical objects using only a straightedge and a compass.
Similarly, origami geometry, a tool-less approach, defines its rules. Huzita-Justin’s rules are the com-
monly agreed rule set on which origami geometry is based. In Table 1, we list some (4 out of 7) of the
Huzita-Justin rules implemented in the Eos system [5] that we use to construct a big wing crane. These
rules, together with newly implemented classical folds (see next section) and software tools of Eos,
allow us to manipulate origami flexibly.
SCSS 2024: The 10th International Symposium on Symbolic Computation in Software Science, August 28–30, 2024, Tokyo, Japan
$ ida@cs.tsukuba.ac.jp (T. Ida)
https://www.i-eos.org/ida (T. Ida)
0000-0002-5683-216X (T. Ida)
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
1
We use the term e-origami to refer to origami innovated by information technology.
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
34
Table 1
Huzita-Justin rules
Rule Command Operation
(O1) HO[𝑃 𝑄] Fold along line 𝑃 𝑄
(O2) HO[𝑃, 𝐺] Fold to superpose point 𝑃 and point 𝑄
(O3) HO[𝑃 𝑄, 𝑅𝑆] Fold to superpose line 𝑃 𝑄 and line 𝑅𝑆
(O4) HO[𝑃 𝑄, 𝑋] Fold along the line perpendicular to line 𝑃 𝑄 that passes through point 𝑋
3. Modeling for e-origami
We give a mathematical notation for the definition of origami and reason about its properties. In origami,
we have faces and two kinds of neighborhood relations between the faces, i.e., superposition and
adjacency. The superposition is a vertical neighborhood relation, and the adjacency is a horizontal one.
A face is a polygon having an attribute of sides. When we denote polygon 𝑃1 . . . 𝑃𝑛 , where we arrange
points 𝑃1 . . . 𝑃𝑛 counterclockwise, we call the plane front side, and the polygon 𝑃𝑛 . . . 𝑃1 is identified
the same as with polygon 𝑃1 . . . 𝑃𝑛 with the attribute of the back side. For any faces, 𝑓 = 𝑃1 . . . 𝑃𝑛 and
𝑔 = 𝑄1 . . . 𝑄𝑛 , 𝑓 is equal to 𝑔 iff 𝑔 is a cyclic permutation of 𝑓 or a cyclic permutation of a reverse of 𝑓 .
We let Π be a finite set of faces, ∽ be a binary relation on Π, called adjacency relation, and ≻ be a
binary relation on Π, called superposition relation. An abstract origami is a structure (Π, ∽, ≻). We
abbreviate abstract origami to AO. We denote the set of AOs by O. An abstract origami system is an
abstract rewriting system (O, ↬) [6] , where ↬ is a rewrite relation on O, called abstract fold.
For 𝒪, 𝒪′ (∈ O), we write 𝒪 ↬ 𝒪′ when 𝒪 is abstractly folded to 𝒪′ . We begin an origami
construction with an initial AO and perform an abstract fold repeatedly until we obtain the desired AO.
Usually, we start an origami construction with a square sheet of paper. This initial sheet of paper is
abstracted as a structure having a single distinguished face denoted by the numeral 1. Then, the initial
AO 𝒪1 is represented by ({1}, ∅, ∅). Furthermore, when we fold face 𝑛, the face is divided into two
faces 2𝑛 and 2𝑛 + 1. We use this convention in this paper and the realization of the data structure of
the e-origami system Eos [7] and the origami language Orikoto of Eos. Suppose that we are at the
beginning of step 𝑖 of the construction, having AO 𝒪𝑖−1 = (Π𝑖−1 , ∽𝑖−1 , ≻𝑖−1 ). We perform an abstract
fold and obtain a next AO 𝒪𝑖 = (Π𝑖 , ∽𝑖 , ≻𝑖 ). Thus, we have the following ↬-sequence.
𝒪1 ↬ 𝒪2 ↬ · · · ↬ 𝒪𝑛
An abstract origami construction is a finite ↬-sequence of AOs. In concrete terms, the operation ↬ can
be a fold by one of Huzita-Justin’s rules, a mountain fold, a valley fold, etc., each requiring arguments
of different kinds. We abuse 𝒪𝑖 to be a name of the origami constructed at step 𝑖.
We now move on to concrete origami. Origami is a term meaning “folding paper.” It also refers
to a sheet of paper used for origami. Folding an origami along a fold line and unfolding the fold to
the previous shape leaves a line segment, called a crease, on the origami. We can construct various
interesting geometric objects when we freely choose fold lines and allow overlaps of faces without
breaking the original sheet. When we impose mathematically plausible rules for choosing fold lines, we
can define an origami geometry that deserves deep mathematical investigation.
