Unspecified Book Proceedings Series

Rigid Flattening of Polyhedra with Slits
Zachary Abel, Robert Connelly, Erik D. Demaine, Martin L. Demaine,
Thomas C. Hull, Anna Lubiw, and Tomohiro Tachi
Abstract. Cauchy showed that if the faces of a convex polyhedron are rigid
then the whole polyhedron is rigid. Connelly showed that this is true evenif finitely many extra creases are added. However, cutting the surface of thepolyhedron destroys rigidity and may even allow the polyhedron to be flattened. We initiate the study of how much the surface of a convex polyhedron
must be cut to allow continuous flattening with rigid faces. We show thata regular tetrahedron with side lengths 1 can be continuously flattened withrigid faces after cutting a slit of length .046 and adding a few extra creases.

1. Introduction
In many real-life situations we want polyhedra or polyhedral surfaces to flatten—
think of paper bags, cardboard boxes, and foldable furniture. Although paper is
flexible and can bend and curve, materials such as cardboard, metal, and plastic arenot. The appropriate model for such non-flexible surfaces is “rigid origami” where
the polyhedral faces are rigid and folding occurs only along pre-defined creases. In
rigid origami, flattening is not always possible, and in fact, often no movement ispossible at all. In particular, Cauchy’s theorem of 1813 says that if a convex polyhedron is made with rigid faces hinged at the edges then no movement is possible(see [6]). Connelly [4] showed that this is true even if finitely many extra creases
are added.
However, cutting the surface of the polyhedron destroys rigidity and may evenallow the polyhedron to be flattened. For example, a paper bag is a box whose
top face has been removed, so the afore-mentioned rigidity results do not apply.
Everyone knows the “standard” folds for flattening a paper bag. Surprisingly, thesefolds do not allow flattening with rigid faces unless the bag is short [2]. Taller bags
can indeed be flattened with rigid faces, but a different crease pattern is required [8].
Many of the clever ways of flattening cardboard boxes involve not only removal of
the top face, but also extra slits and interlocking flaps in the bottom face.
In this paper we initiate the study of “rigid flattening” of a polyhedron: continuous flattening with rigid faces after the addition of finitely many cuts and creases.
1991 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20.
Key words and phrases. folding, polyhedra, flattening, rigid origami.
c 2014 the authors

1

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ABEL, CONNELLY, DEMAINE, DEMAINE, HULL, LUBIW, AND TACHIWe require that the final flat folding be a flat folding of the original polyhedron,i.e. that every cut closes up at the end of the flattening process.
We can use previous results to show that every convex polyhedron has a rigid
flattening. Without the requirement about cuts closing up, we could just cut everyedge of the polyhedron and move the faces to the plane. Alternatively, we could keep
the surface connected and use the “continuous blooming” of the source unfoldingof a convex polyhedron [5].
With our requirement that cuts close up, the final state is a flat folding of theoriginal polyhedron, so we first need to know that every convex polyhedron has aflat folded state. There are three proofs of this result: Bern and Hayes [3], using
a disk-packing method that applies to any polyhedral surface; Itoh et al. [7] via
a continuous motion; Abel et al. [1] via an easily-computable continuous motion
resulting in a flat folding that respects the straight-skeleton gluing. Using theseresults, we can obtain a rigid flattening by just cutting every fold in the flat folded
state. Note that the surface becomes disconnected.
It is an open question whether every convex polyhedron has a rigid flattening
using cuts that do not disconnect the surface. More generally, we might ask tominimize the length of the cuts. Another interesting question is whether there is arigid flattening with only one degree of freedom.
In this paper we begin exploring these ideas by studying the regular tetrahedron. We show that a surprisingly small cut allows rigid flattening. Specifically,
if the tetrahedron has side length 1, a cut of length .046 suffices. We explicitly
specify the few extra creases that are needed. There is one degree of freedom during the flattening. We use Mathematica to model the motion and verify that noself-intersections occur.
We argue that our particular slit cannot be reduced in length, but it is possiblethat a smaller slit in a different position works. In fact it is even possible that theslit length can approach 0 while the number of creases grows. We discuss these andother open questions in the final section of the paper.
2. Flattening a Regular Tetrahedron
In this section we show that a regular tetrahedron with side length 1 can be
rigidly flattened with a cut of length .046. We specify the cut and the extra creases,
and verify in Mathematica that the result folds flat rigidly without self-intersections.

