芸術科学会論文誌 Vol.4 No.4 pp.151-158

CG Image Generation of Four-Dimensional Origami
Akira Inoue

Ryouko Itohara
Kuniaki Yajima
Keimei Kaino
Sendai National College of Technology
yajima@cc.sendai-ct.ac.jp kaino@cc.sendai-ct.ac.jp
Abstract

To produce four-dimensional (4-D) origami, we fold a solid material along flat planes in a 4-D space. A 4-D space has afourth axis that is perpendicular to a three-dimensional space. Because a computer graphics (CG) image of a 4-D origami
must be drawn on a three-dimensional screen to visualize it, we will produce its CG image using a 4-D painter’s algorithmwith a stereogram. First, we will show how to fold a solid material in a 4-D space. After defining front and a back sides of thissolid in a 4-D space, we will make mountain and valley folds and thereby produce a “4-D Noshi” CG image. Secondly, wewill show how to fold a regular tetrahedron flat by bisectors of its dihedral angles and make a 4-D bird base from a double
tetrahedron. Finally, we will produce CG images of this 4-D bird with opened wings using a stereogram.

Keywords: Four-dimensional origami, Four-dimensional painter’s algorithm, Four-dimensional space,Stereogram, View space

４次元折り紙の CG 画像生成
井上亮

糸原良子
矢島邦昭
仙台電波工業高等専門学校

海野啓明

概要
４次元折り紙では，４次元空間において立体を折り面で折る．４次元空間とは，この３次元空間とそれに直交する第４軸により張られた空間である．４次元折り紙の CG 画像は３次元スクリーンで描くのが合理的なので，本論では４次元 CG 画像を４次元ペインタアルゴリズムとステレオグラムを用いて生成する．最初に，４次元空間における立体の折りたたみについて説明する．この立体の表裏を定義すると，折り紙の山折りと谷折りが決まる．それら用いて「４次元のし」を折り，ステレオ画像を生成する．次に，正４面体を稜角２等分折りにより折りたたみ，それを利用して二重４面体から４次元鶴を折る．
最後に，この４次元鶴が翼を開いたところの CG ステレオ画像を数種類生成し，本手法の有効性を示す．

キーワード: ４次元空間， ４次元折り紙， ４次元ペインタアルゴリズム， ３次元スクリーン， ステレオグラム

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1. Introduction
Origami has much to offer as an instrument forexperimenting with scientific and educational ideas. Several
studies of computer processing of origami have beenreported. Uchida and Itoh [1] explored a method to derive asequence of origami folding processes. Its data structure isbasically represented as a binary tree. Miyazaki et al. [2]developed a virtual manipulation system for folding origami.
A three-dimensional (3-D) origami is made of
two-dimensional sheets of paper that are folded in 3-D space.
Similarly, a four-dimensional (4-D) origami is made ofpolyhedra, such as tetrahedra and cubes. Those are folded in
a 4-D space in which a fourth axis, the u-axis, isperpendicular to a usual 3-D xyz-space. Figure 1 shows amodel of this 4-D space in which the basic plane represents
the 3-D space. We call this 3-D space the u=0 hyper-plane.
Kawasaki [3] reported about high-dimensional flat origamis.
Miyazaki [4] showed how to fold a 4-D analogue of “Noshi”
from a regular octahedron, which is a dual of a cube. Thosestudies do not address two paper surfaces, for the sake ofsimplicity.
First, for CG image generation of 4-D objects, we
propose 4-D painter’s algorithm. Using that algorithm,
foreground objects are painted on top of background objects
on a view space. We construct the view space using astereogram. Secondly, we show how to fold a solid along aflat plane. By cutting and folding a parallelepiped, we canproduce 4-D origamic architecture [5]. Its CG image isrepresented by two kinds of hidden lines. Thirdly, we define
the front and a back of an origami material in a 4-D space.
Then we can make both mountain and valley folds in 4-D
space and show a CG image of a 4-D origami “Noshi.”
Finally, we show how to fold a 4-D origami bird from adouble tetrahedron. Using a stereogram, we make CG
images of a 4-D bird when it is opening its wings. Texture
mapping on the surfaces of these opening wings gives on
impression of solidity of the 4-D origami bird.