Euclidean (plane) geometry constructs geometrical objects using only a straightedge and a com-
pass. Similarly, origami geometry, a tool-less approach, defines its rules. Huzita-Justin’s rules are the
commonly agreed rule set on which origami geometry is based. In Table 1, we list some (4 out of 7)
Huzita-Justin rules that we use to construct a big wing crane. These rules, newly implemented classical
folds (to be discussed in the next section), and our software tools allow us to manipulate origami flexibly.
35
4. Classical Folds
4.1. Mountain fold and Valley fold
Understanding the fundamental fold operations of the mountain and valley fold is crucial in origami.
These folds are named after the resemblance of the crease to a mountain ridge and a valley lap. The
crease is formed by the unfold operation that follows the mountain (or valley) fold. The result of the
valley fold is shown in Fig.1(b). The origami is now two-layered, but the back layer is not visible since
the upper triangle face completely overlaps the lower triangle face. Next, we unfold the origami shown
in Fig. 1(c). Unfold does not mean "undoing," although we recover the shape of the origami to the one
in Fig.1(a) except for the dotted line segment CA. We call the line segment valley crease. If we have a
rich imagination, the crease looks like a lap in the valley.
(a) before valley fold (b) after valley fold (c) after unfold
Figure 1: Valley fold
Similarly, we have a mountain fold below.
(a) before mountain fold (b) after mountain fold (c) after unfold
Figure 2: Mountain fold
The mountain and valley folds are similar to the Huzita-Justin rule (O1), with the important difference
that rule (O1) operates on all the stacked faces. In contrast, the mountain and valley folds apply to
automatically selected faces.
The terms “valley” and “mountain” are sometimes misleading to beginner origami hobbyists since
those folds are not necessarily immediately followed by an unfold. Hence, valley and mountain creases
may not appear unless the origami is entirely unfolded. More geometrically clear terminology is desired
for origami geometers.
4.2. FO
The above observation led us to define command FO. FO, standing for Fold Origami, is a command
FO[𝜃][faces, ray] for rotating target faces along ray by angle 𝜃. It is a generalization of the mountain
and valley folds. The implementation of FO contains an algorithm to select the target faces to be rotated
based on faces.
Note that FO is defined as a Curried function. Using FO, we could define ValleyFold and MountainFold
as follows:
ValleyFold = FO[-𝜋 ]; MountainFold = FO[𝜋 ];
With 𝜃 other than ±𝜋, FO constructs a 3D origami in general. Usually, we use FO[𝜃] towards the ending
steps of the construction.
36
4.3. Inside reverse and outside reverse folds
The inside reverse and outside reverse folds are often used in paper folding. Both work on a pair of
superposed faces that share an edge. When the cut-and-glue technique is introduced, both folds are
realized by combinations of the valley and mountain folds. However, each uses the valley and mountain
folds on opposite faces.
4.3.1. Inside reverse fold
Below we show a simple example of an inside reverse fold. Let
𝒪1 ↬* 𝒪4 ↬* 𝒪6 ↬ 𝒪7 ↬ 𝒪8
be a construction sequence of the example. Each origami 𝒪𝑖 , 𝑖 ∈ {1, 4, 6, 7, 8} is visualized in Fig. 3.
(a) 𝒪1 (b) 𝒪4 (c) 𝒪6
(d) 𝒪7 (e) 𝒪8 (f) 𝒪8′ with wider gap
Figure 3: Inside reverse fold
Origami 𝒪4 is consructed from 𝒪1 by a sequence of commands; VallyFold, ValleyFold, and Unfold. It
is a double-layered stack of faces. On the top layer are two faces, i.e., face CFE and face AEFD. On the
second one, i.e., the bottom layer, are two faces of the same shapes as the ones above but in opposite
orientations. Each face is identified by a unique face ID, automatically assigned by the system. In this
elementary example, we do not have to be concerned with face IDs since it is unnecessary to specify to
which faces we apply the inside reverse fold. To the example 𝒪4 of Fig. 3(b) we apply the following
command:
InsideReverseFold["CE", "FE"]
The first argument, CE, specifies the edge to be cut. It is specified as the type ray. The second argument,
FE, is the ray along which mountain and valley folds are performed. The above inside reverse fold
command performs the following operations:
1. Check if the inside reverse fold is feasible. Namely, check if ∠CFE ≤ 𝜋/2.
2. Cut the edge CE. As a result, point C is split into C and C1 (not shown). The result is 𝒪5 . Origami
𝒪5 is not shown in Fig. 3.