D

C
A
x

c
p

B

Figure 1. Flat folding of a regular tetrahedron: (left) the creases; (right) part way
towards the flat folding (shown schematically, since the faces will not really be flatin this configuration).

RIGID FLATTENING OF POLYHEDRA WITH SLITS

3

In order to describe the cut and the extra creases we first explore a rigid flattening using a longer cut. The most natural flattening of a regular tetrahedron on
vertices A, B, C, D uses creases as shown in Figure 1: faces ACD and BDC are
intact; face ADB has one crease bisecting the angle at D and arriving at point pof the opposite edge AB; and the final face ABC has four creases to its centroid
x—three from the vertices and the fourth crease, c, from p to x. We call c thecentroid normal. All the creases are valley folds except c which is a mountain fold.
This flat folding yields a rigid flattening if we cut the centroid normal c andadd two mountain creases that go from the vertices A and B to the cut c and bisect
the angles ∠xAp and ∠xBp respectively. See Figure 2. (In fact, mountain creases
from A and B to the midpoint of c would also work.)
We argue that this flattening has only one Degree of Freedom (DOF). Suppose
face BCD is fixed in 3-space. Faces ACD and BCD are rigid and the one degreeof freedom is the angle between them. Given a value for that angle, the positionsof x and p are fixed in 3-space. (The fact that Cx and Dp are valleys rules out theother possible position for each). This in turn fixes the positions of the final two
mountain creases.

Figure 2. Rigid flattening of a tetrahedron after cutting the centroid normal c.
Our rigid flattening with a small slit is based on the one shown in Figure 2 but
uses a shorter cut along the centroid normal. The cut goes from p, the midpointof edge AB to a point q on the centroid normal. See Figure 3. The final lengthof pq will be .046, although we will discuss other possibilities. The triangles Apq
and Bpq are called the flaps. Creases Ax, Bx, and Cx remain valleys. Crease xq is
a mountain. Creases Aq and Bq will alternate between mountain and valley folds
during the rigid flattening. Point m is placed on the centroid normal segment px
and on the angle bisector of ∠qAp. We would like to add mountain creases Am
and Bm, but this plan needs some refinement.
There are two limitations on the length of the cut pq. The first one can be remedied, but the second one is more fundamental and makes it impossible to shorten
the cut below .046. The second limitation is described below in Subsection 2.2.
Here we address the first limitation.
The first limitation is that as the slit becomes smaller the flaps can interferewith each other during the folding process. In particular, the two copies of m will
collide in the rigid unfolding. This can be remedied by adding pleat folds to theflaps so that the two sides of the cut fold out of the way. In order to achieve the cut
length of .046 we place pleat folds as shown in Figure 4. The pleat creases emanatefrom points p and q, with the the largest pleat crease at p forming angle ∠qpr = 45◦
and the smaller pleat angles at p, going counter-clockwise in order, are 15◦ , 15◦ , 5◦ ,

4

ABEL, CONNELLY, DEMAINE, DEMAINE, HULL, LUBIW, AND TACHI

A

C

x

q m

p

B

Figure 3. The plan of the creases to allow rigid flattening of a tetrahedron after
cutting segment pq. (Other faces remain the same.) The flaps Apq and Bpq are
shaded.
10◦ and then repeat in reverse order. This choice of angles is made so as to avoid
collisions between the pleats across the slit when folding; merely quadrisecting the45◦ angles at p causes collisions near the flat-folded state. Such pleats are effectivein addressing the first limitation because they break the line Am (in Figure 3) sothat instead of being a long mountain crease, it is now a shorter mountain, andthen short valley-mountain-valley-mountain creases as we approach the slit. This
makes Am contract into a zig-zag near the slit, which keeps the two copies of maway from each other during the folding process.
A