(a) 3-D space(b) 4-D space
Figure 1. 3-D space and 4-D space

2. Basic algorithms for four-dimensional CG
2.1 Four-dimensional painter’s algorithm
We propose a 4-D painter’s algorithm. This algorithm is amethod of projecting the 4-D objects on a view space that isa 4-D analogue of a view plane. Let us put a small

hyper-sphere near the view space and a large hyper-sphere
far away from the view space, as shown in Fig. 2(a), andproject them on the view space. We begin with the large
hyper-sphere given as
x2 + y2 + z2 + (u-3) 2 = 2,
whose cross section is a sphere. With decreasing u from 3
to 3- 2 , we paint these spheres given as
x2 + y2 + z2 = 2-(u-3) 2
successively to the view space.
Next, we project the small hyper-sphere given as
x2 + y2 + z2 + u 2 = 1 .
Decreasing u from 1 to 0, we paint its cross sections,given as
x2 + y2 + z2 = 1-u2,
successively on top of the paint background. The resultantsphere we have painted looks like an egg, as shown in Fig.
2(b). Because we cannot view its interior, we paint thissphere translucent.

(a) Two hyper-spheres andtheir cross sections(b) View space

Figure 2: Parallel projection

2.2 Stereogram for view space
Figure 2(b) shows that we project 4-D objects onto theview space. It is easy for us to imagine its 3-D structure, but
hard to imagine its 4-D structure when 3-D objects are
drawn in a view plane. For 4-D CG image generation, werealize a view space using a holographic stereogram [6]. In
this paper, we use a simpler stereogram method to represent
the view space.
Stereogram is a picture that gives the impression ofsolidity. Various methods for stereograms exist: a parallel
method, a crossing method, use of anaglyphs, andstereoscopy. We can make a digital camera set for anaglyphs
with which a pair of photographs – one for each eye – aretaken. Figure 3 shows an anaglyph image of origamicarchitecture [5]. Using eyeglasses with red and blue filters
for anaglyphs, we can view this picture as if it were a 3-D
image.
Parallel and crossing methods are simpler for CG image
generation. Figure 4 shows a pair of crossing method imagesof a parallelepiped, which is rotated along the xz plane by aright angle. We shall give a brief explanation on thetransformation of objects in 4-D space in the next section.

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The transformation matrix is a rotation in the yz plane.
The 4-D origami comprises polyhedra such as tetrahedra
and cubes. Figure 4 shows the presumption that aparallelepiped is put on the u=0 hyper-plane and folded
about a fold plane. In Fig. 5(b) a point p=(x, y, z, u) for z>0
is rotated about a fold plane f by an angle θ. Because anormal vector n to f is parallel to the z-axis, the rotation is inthe zu plane. Its new position p’=(x’, y’, z’, u’) is
determined by multiplication by a matrix [7] as⎛1⎜⎜0p' = p ⎜0⎜⎜0⎝00100 cosθ0 − sin θ0 ⎞⎟0 ⎟sin θ ⎟⎟cosθ ⎟⎠(2)

We apply this folding in 4-D to produce 4-D origamicarchitecture [5]. We must represent two kinds of hiddenlines of its image to make its CG wire-frame image.

Figure 3: Anaglyphic image of origamic architecture [5](a) 3-D origami

Figure 4: Folding of parallelepiped

2.3 Rotation in a two-dimensional plane
A 3-D origami is made of a sheet of paper that is folded in
3-D space. Presume that a rectangular paper is put on thez=0 plane and folded about a fold line. In Fig. 5(a) a point
p=(x, y, z) for y>0 is rotated about a fold line l by an angle θ.
Its new position p’=(x’, y’, z’) is given as0⎛1⎜p' = p ⎜ 0 cos θ⎜ 0 - sin θ⎝0 ⎞⎟sin θ ⎟cos θ ⎟⎠(b) 4-D origami
Figure 5: Folding of 3-D and 4-D origami

(1)

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2.4 Hidden lines and origamic architecture
We have shown how to project 4-D objects onto the view
space using the 4-D painter’s algorithm. We cannot view thecontents of 3-D objects from outside. In wire-frame
drawings, all parts of objects are always displayed butprojected 4-D objects can easily become anindistinguishable mess of line segments. Let us consider theproblem of removing those lines that are hidden by parts of
the object.
Figure 6(a) shows a cutting and folding pattern of thesimplest 3-D origamic architecture. In its completed figureof Fig. 6(b), hidden lines represented by broken lines exist.
Figure 7(a) shows a pattern of simplest 4-D origamicarchitecture, in which four sides of the red parallelepiped are
cut and its three faces of abcd, opqr, and efgh are creases.
After folding, we obtain the completed figure of Fig. 7(b),which depicts two kinds of hidden lines. Broken lines areapparent because the figure is visible if rotated properly, but
dot-dashed lines are not visible because they are inside of
the 3-D figure [7].
3. Four-dimensional origami(a) Pattern

3.1 Mountain and valley folds
We take a solid in a u=0 hyper-plane and move this solid a
bit upward in the direction of the u axis. In this pair of solids,
we regard the upper solid as a front side and the lower one asa back side; we define its normal vector n as a direction of
the relative movement (0,0,0,1). Let us call such a pair ofsolids a 4-D origami. Then folds come as two types:
mountain folds, which are convex, and valley folds, which
are concave. Let us put a regular octahedron in the u=0
hyper-plane with the front painted white and the backpainted red.

