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Simple Flat Origami Exploration System with Random Folds
Naoya TSURUTA, Jun MITANI, Yoshihiro KANAMORI and Yukio
FUKUI

1. Introduction
Recent origami design techniques have made possible the design of realistic
models. A modern realistic model with complex folds can represent the feature ofan object in detail. The circle/river method [Lan96, Meg94] is a powerful toolfor designing such complex origami models. This algorithm calculates the crease
pattern to fold a tree structure that represents the skeleton of the target object.
Another algorithm proposed by Tachi [Tac09] gives the crease pattern to fold anarbitrary 3D shape of a topological disk. However, there are no practical design
tools suitable for simple origami models, which are made with a small number offolds. Existing design algorithms usually result in a complicated crease pattern. It
seems that simple origami models are designed by a heuristic approach.
We proposed an enumeration based approach for exploring simple origami models [TMKF12]. By limiting the folding operations and the number of folds, 136, 284
diﬀerent folded shapes were obtained with up to four times folds. In order to ﬁnd,from a set of obtained pieces, the pieces that look like the shape of something,
the pieces that are similar to an input polygon can be extracted by comparing thecontour of folded pieces and the input polygon. Although this enumeration based
approach can discover computer-generated origami models, the folding operations
and the number of folds are strictly limited due to combinatorial explosion.
To address this problem, we have proposed an interactive system for exploring
simple origami models by random generation of folded pieces [TMKF13]. The
key idea is to assist “the action of labeling” (“M itate” in Japanese). The action oflabeling in origami is to recognize a folded shape as another object (such as animals,insects, and ﬂowers) because of its color and physical appearance. The systemgenerates origami pieces with several folds automatically, and displays them, so thatthe user can focus on the labeling action. Every time the user requests another set ofpieces, diﬀerent shapes appear, because the generation process includes randomness.
We could discover several origami models. However, the generation algorithm inthis prototype system cannot accommodate particular types of folds that have one
degree of freedom (DOF) and there was no discussion about discovered models.
This paper proposes an improved generation method. The previous system usedfolds both with and without landmark points. A fold with landmark points refers
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NAOYA TSURUTA, JUN MITANI, YOSHIHIRO KANAMORI AND YUKIO FUKUIq lpp
(a)

(b)

Figure 1. Examples of the folds without and with DOF, respectively: (a) a fold places a corner to another corner; (b) a fold placesa corner onto an edge of the paper
to a fold such as the corner-to-corner fold. A fold without landmark points refers toa fold placed on a meaningless location. We add the fold that has one DOF, suchas a corner-to-edge fold. A corner-to-edge fold uses landmarks, but the location
has one DOF. This new generation method gives various folded pieces that do notappear in the previous system. We also provide another generation method thatconsiders symmetry. The resultant folded shapes diﬀer greatly and look similar tohuman designed models. Finally, we discuss the models discovered by our system.
We introduce three characteristic groups that are easily recongnized.
2. Random Generation of Simple Folded Pieces
To generate a folded piece automatically, we randomly determine a fold, calculate the folded shape, and repeat this process a given number of times. A foldis composed of three parameters; the location of the fold, the folding direction(mountain or valley), and the number of layers to be folded. The problem is how
to determine these parameters. Determining them completely at random will notgenerate commonly used folds, such as the corner-to-corner fold. Thus, we limit
these parameters to generate meaningful folded pieces as if made by a human.
2.1. Location of Fold. The location of a fold is described by a combinationof points and lines. Initially, an unfolded origami paper has four corner points andfour edges. We randomly choose a required number of landmarks. For example, wepick two points at random in the case of the fold that places a point onto anotherpoint.
Our previous system [TMKF13] used the 6th Huzita-Justin axiom [Huz91,Jus91] and the random sliding of a fold. All the creases were obtained by the 6thaxiom that places point p1 onto line l1 and point p2 onto line l2 . The obtained fold
was slid in a random direction in the last n-steps. This random sliding simulates
judgementf old (Gurai − ori in Japanese), which is a folding operation that does
not use any references. This folding often appears in later folding steps in theartists’ model. Although we could ﬁnd new origami models (shown in Section 5),
this method cannot deal with a fold that has one DOF, such as the corner-to-edge
fold. Figure 1 shows examples of a fold with no DOF and a fold with one DOF.
Instead of randomly sliding the folds, we add the following three operations
that have one DOF as shown in Figure 2:
S1 : Fold a line x which passes through a given point A.