r

C

q

p

B

Figure 4. Crease pattern for rigid flattening of a regular tetrahedron after cutting
a slit of length .046. (left) Crease pattern on face ABC. (right) close-up of thecircled slit region (red lines are mountain folds, blue lines are valleys, and thedashed creases switch between the two).
This completes the description of the cut and the extra folds to enable rigid
flattening of a regular tetrahedron with a slit length of .046. There is still one
degree of freedom because the folding of the degree four vertices of the pleats will
be determined by the neighboring creases adjacent to vertices A and B. Figure 7
shows 4 frames of the rigid flattening. Note in particular that frame 3 shows how
the flaps have folded out of the way and avoided colliding. With careful observation,
one can see that the fold Aq is a mountain in frame 3 and a valley in frame 4.

RIGID FLATTENING OF POLYHEDRA WITH SLITS

(b)

(a)

0.5

1.0

5

1.5

2.0

2.5

3.0

-0.00005



x0=H0,0,0L

-0.00010



pê4


-0.00015<-- x2=H0,0,-1L

p

p

12

12

p

18
p
36

x3

x4

x5 x6



(c)



0.5

      --   

1.0

1.5

2.0

2.5

3.0

-0.00001-0.00002

x7

-0.00003

-0.00004-0.00005-0.00006-0.00007



(d)


0.5

      --   
1.0

1.5

2.0

2.5

3.0

-0.00001-0.00002

x1

-0.00003

-0.00004-0.00005-0.00006-0.00007      --   

Figure 5. (a) Schematic of the pleat “sink” points. (b)-(d) Graphs of the dot
product between the sink point vectors and the normals to the sides of the conefrom x2 to the triangle made by x0 , x1 , and x3 .
2.1. Checking potential collisions. We have verified the rigid flattening of
this model in Mathematica using a kinematics model for the regions between thecreases. Figure 5(a) shows a detail of the pleats near the slit in the crease pattern,which we could also refer to as a sink to borrow origami terminology. The points x0 ,x1 , and x2 correspond to the points p, q, and A from Figure 4, respectively, exceptthat we imagine them to be in the xz-plane. To argue that the sink points x4 -x7
do not collide with other parts of the model, we examine the polygonal cone madeby x2 (the cone point) and the triangle x0 x1 x3 . If the sink points remain inside
this cone throughout the folding process then they will not collide with other sidesof the folding tetrahedron. To this end, we graph the dot products of the outwardpointing vectors normal to the sides of this cone and the sink point vectors; so longas these dot products are nonpositive, the sink points will not penetrate the planes
made by the cone sides and thus will remain inside the cone. Graphs of these dot
products are shown in Figure 5(b), (c), and (d), where the horizontal axis is thefolding angle 0 ≤ t ≤ π, and the length of the slit is taken to be approx. 0.046. Notethat the dot products all remain negative with the slight exception of the point x7
in graph (d) at the end of the folding process (t ≈ π). This is because when thecone is nearly flat the sink pleat made by x7 is inclined upward and escapes thecone. However, examining this case shows that x7 quickly folds flat and remains
clear of any collisions as t → π.

6

ABEL, CONNELLY, DEMAINE, DEMAINE, HULL, LUBIW, AND TACHIWe also must show that none of the sink points collide with their mirror-image
counterparts on the other side of the sink. As mentioned previously, if the angles ofthe sink pleats are not chosen with care, then such collisions will occur and obstruct
the rigid folding of the model. One way to check this with our current model is tosee if during the folding the sink points pass through the plane through p = x0 ,C, and perpendicular to AB in Figure 4. In our Mathematica model, we had thisplane be the xy-plane throughout the folding, so all we need to do is to plot thez-coordinates of the sink points x4 -x7 . This is shown in Figure 6(a), where theslit length is taken to be 0.046. Since the z-coordinates remain negative, the sinkpoints will not collide with their counterparts on the other side of the slit. Note,however, that x7 does touch its mirror-image at the folding angle t ≈ 0.62, where
the slit closes up before opening again, as seen in Figure 6(b).