(b) Completed figure
Figure 7: 4-D origamic architecture and hidden lines(a) Pattern(b) Completed figure
Figure 6: 3-D origamic architecture and hidden lines

Figure 8(a) shows the 4-D image where this octahedron ismountain-folded by an angle θ less than a right angle. An
image of the folded octahedron is shown in Fig. 8(b) wherethe point C’ moves to the origin from the point C when theviewpoint is set above in the u axis shown in Fig. 8(a). Whenrotated by an angle θ larger than a right angle, as show in Fig.
8(c), the rotated part of the octahedron turns red, as shown inFig. 8(d).
We fold an origami “Noshi” from a square after
alternately making mountain folds and valley folds.
Similarly, we can make a 4-D origami “Noshi” from a 4-D
origami of a regular octahedron [4]. Figure 9 shows animage of 4-D origami “Noshi” by the crossing method ofstereograms. To view the entire 4-D “Noshi”, we render it154

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translucent and paint its front parts white and its back part
red.
frontviewpointback(a) 0 ≤ θ < 90

front(b) 0 ≤ θ < 90

viewpointback(c) 90 ≤ θ ≤ 180(d) 90 ≤ θ ≤ 180

Figure 8: Folded 4-D origami of a regular octahedron and its
projection

3.2 Folding tetrahedron
We will show how to fold a regular tetrahedron using theincenter theorem of a tetrahedron. Figure 10(a) shows thistetrahedron ABCD, where I is the incenter, M and N are themidpoints of AC and CD, respectively, and O, S and T are
the tangency points at which the inscribed sphere touches
the faces of ABCD. Each time the tetrahedron is folded, it isdivided into two or more small tetrahedra. Those new small
tetrahedra lie at different heights along the u direction andare represented by a tree structure with nodes [1]. In Table 1,heights of the small tetrahedra are given as integral values
related to a scale ε of the thickness of the 4-D origami
material, whose information indicates the stacking order ofthe small tetrahedra along the u direction. The procedure forfolding the tetrahedron ABCD is as follows [8]:(1) Fold the top half of ABDM to the bottom BCDM
(Fig. 10(a)).
(2) Inside reverse-fold BCMI and DCMI whilecrimp-folding CTMI (Fig. 10(b)).
(3) Complete the fold (Fig. 10(c)).
First, the vertex A is superposed to C in the 4-D space.
Then the midpoint M is superposed to N and the tangency
point T to the origin O. We can see that the four flat faces of
the regular tetrahedron lie on the basal plane BCD and thatthe other two tangency points are superposed to O.
We continuously fold the regular tetrahedron ABCD bysimultaneously rotating small tetrahedra from 4 to 11 inTable 1 in the 4-D space. Let us rotate MABD around thefold plane IBD by an angle θ and IBCM by an angle φsimultaneously and move the point M to M’ (N). Using thecondition AM’= CM’=AM=CM, we obtain tan (φ /2) = 2 tan(θ/2) and a diagram of φ as a function of θ, as shown in Fig.
11. That relationship suggests that the regular tetrahedron iscontinuously folded.
Table 1: Data structure of the tetrahedron
Tetrahedron Height FatherSons
Front / Back
1
ABCD
0Root
2,3
Front
2 MBCD
0
14,5,6,7
Front
3 MABD
7ε
18,9,10,11
Back
4
IBCD
02
Front
5ICDO
1ε2
Back
6ICNO
2ε2
Front
7IBCN
3ε2
Back
8 IA’BM
4ε3
Front
9 IA’MO
5ε3Back
10 IA’DO
6ε3
Front
11 IA’BD
7ε3Back
A → A’(C), M → N, S, T → O

Figure 9: 4-D “Noshi” by crossing method155

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180160140φ[deg]

1201008060402000(a)20406080

100120140160

180

θ[deg]

Figure 11: Relation to the fold-angle of the regulartetrahedron

(b)