SIMPLE FLAT ORIGAMI EXPLORATION SYSTEM WITH RANDOM FOLDSxlxx

3AAl(a)(b)(c)

Figure 2. Folds with one DOF: (a) a fold that passes througha point; (b) a fold that is perpendicular to a line; (c) a fold thatplaces a point onto a line
S2 : Fold a line x which is perpendicular to a given line l.
S3 : Fold a line x which places a given point A onto a given line l.
These operations are three of the ﬁve one-fold alignments of points and lines described by Alperin and Lang [AL09]. Although any line in the plane has threedegrees of freedom to be speciﬁed, these use only one DOF. Hence, we randomlydetermine the remaining parameter when we use these operations. The remaining
parameter is an angle in the case of S1 and a location on an edge in the cases of S2and S3 . Note that several Huzita-Justin axioms may have two or more solutions,but we assume that the folds have no DOF because the solutions are discrete.
2.2. Folding Direction and the Number of Layers to be Folded. The
folding techniques used in our system are simple valley folds and mountain folds.
We randomly choose one of these.
After a location of the fold and a folding direction have been determined, wedetermine the number of layers to be folded. First, we choose a number from 1 tom at random. m is the total number of layers at the current state. Then, we try
to fold m layers from the top in the case of valley folds. There are cases where
we cannot fold m layers together when the fold intersects with existing creases,
because faces adjacent to the existing creases must be folded at the same time.
In such cases, we fold them together with the next layer and repeat this until weobtain a valid shape. Whether the fold is valid or not can be checked by testing
the ﬂat foldability for the inner vertices. The resultant folded shape is globally
ﬂat-foldable as long as we fold the paper from the top-most layer. We perform thesame process from the bottom-most layer in the case of a mountain fold.
3. Random Generation of Symmetric Pieces
Here we describe another generation method that considers symmetry. This
generation method is an extension of the simple folded piece generation we describedabove. We add another parameter, that is, the type of the symmetry axis. Thesymmetry axis has two types: book and diagonal (Figure 3). We randomly choose
one of these at the very beginning of the generation process.
We use multiple folds or the squash fold to make symmetric shapes in one
folding step. Both folding operations are calculated based on a single fold. The
determination of which folding operation to use is depends on a positional relationship between a fold and symmetry axis. This relation can be categorized according
to whether they intersect. Figure 4 shows a case where they do not intersect. Theﬁgure shows the right half of the square paper. If the ﬂaps folded by an obtained

4NAOYA TSURUTA, JUN MITANI, YOSHIHIRO KANAMORI AND YUKIO FUKUI(a) Book(b) Diagonal

Figure 3. Two types of symmetry

Figure 4. An entire folding operation that has two simple folds.
The folds do not intersect the symmetry axis.

Figure 5. An entire folding operation becomes squash fold when
the fold intersects with the symmetry axis.

fold intersect the symmetry axis, the ﬂaps are folded again by the axis. Thus theentire folding operation has at most two simple folds.
Figure 5 shows a case of intersection. The ﬂaps folded by an obtained foldintersect the symmetry axis, except for the case of the fold that is perpendicular tothe axis. The folded part is folded back by the axis, and then, folded back again
by the bisector of the angle between the axis and its edge so that the edge lieson the axis. The entire folding operation becomes a squash fold. If the fold isperpendicular to the axis, the entire folding becomes a simple horizontal fold.
The folded shape keeps symmetry as long as we use the above two folding
operations. Note that these are not the only way to maintain the folded shapes’symmetry.