3. Valley fold along ray FE. The face C1 EG is moved. We impose a rule that faces to the right of a fold
line (interpreted as a ray) are moved by a fold. The result is 𝒪6 ,
4. Mountain fold along ray FE. The face CFE is moved. The result is 𝒪7 .
5. Glue the moved edges CE and C1 E to form a new edge CE. The result is 𝒪8 .
37
Note in passing that performing Steps 2 and 3 above in sequence would only be possible when we
cut and separated the shared edge EC. As we construct origami in a virtual space, we can cut CE and
glue the moved CEs without complication. After following the above steps, the obtained origami is
shown in Fig. 3(e) and (f); the latter illustrating the ins and outs of the origami object. Our specially
designed viewer generates this graphics image, providing a comprehensive view of the origami.
On the other hand, in the case of the outside reverse fold, we make a polygonal cover on the faces.
4.3.2. Outside reverse fold
An outside reverse fold is similar to an inside reverse fold. The difference is that the outside reverse
fold applies mountain and valley folds to different faces in opposite layers. We refer to the construction
sequence
𝒪1 ↬ 𝒪2 ↬ · · · ↬ 𝒪8 ,
where 𝒪𝑖 , 𝑖 = 1, . . . , 8 are visualized in Fig. 4(a)∼(e). We apply the following command:
OutsideReverseFold["CE","FE"] (1)
to 𝒪4 , and obtain 𝒪8 . Note that constraint 𝜋/2 ≤ ∠𝐶𝐸𝐹 ≤ 𝜋 should be satisfied; otherwise the
execution of (1) fails. The origami construction sequence in Fig. 4 is now self-explanatory. Figure 4 (f)
shows the origami structure of 𝒪8 more clearly.
(a) 𝒪1 (b) 𝒪4 (c) 𝒪6
(d) 𝒪7 (e) 𝒪8 (f) 𝒪8′ with wider gap
Figure 4: Outside reverse fold
4.3.3. Other classical folds
In addition to the classical folds we have discussed so far, we analyzed the folding algoriths of the
following classical folds well-known in the origami community and implemented them: SquashFold,
RabbitEarFold, SwivelFold, InsideCrimpPleatFold, and OutsideCrimpPleatFold, using the cut-and-glue
technique. They are given in the Appendix.
5. Construction of a big wing crane
5.1. Overview
A crane origami is an exciting example. It is one of the best-known artworks, requiring the classical
folds we discussed. Making a delicate crane by hand is challenging for beginner origami hobbyists. In
38
the context of e-origami, this example poses another challenge in modeling a new class of folds. We
will present the construction of a big wing crane origami. However, instead of describing in natural
language with guiding icons and annotation, we present an algorithm for constructing a crane origami.
This algorithm provides a step-by-step prescription for folding the crane in a programming language.
We present a big wing crane, a well-known flying crane origami variation [4]. Algorithmically, it is
simpler than the ordinary, well-known ones.
We start with a rhombus-shaped piece of paper. We have included entire program codes in a separate
webpage (www.i-eos.org/orikoto-program-of-a-big-wing-crane) to help the reader better understand
the internal origami structure. These codes are not just for show-they allow us to visualize the intricate
folds and their arrangement, making the construction process more accessible. In this construction,
we specify the origami faces of textured patterns on the completed work, adding an artistic dimension
to the origami. This approach, requiring a special texture mapping algorithm, opens up new creative
possibilities in origami modeling. To complete the artwork shown below, we need 71 steps. In one step,
an origami object will make one structural change. For example, we count one structural change of
the origami object in one application of a valley fold, a mountain fold, and one of the Huzita-Justin
folds. One inside reverse fold requires four substeps, i.e., cut, valley fold, mountain fold, and glue. The
number of steps for the entire construction is surprisingly large at first sight, but we should observe
that the origami objects have several symmetries. Therefore, a similar code sequence is repeated more
than once. The number of crucial operations is limited. While the space is limited in this paper, we
assure the readers that they grasp the essence of the construction by showing carefully selected crucial
steps and output graphics of the origami construction process.
The construction consists of the following four stages.
(1) bird base construction,
(2) leg construction,
(3) bill construction,
(4) wing construction.
We will briefly explore the steps in each stage.
5.2. Bird base construction
LLet 𝜏 be a construction 𝒪1 ↬* 𝒪𝑘 , where 𝑘 > 1 and 𝒪𝑘 has a certain distinguidh feature 𝒜, we call
𝜏 an 𝒜-base construction. We want to construct a bird-like origami 𝒪𝑘 . In this stage, we will construct
a bird base, i.e., 𝒜 is a bird and 𝑘 = 49. We also call 𝒪𝑘 a bird base.
