FoldingFolding && Unfolding:Unfolding: FoldingFolding PolygonsPolygons toto ConvexConvex PolyhedraPolyhedra

JosephJoseph O’RourkeO’Rourke SmithSmith CollegeCollege FoldingFolding andand UnfoldingUnfolding TalksTalks

Linkage folding Tuesday Erik Demaine

Paper folding Wednesday Erik Demaine

Folding polygons into Saturday Joe O’Rourke convex 1 polyhedra Unfolding Saturday Joe O’Rourke polyhedra 2 OutlineOutline

 Reconstruction of Convex Polyhedra  Cauchy to Sabitov (to an Open Problem)  Folding Polygons  Algorithms  Examples  Questions OutlineOutline11

 Reconstruction of Convex Polyhedra  Cauchy to Sabitov (to an Open Problem)  Cauchy’s Rigidity Theorem  Aleksandrov’s Theorem  Sabitov’s Algorithm OutlineOutline22

 Folding Polygons  Algorithms  Edge-to-Edge Foldings  Gluing Trees; exponential lower bound  Gluing Algorithm  Examples  Foldings of the Latin Cross  Foldings of the Square  Questions  Transforming shapes? ReconstructionReconstruction ofof ConvexConvex PolyhedraPolyhedra

 graph Steinitz’s Theorem  face angles  edge lengths  face areas Minkowski’s Theorem  face normals  dihedral angles  inscribed/circumscribed Minkowski’sMinkowski’s TheoremTheorem

(a) (b) ReconstructionReconstruction ofof ConvexConvex PolyhedraPolyhedra

 graph  face angles Cauchy’s Theorem  edge lengths  face areas  face normals  dihedral angles  inscribed/circumscribed Cauchy’sCauchy’s RigidityRigidity TheoremTheorem

 If two closed, convex polyhedra are combinatorially equivalent, with corresponding faces congruent, then the polyhedra are congruent;  in particular, the dihedral angles at each edge are the same.

Global rigidity == unique realization SameSame facialfacial structure,structure, noncongruentnoncongruent polyhedrapolyhedra SphericalSpherical polygonpolygon

v v

(a) (b) SignSign Labels:Labels: {+,-,0}{+,-,0}

 Compare spherical polygons Q to Q’  Mark vertices according to dihedral angles: {+,-,0}.

Lemma: The total number of alternations in sign around the boundary of Q is ≥ 4. The spherical polygon opens.

(a) Zero sign alternations; (b) Two sign alts. SignSign changeschanges EulerEuler TheoremTheorem ContradictionContradiction

Lemma
≥ 4 V +

+ - v -

+ - f +

+ - FlexingFlexing toptop ofof regularregular octahedronoctahedron Steffen’sSteffen’s flexibleflexible polyhedronpolyhedron

14 triangles, 9 vertices http://www.mathematik.com/Steffen/ TheThe Bellow’sBellow’s ConjectureConjecture

 Polyhedra can bend but not breathe [Mackenzie 98]  Settled in 1997 by Robert Connelly, Idzhad Sabitov, and Anke Walz  Heron’s formula for area of a triangle: A2 = s(s-a)(s-b)(s-c)  Francesca’s formula for the volume of a tetrahedron  Sabitov: volume of a polyhedron is a polynomial in the edge lengths. OutlineOutline11

 Reconstruction of Convex Polyhedra  Cauchy to Sabitov (to an Open Problem)  Cauchy’s Rigidity Theorem  Aleksandrov’s Theorem  Sabitov’s Algorithm Aleksandrov’sAleksandrov’s TheoremTheorem (1941)(1941)

 “For every convex polyhedral metric, there exists a unique polyhedron (up to a translation or a translation with a symmetry) realizing this metric." Pogorelov’sPogorelov’s versionversion (1973)(1973)

 “For every convex polyhedral metric, there exists a unique polyhedron (up to a translation or a translation with a symmetry) realizing this metric.“  “Any convex polyhedral metric given ... on a manifold homeomorphic to a sphere is realizable as a closed convex polyhedron (possibly degenerating into a doubly covered plane polygon)." AlexandrovAlexandrov GluingGluing (of(of polygons)polygons)

 Uses up the perimeter of all the polygons with boundary matches:  No gaps.  No paper overlap.  Several points may glue together.  At most 2 angle at any glued point.  Homeomorphic to a sphere.

