3D Origami Design based on Tucking Molecule
Tomohiro Tachi
The University of Tokyotachi.tomohiro@gmail.com

Abstract
This paper presents a new approach for folding a piece of paper into an arbitrary
three-dimensional polyhedral surface based on ”tucking molecules.” The approachis to align the polygons of the three-dimensional surface onto a convex region of aplane and fill the blank area between the polygons with ”tucking molecules.” Tucking
molecules are aligned between segments and vertices of the polygons, and ”paste”
them together by being folded flat. We propose a novel method for generatingcrease pattern for tucking molecules. Necessary and sufficient conditions to enable
construction of the required surface are also shown. Additionally we show that thenecessary conditions can be represented in inequality conditions by using the ”tuckproxies.”

1

Introduction and Previous Methods

Designing arbitrary three-dimensional surface by origami is one of the ultimate objectivesof origami designing. There have been many approaches for designing three-dimensionalmodels.
Most popular way of designing a three-dimensional origami model is by shaping aflat-folded uniaxial base with controlled lengths of flaps. Methods for designing uniaxial
base is mainly investigated by Toshiyuki Meguro [1], Fumiaki Kawahata [2], and RobertLang [3], [4]. This method is good for organic models such as insects and some animals.
However it is not possible to specifically control the three-dimensional shape because theonly controllable parameters when designing a base are lengths of the flaps.
Many origami artists are also engaged in two-dimensional design of origami bases.
A design method for 2D origami base based on placing facets are recently investigatedby Toshiyuki Meguro [5]. For designing the silhouette of an arbitrary two-dimensional
polygon, a proof and a method is shown by Demaine et.al. [6], and another method is
proposed by Masahiko Tanaka [7].
Demaine et.al. [6] proved that any polyhedron can be folded from a sheet of paper.
The method used for the proof is based on wrapping a strip of paper around a polyhedron
and it is practically not possible to design real models by the method.
Some arbitrary polyhedral models have been designed. Both roofs by the author [8]and a frog by Masahiko Tanaka [7] use symmetry of tucks which constrains the alignmentof the polygons on paper.
In this paper, we propose a new method for designing an arbitrary polyhedral surface(Figure 1). Instead of using the symmetry of tucks between edges, we propose ”edgetucking molecules,” which can be generated for a given alignment of edges. We further

Figure 1: A 3D teapot can be folded (right) from the crease pattern (middle) constructedfrom a teapot represented by a triangle mesh (left)
investigate the conditions for three-dimensional curvature, which is the most distinctive
part of 3D origami design.

2

Method Overview

2.1

Tucking Molecules

Our approach is to align polygons of the polyhedral surface onto a convex region of a planeand fill the area between polygons with ”edge-tucking molecules” and ”vertex-tuckingmolecules.” Figure 2 shows how the paper is tessellated into polygons and molecules.

Vertex-Tucking Molecules
Edge-Tucking Molecules
Surface Polygons

Figure 2:
Edge-tucking molecules are aligned between pairs of edges of the aligned surface polygons. The two edges are moved to a three-dimensionally identical position by folding theedge-tucking molecule between them (Figure 3(a)). Vertex-tucking molecules are aligned
between vertices corresponding to one vertex of the three-dimensional surface polygons.
The vertices are moved to an identical position by folding the vertex-tucking moleculealigned between them (Figure 3(b)). Hence edge-tucking molecules and vertex-tucking
molecules are used for pasting pairs of edges and combinations of vertices respectively.

2.2

Conditions

There are two kinds of conditions for aligning polygons and tucking molecules. One is what
we call ”2D conditions,” which is mainly related to aligning completely unfolded molecules