Let 𝜏 be a construction 𝒪1 ↬* 𝒪𝑘 , where 𝑘 > 1 and 𝒪𝑘 has a certain distinguidh feature 𝒜, we call
𝜏 an 𝒜-base construction. We want to construct a bird-like origami 𝒪𝑘 . In this stage, we will construct
a bird base, i.e., 𝒜 is a bird and 𝑘 = 49. We also call 𝒪𝑘 a bird base.
The first 11 steps in 𝜏 produce 𝑂1 , . . . , 𝑂11 shown in Fig. 5. We start with an initial origami of a
rhombus shape to make the crane’s wings bigger than the crane made from the square initial origami.
Applying rule (O2) to 𝒪1 and 𝒪2 , we obtain 𝒪3 . 𝒪3 is quad layered. We apply rule (O3) to fold 𝒪3
along the bisector of ∠BCE, and then unfold 𝒪4 to obtain 𝒪5. Steps 4 and 5 aim to construct a valley
crease F4E that will be used in the inside reverse fold on 𝒪5 . 𝒪5 has three other creases, F3E, F2E, and
F1E, at the intersection of the bisector and the hypotenuses of the right triangular faces on four layers.
The subsequence of the construction 𝒪5 ↬* 𝒪9 ↬ 𝒪10 shows the first application of the inside
reverse fold on 𝒪5 :
InsideReverseFold}["BE", "F4E"].
The application of InsideReverseFold requires four substeps, i.e., cut, valley fold, mountain fold, and
glue, and returns 𝒪9 . The top view of 𝒪5 and 𝒪9 appears the same, but two triangular faces BF3E and
BF4E have been moved below face CEF4 in 𝒪9 by the application. When we move face CEF4 at step 10,
we observe the difference. By turning over 𝒪10 , we have origami 𝒪11 seen from the backside2 .
2
We have a command TurnOver[𝑟𝑎𝑦], which is a variation of FO[𝜋][𝑟𝑎𝑦].
39
(a) 𝒪1 (b) 𝒪2 (c) 𝒪3 (d) 𝒪4
(e) 𝒪5 (f) 𝒪9 (g) 𝒪10 (h) 𝒪11
Figure 5: Subsequence 1 of the bitd base
Similarly, we apply another inside reverse fold to the right half of 𝒪11 , and by following the con-
struction sequence 𝒪11 ↬* 𝒪15 ↬ 𝒪16 ↬ 𝒪17 .
(a) 𝒪15 : by insert reverse (b) 𝒪16 : by (O1) (c) 𝒪17 : by turn over
fold
Figure 6: Subsequence 2 of the bird base
The rest of the subsequence in the bird base is given in Figs. 7 and 8 with annotation in the title of
each sub-figure. Each sequence of figures are the visualization of
40
↬* 𝒪23 ↬ 𝒪29 ↬ 𝒪30 ,
and
↬* 𝒪36 ↬ 𝒪42 ↬ 𝒪43 ↬* 𝒪46 ↬ 𝒪48 ↬ 𝒪49 .
(a) 𝒪23 : by insert reverse(b) 𝒪29 : by insert reverse (c) 𝒪30 : by turn over
fold fold
Figure 7: Subsequence 3 of the bird base
At the end of the bird base construction, we obtain 𝑂48 and 𝒪49 . The latter is the bird base. The bird
base is crucial in origami crane construction, whether for the big wing crane or for a commonly known
classical crane. This slim diamond-shaped origami piece has a crack in the middle of the upper half
shown in 𝒪49 in Fig. 8(f). This piece is quad-layered, and each double layer is vertically symmetric as
well as horizontallly symmetric with respect fo the axis AB.
5.3. Leg construction
The leg construction subsequence is the following:
↬* 𝒪53 ↬* 𝒪57
*
↬* 𝒪61
We apply the inside reverse fold to the right and left parts of the bird base 𝑂49 . Note that the inside
reverse fold can be applied to multiple layers of faces.
5.4. Bill construction
The bill construction subsequence is the following:
↬* 𝒪61 ↬* 𝒪63 ↬* 𝒪67 ↬ 𝒪68
Here, we make crease Z1Z2, where we choose by the designer’s preference arbitraly positions of points
Z1 and Z2 on the edges. The crease determines the twist of the bill. We apply the inside reverse fold on
𝑂62 . Then, we rotate 𝑂66 along R1L1 by 𝜋 and obtain 𝑂68 .