SIMPLE FLAT ORIGAMI EXPLORATION SYSTEM WITH RANDOM FOLDS

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Figure 6. Interface of our system.
4. Our Proposed System
Our system generates dozens of folded pieces and displays them simultaneously.
Figure 6 shows the interface of the system. The user chooses one of the displayedpieces if it looks like a desirable shape such as an animal. The system generates
diﬀerent shapes every time the user requests another set of pieces. The user canperform the following operations:• Change the number of folds• Request another set of pieces• Rotate, turn over, or change the color of paper• Label the selected piece and see its diagram• Register the labeled piece to an online database server• Set the initial state of the generation process
The last user operation changes the initial state of the generation process. The
user can choose the initial state from the diagram. In other words, the ﬁrst n steps
can be ﬁxed by the user.
5. Results
We implemented our system as a Flash application with Action Script. The online origami model database is implemented by MySQL and PHP. Our experiment
revealed that a folded piece to be labeled appears after between 10 and 20 retrials
on average. The computation time increases exponentially with the number of folds
because the number of layers is approximately doubled by a single mountain foldor valley fold. Our system runs in real time on a PC with Intel Core i7 2.6 GHz
CPU with the settings of six folds and twenty pieces. Figure 7 is an example modelthat we found.
We have also tried to ﬁnd pieces that resemble existing origami models designed by origami artists. Figure 8 shows a well-known simple origami model “2fold Santa” designed by Paula Versnick [Ver], an origami piece generated by ournew generation method that includes folds with one DOF, and an origami piece
generated by previous method using random sliding. The ﬁrst fold of Paula’s Santa
is made by placing a corner onto nearly the middle point of the edge. This foldhas one DOF. The folds with one DOF makes the folded shape neatly and easierto fold in comparison with those that have two DOF. Our new generation methodgenerates more attractive origami pieces.

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NAOYA TSURUTA, JUN MITANI, YOSHIHIRO KANAMORI AND YUKIO FUKUIFigure 7. Auto-generated diagram of manatee.

(a) 2 fold santa(b) With one DOF fold(c) Without one DOF fold

Figure 8. Comparison of Paula Versnick’s 2 fold santa [Ver] andgenerated pieces with and without one DOF fold.

Figure 9 shows models stored in the online database. These models were generated by the previous generation method we brieﬂy discussed in Section 2.1. We
found that there were similar models, and have extracted three interesting groups.
The ﬁrst row in Figure 9 shows there are three bird models. The paper is folded
in half into a triangle in the ﬁrst step, and the corner becomes the beak of a bird.
The second group is hooded persons and four models are included, as seen in thesecond row. The inside-out representation is used in these models. Inside-out is atechnique that uses both sides of the origami paper and is often used for penguins,pandas, and texture of zebras. This technique is useful for both simple and complex origami. In simple origami, the inside-out technique enables a wide range ofexpression with a smaller number of folds. The third group contains two boats anda yacht. Boats and yachts are very typical objects in simple origami. We often see
these kinds of models in origami books for children. The inside-out representationis also used in the yacht model. We could not ﬁnd any characteristics for the models in the fourth or later lines. In other words, these models are rare and hard todiscover.
We held a workshop for four small children aged from 5 to 7 years. After
10 minutes of instruction, each child could ﬁnd about 7 models in 40 minutes.
Figure 10 shows four of the discovered models. Our exploring system was enjoyablefor the small children. However, several steps in the auto-generated diagram are
diﬃcult to understand due to the lack of annotation and of distortions that indicate
the relationships between the layers.
Figure 11 shows four examples of symmetric models generated by the method
we mentioned in Section 3. These models were obtained by two times symmetricfolding. Note that a single symmetric folding may contain multiple folds. Thesemodels seem as if they could have been designed by humans. Considering symmetry
is eﬀective for generating well-arranged folded pieces. Adding other rules will be

SIMPLE FLAT ORIGAMI EXPLORATION SYSTEM WITH RANDOM FOLDS(a) 4 fold Bird(d) 3 fold Profile ofMother Teresa(b) 6 fold Penguin(e) 4 fold Sister(h) 5 fold Boat(c) 4 fold Swan(f) 3 fold Sister(i) 4 fold Boat7(g) 3 fold Red hood(j) 5 fold Yacht(k) 4 fold Dog’s face(l) 3 fold Dog(m) 5 fold Old dog(n) 6 fold Rhino(o) 4 fold Pencil(p) 6 fold Wallet(q) 5 fold Tie(r) 4 fold Rhino
warning sign(s) 3 fold Cicada(t) 3 fold Grasshopper(u) 4 fold Rose(v) 6 fold Mask

Figure 9. Example models stored in the online database

an interesting challenge; for example, considering rotation symmetry may generate
a lot of ﬂowers.