(a)

0.5

1.0

1.5

2.0

2.5

3.0

-0.005x4-0.010x5Out[62]=x6-0.015x7-0.020The z-coordinates of the sink points, showing that x4-x7 do not pass through the xy-plane.

(b)t=0.565

t=0.62t=0.757

Figure 6. (a) Plots of the z-coordinate of the sink points in our Mathematica
model for slit length 0.046. (b) Close-ups of the slit opening and closing neart = 0.62.
Therefore, with the pleats in Figure 4 included, the two sides of the cut will
not intersect throughout the folding process. Interested readers can download
and examine the Mathematica code for this model at http://mars.wne.edu/∼thull/
rigidtet/tet.html.
2.2. A limitation on the cut length. In this section we show that the cut
length cannot be shorter than 0.046 if we place the cut and the creases as shownin Figure 3 and allow extra folds only in the flaps. Note that this is a very limited
result. It is quite possible that there is a rigid flattening using a shorter cut ina different position, or even in the same position but with extra folds outside theflaps.
Consider the creases in Figure 3. We will ignore the flaps—just cut them outof the surface. The remaining surface consists of 8 rigid triangles: 4 on face ABCof the tetrahedron, plus 2 on face ABD, plus the 2 intact faces. There is only one

RIGID FLATTENING OF POLYHEDRA WITH SLITS

1

2

3

4

7

Figure 7. Rigid flattening of a regular tetrahedron after cutting a slit of length.046, showing 4 frames of the flattening, each with the detail of the slit region shownin the circular close-up.

8

ABEL, CONNELLY, DEMAINE, DEMAINE, HULL, LUBIW, AND TACHI

degree of freedom during the rigid flattening. Points C, D, x, p, q remain on thesame plane. A cross-section in that plane is shown in Figure 8.
D

D

D

C

C

C

q

q p

C

x

x

x

D

p

q

x

p

D

x

C

q p

q

p

Figure 8. Cross sections in the plane of C, D, x showing how points p and q change
distance during the rigid flattening process.
During the flattening process the distance between points p and q changes. Our
main observation is that the cut pq must be long enough to accommodate this. In
particular, if the flaps are included, they prevent p and q from being farther apartthan the length of the cut. This is true even if the flaps are completely flexible.
We wrote a Mathematica program to compute the distance between points p
and q in the plane of C, D, x during the rigid flattening of the 8 rigid triangles
described above. As can be seen in Figure 9, if the cut length is less than 0.0461201
then at some point during the rigid flattening, points p and q will be further apartthan the cut length. Thus the minimum possible cut is approximately 0.0461201.
dist. between endpoints of the cut
0.04
0.03
0.02
0.01
0.2

0.4

0.6

0.8

1.0

1.2

progress of flattening

Figure 9. A graph of the distance between points p and q (y-axis) during therigid flattening (x-axis) of the crease pattern in Figure 3 when the cut length is
0.0461201. Observe that the graph has a local maximum at an intermediate point of
the flattening and this local maximum reaches the original cut length of 0.0461201(indicated by the horizontal line). If we decrease the cut length, the new graph has
a local maximum that exceeds the cut length.

3. Discussion and Open Problems
Our investigation of rigid flattening of a regular tetrahedron leaves many open
questions:(1) Does every convex polyhedron have a rigid flattening using cuts that leave
the surface connected?