5.5. Wing construction
Finally, we apply the command FO twice to make the origami three-dimensional. We apply FO[-(3/8) 𝜋]
to those faces that constitute the wings. We complete the construction to bring the bill partly to the
right, as this is our preferred posture. Our viewer can manipulate the origami object. The viewer and
41
(a) 𝒪36 : by insert reverse (b) 𝒪42 : by insert reverse (c) 𝒪43 : by turn over
fold fold
(d) 𝒪46 : bring inner face to (e) 𝒪48 : bring inner face to (f) 𝒪49 : by turn over
outside outside
Figure 8: Subsequence 4 of the bird base
Mathematica’s graphics functions [8], for example, remove the point names and change the size of the
image and lighting.
↬* 𝒪69 ↬ 𝒪70 ↬ 𝒪71
5.6. Texture mapping and adjustment of posture
The following are the results of our artwork. Figure 11(a) is the polished version of 𝒪71 . We removed
the point names, enlarged the image, and readjusted the posture of the crane. Figures 11(b) and (c)
are obtained by adding textures to the image of 𝒪71 , using the functionality of texture mapping of
Mathematica.
6. Concluding remarks
We have shown that the cut-and-glue technique simplifies modeling the classical folds and, hence,
the implementation in the e-origami environment. Using a cut operation, we have shown that the
seemingly complex inside (and outside) reverse fold is reduced to a sequence of FO folds. We illustrated
other popular classical folds, such as a rabbit ear fold, are realized similarly.
We constructed a big wing crane origami as a nontrivial application of the newly defined inside
reverse fold. The design is coordinate-free yet allows for the freedom of choice of wing angles and
42
(a) 𝒪53 : creae X1F3 (b) 𝒪57 : by insert reverse (c) 𝒪61 : right leg construc-
fold tion
Figure 9: Leg construction
(a) 𝒪61 (b) 𝒪63 (c) 𝒪67 (d) 𝒪68
two positions of bending points of a bill. The construction proceeds as step-by-step fold operations
described in the Orikoto origami programming language. Thus, creating a big wing crane is purely an
algorithmic process. Furthermore, the added feature of texture mapping of our system, Eos, makes the
final product more original artwork.
The algorithm’s implementation in the earlier paper [1] was extensively tested and extended for more
capabilities. The newer Eos version and the construction program are published at the website [9].
References
[1] T. Ida, H. Takahashi, A new modeling of classical folds in computational origami, in: P. Janičić,
Z. Kovács (Eds.), Proceedings of the 13 th International Conference on Automated Deduction in
Geometry, volume 352 of EPTCS, Elsevier Inc., 2021, pp. 41–53.
[2] H. Huzita (Ed.), Proceedings of the First International Meeting of Origami Science and Technology,
Ferrara, Italy, 1989.
[3] J. Justin, Résolution par le pliage de l’équation du 3e degré et applications géométriques, L’Ouvert
(1986) 9 – 19.
[4] K. Fushimi, M. Fushimi, Geometry of Origami, Nippon Hyoron sha Co. Ltd., 1979. (in Japanese).
[5] T. Ida, An introduction to Computational Origami, Texts and Monographs in Symbolic Computation,
Springer International Publishing Switzerland, 2020.
43
(a) 𝒪69 : by FO (b) 𝒪70 : by turn over (c) 𝒪71 : by FO
Figure 10: Opening wings
(a) final (b) textured crane (c) another textured crane
Figure 11: Monotone and textured cranes
[6] M. Bezem, J. W. Klop, Abstract reduction systems, volume Term Rewriting Systems, Cambridge
University Press, 2003, pp. 7 – 23.
[7] T. Ida, D. Tepeneu, B. Buchberger, J. Robu, Proving and Constraint Solving in Computational
Origami, in: Proceedings of the 7th International Symposium on Artificial Intelligence and Symbolic
Computation (AISC 2004), volume 3249 of Lecture Notes in Artificial Intelligence, 2004, pp. 132–142.
[8] Wolfram Research, Inc., Mathematica, 2023.
[9] Eos project, 2024. URL: https://www.i-eos.org.
A. Classical folds realizable by cut-and-glue technique
• Squash fold
SquashFold = InserReverseFold; (O1)
• Inside crimp pleat fold
• Outside crimp pleat fold
44
↬*
(a) before (b) after
↬*
(a) before (b) after
• Rabbit ear fold
↬*
(a) before (b) after
• Swivel fold
↬*
(a) before (b) after
45