(a) Edge-tucking molecule(b) Vertex-tucking molecule

Figure 3: (a)An edge-tucking molecule make positions of two segments identical. (b)Avertex-tucking molecule make positions of vertices identical.
on unfolded paper. The other is ”3D conditions,” which comes from the relations between
polygons and molecules when the model is folded. The followings are the conditions fordesigning polyhedral surface using tucking molecules.
Condition 1 Polygons and tucking molecules properly tessellate convex planer area.
Condition 2 Crease patterns can be generated and do not intersect polygons or eachother.
Condition 3 The tucks and the polygons do not intersect each other when folded.
Condition 4 The tucks have enough angle to get the right curvature of the required surface when folded.
Condition 1 and 2 are 2D conditions, and Condition 3 and 4 are 3D conditions.
As origami models are designed by placing molecules and generating crease pattern on
two dimensional paper, 3D conditions must also be written down as conditions on crease
pattern. We use ”tuck proxies” to assume three-dimensional positions of the folded tucks,and get sufficient angle inequality conditions on crease pattern.
The procedure and the conditions for proper tessellation are shown in Section 3. The
method for generating a crease pattern of the edge-tucking molecule for a given pairededges is shown in Section 4.1 and their conditions in Section 4.2. The method for generating a crease pattern of the vertex-tucking molecule for a given combination of vertices
is shown in Section 5.1 and their conditions in Section 5.2. Then 3D conditions are shownin Section 6. Finally, Section 7 shows examples of the application of this method.

3Tessellation

First, a polyhedral surface is cut so that it is topologically equivalent to a disk if it is notso. Then, edge-tucking molecules and vertex-tucking molecules are added to the surface
to split edges and vertices. The order of vertices around the molecules are set according
to surface direction.
The polygons and added molecules are mapped onto a convex region of a plane without
intersection of the elements or changing of the connectivity. The surface polygons are kept

congruent after mapping, and the tucking molecules are deformed to fit the blank area
between the aligned polygons.
The followings are the conditions for properly tessellating planer area by molecules.
Condition 5 Any mapped tucking molecule does not cross itself.
Condition 6 Signed area of any tucking molecule is not negative.
Condition 7 The mapped boundary of the surface is convex.
Figure 4 shows an example of how Condition 5 and 6 help avoid intersections ofmolecules.

(a)

(b)

(c)

(d)

Figure 4: Conditions for properly tessellation. (a) Ok. (b) The vertex-tucking molecule
is crossed. (c) The edge-tucking molecule is crossed. (d) The vertex-tucking moleculesand edge-tucking molecules have negative area.

4

Edge-Tucking Molecules

4.1

Generating Edge-Tucking Molecule

Figure 5 shows the procedure of generating an edge-tucking molecule. An edge-tucking
molecule is a quadrilateral area (AA0 B0 B) surrounded by the paired edges (AB and A0 B0 )and segments connecting corresponding vertices of the paired edges (AA0 and BB0 ).

AA
MA

LAA'

BC

BCA'MBLBB'

MALAMB
D

LB
B'

ValleyMountain

Figure 5: Generating procedure of crease pattern for an edge-tucking molecule. Right
image shows the resulting crease pattern.
The basic idea is to make a valley fold along perpendicular bisectors of AA0 and BB0
so that point A is folded to A0 and B to B0 , and make a flat-foldable crease pattern thatconnects these perpendicular bisectors. The followings are the procedure.

1. Define LA as the perpendicular bisector of segment AA0 and define LB as the perpendicular bisector of segment BB0 .
2. Denote the midpoints of segment AA0 and of BB0 by MA and MB respectively.
3. If AB0 = A0 B, then LA is on LB and the crease pattern can be get by drawing avalley crease on segment MA MB . Else, assume that A0 B < AB0 . (The crease pattern
can be constructed in the same way when A0 B > AB0 .)4. Define point C on LA inside quad AA0 B0 B.
5. Define point D on LB so that 6 MA CA0 + 6 DCB = π
6. Draw crease pattern according to the right image of Figure 5.
In step 5, Kawasaki’s theorem [9] is used for flat-foldability of vertex C. Note that
Kawasaki’s theorem is satisfied also in vertex D. This is always satisfied if you can define
point D on LB .