RIGID FLATTENING OF POLYHEDRA WITH SLITS

9(2) Is there such a rigid flattening with one degree of freedom?
(3) Does a regular tetrahedron with unit side lengths have a rigid flattening
using a cut of length less than .046? Can the cut length approach 0 (asthe number of extra creases grows)?
(4) Does every box have a rigid flattening using one straight cut? We suppose
the answer is yes if you cut almost all the way around one equator andapply the rigid flattening for shallow paper bags [2], but will a shorter cut
suffice?
Regarding the main open question (1), it may be easier to start from the flat
folded state and ask what slits (plus extra creases) allow the flat folded state to
unfold. Based on the example in Figure 2, where the regular tetrahedron is slit fromthe centroid of one face to an edge normal, we conjecture that it suffices to slice
all the mountain folds in the flat folding. This is not true for polyhedral surfaces
in general (the square twist is a counterexample), but might be true for convex
polyhedra, or at least for straight skeleton flat foldings of convex polyhedra [1]. Asa starting point, what happens if we cut the mountain folds and the edges of thepolyhedron.
Acknowledgments
This work was begun at the 2013 Bellairs Workshop on Computational Geometry, co-organized by Erik Demaine and Godfried Toussaint. We thank the otherparticipants of the workshop for stimulating discussions.
This research could not have been accomplished without the Rigid Origami
Simulator written by Tomohiro Tachi which allowed us to test rigid foldability of
convex polyhedra with various cuts and creases.
Research of E. Demaine and M. Demaine supported in part by NSF grant EFRIODISSEI-1240383 and NSF Expedition grant CCF-1138967. Research of T. Hullsupported by NSF grant EFRI-ODISSEI-1240441 “Mechanical Meta-Materials from
Self-Folding Polymer Sheets”. Research of A. Lubiw supported by the Natural Sciences and Engineering Research Council of Canada. Research of T. Tachi supportedby the Japan Science and Technology Agency Presto program.
References[1] Z. Abel, E. D. Demaine, M. L. Demaine, J.-i. Itoh, A. Lubiw, C. Nara, and J. O’Rourke.
Continuously flattening polyhedra using straight skeletons. Proc. 30th Annual Symposium on
Computational Geometry (SoCG), pp. 396–405, 2014, doi:10.1145/2582112.2582171.
[2] D. J. Balkcom, E. D. Demaine, M. L. Demaine, J. A. Ochsendorf, and Z. You. Folding paper
shopping bags. Origami4 : Proceedings of the 4th International Meeting of Origami Science,
Math, and Education (OSME 2006), pp. 315–334. A K Peters, September 8–10, 2006.
[3] M. Bern and B. Hayes. Origami embedding of piecewise-linear two-manifolds. Algorithmica
59(1):3–15, 2011, doi:10.1007/s00453-010-9399-8.
[4] R. Connelly. The rigidity of certain cabled frameworks and the second-order rigidity ofarbitrarily triangulated convex surfaces. Advances in Mathematics 37(3):272–299, 1980,
doi:10.1016/0001-8708(80)90037-7.
[5] E. Demaine, M. Demaine, V. Hart, J. Iacono, S. Langerman, and J. O’Rourke. Continuous
blooming of convex polyhedra. Graphs and Combinatorics 27(3):363–376, 2011,
doi:10.1007/s00373-011-1024-3.
[6] E. D. Demaine and J. O’Rourke. Geometric Folding Algorithms: Linkages, Origami,
Polyhedra. Cambridge University Press, 2007.

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ABEL, CONNELLY, DEMAINE, DEMAINE, HULL, LUBIW, AND TACHI[7] J.-i. Itoh, C. Nara, and C. Vı̂lcu. Continuous flattening of convex polyhedra. Computational
Geometry, pp. 85–97. Springer, Lecture Notes in Computer Science 7579, 2012,
doi:10.1007/978-3-642-34191-5 8.
[8] W. Wu and Z. You. A solution for folding rigid tall shopping bags. Proceedings of the Royal
Society A: Mathematical, Physical and Engineering Science 467(2133):2561–2574, 2011,
doi:10.1098/rspa.2011.0120.
Abel, Demaine, Demaine: CSAIL, MIT, Cambridge MA, USA
E-mail address: {zabel,edemaine,mdemaine}@mit.edu
Connelly: Department of Mathematics, Cornell University, Ithaca NY, USA
E-mail address: rc46@cornell.edu

Hull: Department of Mathematics, Western New England U., Springfield MA, USA
E-mail address: thull@wne.edu
Lubiw: School of Computer Science, University of Waterloo, Waterloo ON, Canada
E-mail address: alubiw@uwaterloo.ca
Tachi: Department of General Systems Studies, The University of Tokyo, Japan
E-mail address: tachi@idea.c.u-tokyo.acjp