8NAOYA TSURUTA, JUN MITANI, YOSHIHIRO KANAMORI AND YUKIO FUKUI(a) Bird(b) Otter(c) Whale(d) Penguin

Figure 10. Example models discovered by small children(a) Rocket(b) Circus tent(c) Japanese imperial palace’s dolls(d) Japanese court
lady wearing a ceremonial kimono

Figure 11. Examples of obtained by the generation of pieces withsymmetry
6. Conclusion
We have proposed an interactive exploration system for simple ﬂat origami.
Our system generates folded pieces with a speciﬁed number of folds, and displaysthem. The user can discover new computer generated origami models by labelingthem, without folding a physical sheet of paper. Folding techniques used in oursystem are limited to the simple valley fold and mountain fold, or the squash fold.
Adding other folding techniques such as reverse folds would increase the variety offolded shapes.
Folds that have one DOF have now been added. These folds enable neat shapes
than the previous system. Note that it is necessary to perform a quantitativeevaluation to determine which method generates more meaningful and recognizablepieces.
The action of labeling in origami may be an interesting research topic. Our
results showed that there are characteristic shapes which are easy to recognize.
Avoiding the folds used to generate those shapes will lead to the discovery of different models. In our system, the labeling depends on a person. If we obtain alarge number of labeled origami pieces, automatic labeling of simple origami may
be possible by artiﬁcial intelligence.
References[AL09]

Roger C. Alperin and Robert J. Lang, One-, two-, and multi-fold origami axioms,Origami 4: Fourth International Meeting of Origami Science, Mathematics, and Education (Robert J. Lang, ed.), A K Peters Ltd, 2009, pp. 371–393.

SIMPLE FLAT ORIGAMI EXPLORATION SYSTEM WITH RANDOM FOLDS

9Humiaki Huzita, Axiomatic development of origami geometry, the First International
Meeting of Origami Science and Technology (H. Huzita, ed.), Italy: Dipartimento diFisica dell’Universita di Padova, 1991, pp. 143–159.
[Jus91]Jacques Justin, Resolution par le pliage de l’equation du troisieme degre et applications geometriques, the First International Meeting of Origami Science and Technology (H. Huzita, ed.), Italy: Dipartimento di Fisica dell’Universita di Padova, 1991,
pp. 251–261.
[Lan96]
Robert J. Lang, A computational algorithm for origami design, the Twelfth Annual
Symposium on Computational Geometry, New York: ACM Press, 1996, pp. 98–105.
[Meg94]
Toshiyuki Meguro, Tobu kuwagatamushi-to ryoikienbunshiho (flying stag beetle andthe circular area molecule method), Oru, Sojusha, Tokyo, 1994, pp. 92–95.
[Tac09]Tomohiro Tachi, 3d origami design based on tucking molecules, Origami 4: FourthInternational Meeting of Origami Science, Mathematics, and Education (Robert J.
Lang, ed.), MA: A K Peters, 2009, pp. 259–272.
[TMKF12] Naoya Tsuruta, Jun Mitani, Yoshihiro Kanamori, and Yukio Fukui, A computer aided
design system for simple origami pieces based on the enumeration of folded shapes(in japanese), Science of Origami (Jun Maekawa, ed.), vol. 2, Koryo Miura, 2012,
pp. 33–44.
[TMKF13], An exploring tool for simple flat origami based on random foldings, International Conference on Simulation Technology (JSST2013), 2013.
[Ver]
Paula Versnick, Orihouse.
[Huz91]