4.2

Conditions for Generating Edge-Tucking Molecule

The followings are the two conditions for a properly aligned quadrilateral to be an edgetucking molecule.
Condition 8 The crease pattern can be generated for each edge-tucking molecule.
Condition 9 The generated crease patterns do not intersect each other.
The necessary and sufficient condition for condition 8 can be written as follows.
Condition 10 Perpendicular bisectors of AA0 and BB0 does not intersect the surface
polygons connected to the edge-tucking molecule.
Figure 6(a) shows the example. This condition is represented as,
AB0 ≥ AB and A0 B ≥ AB(= A0 B0 )(1)

Instead of Condition 9, we can use the following sufficient condition, which is useful
because it can be determined only from the shape of each molecule.
Condition 11 The generated crease pattern is inside the quadrilateral of the molecule.
Condition 11 can be represented by angle inequality (Figure 6(b)). Assume that weput point C (a point on LA from which the new crease line is generated) on MA , which is
the most preferable position to make the crease pattern inside. The condition is satisfied
if and only if the point D is inside the quadrilateral AA0 B0 B. This means,
6BMA MB ≤ 6 BMA D(2)Because point D is defined by flat foldability around vertex C,
6BMA D = π − 6 LA MA A = π/2(3)Hence, Condition 11 is represented as,6BMA MB ≤ π/2(4)

A
AA

BA

B MA=C B
LA

B
MA=CA'LAA'
B'

B'A'A'

B'(a) Left: Ok. Right: A0 B < ABMB DMBDB'(b) Left: CP is inside. Right: CP is outside

Figure 6: Conditions for an edge-tucking molecule. (a) Conditions to generate crease
pattern. (b) Conditions for making the crease pattern inside the molecule.

5Vertex-Tucking Molecules

5.1

Generating Vertex-Tucking Molecule

Figure 7 shows how the crease pattern for a vertex-tucking molecule is generated. The
procedure is as follows.
1. Draw Voronoi diagram with valley crease using the vertices of the molecule as generating points.
2. Connect vertices of each Voronoi polygon to the generating points with mountaincrease.
3. Crimp or rabbit ear fold triangles which is not connected to edge-tucking molecules.
The procedure 3 can be omitted in real design. For this is for making the folded shapeof the molecule composed of tucks connected to one axis (Figure 10), which is useful for
calculating 3D conditions described in Section 6. This kind of molecule based on Voronoi
diagram has recently been used for some origami designs. Though the procedure is notprecisely described, Robert Lang is showing the same molecule for the connection of strip
grafts [4, p.152].

Figure 7: Generating process of crease pattern for a vertex-tucking molecule.

5.2

Conditions for Generating Vertex-Tucking Molecule

The followings are the two conditions for generating vertex-tucking molecules.
Condition 12 The crease pattern can be generated for each vertex-tucking molecule.
Condition 13 The generated crease patterns do not intersect each other.
Condition 12 is represented as follows.
Condition 14 The boundary of vertex-tucking molecule is on the edges of Delaunay triangles dual to the generated Voronoi diagram.
Figure 8(a) shows how this condition works. If a segment on the boundary is not on theDelaunay triangles, an edge-tucking molecule cannot connect to the segment because theperpendicular bisector does not cross the segment.
For Condition 13 we use the following sufficient condition instead of the necessary andsufficient condition.
Condition 15 The generated crease pattern is inside the molecule.
Which is also written as,For any edge Vi Vj on the boundary of the molecule and for any vertex Vk (k 6= i, j) of themolecule,6 Vj Vk Vi ≤ π/2(5)Note that (5) is also a sufficient condition for Condition 14 as well as for Condition 13.
Figure 8(b) shows the example. In the right image, 6 V1 V2 V0 > π/2 and the crease
pattern get outside the edge V1 V0 .

V3V0
V2

good bad(a) Left: Ok. Right: The boundary is noton the edges of Delaunay triangles.

V1(b) Left: CP is inside. Right: CP is outside.

Figure 8: Conditions for a vertex-tucking molecule. (a) Condition to generate creasepattern. (b) Conditions for making the crease pattern inside the vertex-tucking molecule.

63D Conditions

3D conditions are conditions on the paper after folded, shown as Condition 3 and 4. Necessary and sufficient conditions for 3D conditions are difficult to be handled by parametersof crease pattern, because the three-dimensional configuration changes not only by thecrease pattern, but also by the folding angles of non flat foldable crease lines.
Thus we assume the tucks (i.e., folded tucking molecules) are on predefined ”tuckproxies,” so that Condition 4 is represented as a condition on parameters of crease pattern(i.e., angles inequalities), while keeping Condition 3.