(c)
Figure 10: Folding of regular tetrahedron

3.3 Four-dimensional bird
Traditional origami is usually folded from a square paper.
Classic examples are the “yakko” (serving man) and the“orizuru” (crane). An orizuru has a neck and a tail withmobile wings. Those are made from four corners of a square.
It is a challenging problem to fold a 4-D crane. For
simplicity, let us fold a 4-D bird from a double tetrahedron,as shown in Fig. 12(a).
A tetrahedron is folded using the incenter theorem so as tocontact three faces (z>0) to the basic plane (z=0) [8]. After
folding both the upper and the lower tetrahedron in the sameway, we obtain a base of the bird shown in Fig. 12(b). From
this base, we can fold a bird with wings. They disappear, asshown in an image of Fig. 12(c), because it is drawn in theu=0 hyper-plane when those wings open along the u-axis.
We can see wings as shown in Fig. 12(d), whereas we canhardly see the body because it has no width when it is drawn
in the z=0 hyper-lane. We can resolve this problem using thestereogram.
Four-dimensional origami can be portrayed in the view
space using a stereogram. Then we project the body of a 4-D
origami bird onto the u=0 hyper-plane, as shown in Fig. 1(b),and present its opening wings using a stereogram. Figure 13
is an image of a wire frame model of the 4-D origami bird;
Fig. 14 is an image of its solid model with translucent wings.
Figure 15 shows anaglyph images of the 4-D origami birdwith opened wings. As we can readily imagine, the wire
frame model is suitable for representing the structure of the4-D origami bird. On the other hand, the anaglyph method
seems not to work good because numerous flat faces exist.
Figure 16 shows that this problem is resolved when wings
are texture-mapped.

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芸術科学会論文誌 Vol.4 No.4 pp.151-158(a) Double tetrahedron(b) Bird base(a) Wire frame model(b) Solid model
Figure 15: Anaglyph image of a 4-D bird(c) Projection on u=0(d) Projection on z=0
Figure 12: Four-dimensional bird(a)(b)
Figure 16: Anaglyph image of 4-D bird using texture
mapping

Figure 13: 4-D bird depicted as a wire frame model

Figure 14: 4-D bird by solid model

4. Conclusions
We have studied 4-D CG algorithms and 4-D origamis
and applied them to produce CG image generation of 4-Dorigami [9]. We have obtained the following results:(1) We proposed the 4-D painter’s algorithm with theview space presented using a stereogram. Subsequently,
we applied this algorithm to produce a CG image of4-D origamic architecture.
(2) We defined a front and a back of 4-D origami
paper for mountain and valley folds and applied thismethod to produce a CG image of 4-D origami“Noshi”.
(3) We partly proved that a regular tetrahedron iscontinuously folded using the incenter theorem of atetrahedron and made a 4-D bird base from a double
tetrahedron. Using the above bird base, we folded a 4-Dorigami bird and made a 4-D origami flapping bird.
(4) We showed that a stereogram with texture

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mapping is helpful to imagine a complex structure of4-D origami.
References[1] T.Uchida and H.Itoh, “Knowledge representation of origami
and its implementation,” IPSJ (in Japanese), 32,pp.1566–1573, 1991.
[2] S.Miyazaki, T.Yasuda, S.Yokoi, and J.Toriwaki, “AnOrigami Playing Simulator in the Visual Space,” The Journalof Visualization and Computer Animation, 7, pp.25–42,
1996.
[3] T.Kawasaki, “On High Dimensional Flat Origamis,”
Proceedings of the First International Meeting of Origami
Science and Technology, pp.131–141, 1989.
[4] K.Miyazaki, “Four-Dimensional Origami,” Proceedings ofthe Second International Meeting of Origami Science andScientific Origami, pp.51–61, 1994.
[5] M.Chateni, ”Origamic Architecture,” Shokokusya, 1983, (inJapanese).
[6] F.Okano,“PresentStatusandExpectationofThree-Dimensional Video Technology,” Japanese Journal ofOptics, 31, pp.140–147, 2002.
[7] K.Miyazaki and N.Odaka, “Science of Graphics,” Asakura,2000, (in Japanese).
[8] K.Kaino, “Folding Tetrahedra and Four-DimensionalOrigami,” Forma, 15, pp.49–56, 2000.
[9] A.Inoue, R.Itohara, K.Yajima and K.Kaino, “CG Image
Generation of Four-Dimensional Origami,” Proceedings ofNICOGRAPH International 2005, pp.173–178, 2005.

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