6.1

Generating Tuck Proxies

We set the widths of tuck proxies as large as possible without intersection with surface
polygons or each other, as tuck proxies define the range in which tucking molecules canbe folded. Here is an example of generating tuck proxies (Figure 9).
1. Define a segment from each vertex to the direction opposite to the surface direction.
The length of the segment is set so that the segment does not intersect the polygons.
2. Connect adjacent segments with strips of two triangles and define them as tuckproxies.
3. If the strips intersect with the surface polygons or each other then shorten thesegment length and redo from 1. Else end.
As we assume that folded tucking molecules are on tuck proxies, width of the foldedtuck must be less than or equal to width of the tuck proxy. The half of the distance
between the corresponding vertices of the tucking molecules can be used for calculating
the width of the folded tucks.

Figure 9: Generating procedure of tuck proxies

6.2Angle Inequality

Sufficient condition for Condition 4 can be represented by inequalities between angles offolded molecules and angles between edges along tuck proxies.
Relations between the angles on unfolded paper and on folded molecules are shownin Figure 10. In this example 6 ABB0 < 6 A0 B0 B, so 6 ABB0 keeps its angle after folded.
General equation is as follows.
min (6 ABB0 , 6 A0 B0 B) + 6 OB BB0 |in 2D = 6 ABOB |in 3D(6)We name this angle (i.e., 6 ABOB |in 3D ) ”tuck angle” and denote it by θi,k for edge iconnecting to vertex k.
Folded tucking molecules are easily folded to fit the tuck proxy and crimped to adjust
tuck angles of each edge (Figure 11). Note that joint axis (i.e., axis to which the tucksare connected) can be moved by folding the tucks, and one of the tuck angles can beincreased. The necessary and sufficient condition for making such adjustment possible forvertex k is as follows
for any edge i and j (i 6= j) connected to vertex k,00
θi,k + θj,k ≥ θi,k+ θj,k(7)AB'''

BA

B

MB
OB

MBB''

B'A'

OB

Figure 10: Angle relations between unfolded paper and folded molecules.
θ4−δ

θ4
θ3

θ1

θ 4'

θ3−δ

θ1+δ

θ3'
θ 1'

δ
θ2

θ2−δ

θ2'

Figure 11: Adjusting tuck angles. (Left to Middle) One of the tuck angles is increased bymoving the joint axis. (Middle to Right) Any tuck angles can be reduced by crimping.
0where θi,k is the original tuck angle calculated by equation (6), and θi,kis the desired tuckangle along tuck proxy.

7Application

We show two ways of application of the proposed method. Figure 12 shows the examples.
By applying this method for an overall model, a complex, three-dimensionally definedmodel can be folded by a sheet of paper. Figure 12(a) is a part of the crease pattern forthe model ”Teapot,” whose folded model is shown in the right picture of Figure 1.
The proposed method is also useful for designing some parts of the model threedimensionally and connect to other parts designed by other methods. As shown in Figure12(b), the connection parts can also be flexibly designed using tucking molecules. Thisapproach helps designers to control the degrees of detail and the complexity of the model.

8Conclusions and Future Works

We proposed a method for designing three-dimensional polyhedral surface using tucking
molecules. The generating procedures for building vertex-tucking molecules and edgetucking molecules were shown. We investigated the conditions for generating the crease
pattern (2D conditions), and for constructing desired three-dimensional configuration (3Dconditions). We have also shown that the sufficient conditions for 3D conditions can be(a) Zoom view of the crease pattern for the teapot.
See Figure 1 for the whole crease pattern and thefolded model.

(b) Connecting 3D design (top) to regular box pleating pattern (bottom).

Figure 12: Examples.
represented by angles and length inequalities on molecules aligned on two-dimensional
paper.
The proposed method can be used for designing complicated three-dimensional models
and also for adding flexibility to other origami designing methods. The future work is toimplement this method and make a tool to help design three-dimensional origamis.

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