Aleksandrov’s Theorem  unique “polyhedron” FoldingFolding thethe LatinLatin CrossCross CauchyCauchy vs.vs. AleksandrovAleksandrov

 Cauchy: uniqueness  Aleksandrov: existence and uniqueness

 Cauchy: faces and edges specified  Aleksandrov: gluing unrelated to creases UniquenessUniqueness

 Cauchy: combinatorial equivalence + congruent faces => congruent  Aleksandrov: “Two isometric polyhedra are equivalent”  The sphere is rigid [Minding]  The sphere is unique given its metric [Liebmann, Minkowski]  Closed regular surfaces are rigid [Liebmann, Blaschke, Weyl]  Uniqueness of these w/in certain class [Cohn-Vossen]  ...  “Isometric closed convex surfaces are congruent” [Pogorelov 73] AlexandrovAlexandrov ExistenceExistence11

 Induction on the number of vertices n of the metric:  from realization of n-1 vertex metric to n vertex metric  by continuous deformation of metrics  tracked by polyhedral realizations ExistenceExistence22 curvature = 2 –
angs

 There are two vertices a and b with curvature less than .  Connect by shortest path .  Cut manifold along  and insert double  that leaves curvature unchanged.  Adjust shape of  until a and b both disappear: n-1 vertices.  Realize by induction, introduce nearby pseudovertex, track sufficiently close metrics. (a)(a) aa flattened;flattened; (b)(b) bb flattened.flattened.

c

c

A=60 θ θ=B=30

α=240 β=300 (a) (b) D-FormsD-Forms

SmoothSmooth closedclosed convexconvex curvescurves ofof samesame perimeter.perimeter. GlueGlue perimetersperimeters together.together. D-formD-form

c1 c2 Helmut Pottmann and Johannes Wallner. Computational Line Geometry. Springer-Verlag, 2001.

Fig 6.49, p.401 PottmannPottmann && WallnerWallner

 When is a D-form is the convex hull of a space curve? Always

 When is it free of creases? Always OutlineOutline11

 Reconstruction of Convex Polyhedra  Cauchy to Sabitov (to an Open Problem)  Cauchy’s Rigidity Theorem  Aleksandrov’s Theorem  Sabitov’s Algorithm Sabitov’sSabitov’s AlgorithmAlgorithm

 Given edge lengths of triangulated convex polyhedron,  computes vertex coordinates  in time exponential in the number of vertices. SabitovSabitov VolumeVolume PolynomialPolynomial

 2N 2(N-1) 2(N-2) V + a1(l)V + a2(l)V + ... 0 + aN(l)V = 0

 2 Tetrahedron: V + a1(l) = 0  l = vector of six edge lengths



2 2 2 a1(l) = ijk (li) (lj) (lk) /144  Francesca’s formula

 Volume of polyhedron is root of polynomial 22NN possiblepossible rootsroots

(a)

(b) GeneralizedGeneralized PolyhedraPolyhedra

 Polynomial represents volume of generalized polyhedra  any simplicial 2-complex homeomorphic to an orientable manifold of genus ≥ 0  mapped to R3 by continuous function linear on each simplex.  Need not be embeddable: surface can self-intersect. SabitovSabitov ProofProof11

e

v1

d

v2 SabitovSabitov ProofProof22

 vol(P) = vol(P’) -  vol(T),  = M1  ... [many steps] ...  polynomial = 0  unknowns:  edge lengths vector l  V = vol(P)  unknown diagonal d SabitovSabitov ProofProof33

 unknowns:  edge lengths l [given]  V = vol(P) [“known” from volume polynomial]  unknown diagonal d  Try all roots for V, all roots for d: candidates for length of d & dihedral at e.  Repeat for all e.  Check the implied dihedral angles for the convex polyhedron. Open:Open: PracticalPractical AlgorithmAlgorithm forfor CauchyCauchy RigidtyRigidty

Find either C a polynomial-time algorithm, C or even a numerical approximation procedure, that takes as C input the combinatorial structure and edge lengths of a triangulated convex polyhedron, and C outputs coordinates for its vertices. OutlineOutline22

 Folding Polygons  Algorithms  Edge-to-Edge Foldings  Gluing Trees; exponential lower bound  Gluing Algorithm  Examples  Foldings of the Latin Cross  Foldings of the Square  Questions  Transforming shapes? FoldingFolding PolygonsPolygons toto ConvexConvex PolyhedraPolyhedra

 When can a polygon fold to a v8 v7 polyhedron?  v9 v6 “Fold” = close up v10 v5 perimeter, no overlap, no gap : v11 v4 v12 v3  When does a v13 v2 polygon have an Aleksandrov v0 v1 (a) (b) gluing? UnfoldableUnfoldable PolygonPolygon

b

a c FoldabilityFoldability isis “rare”“rare”

Lemma: The probability that a random polygon of n vertices can fold to a polytope approaches 0 as n J 1. PerimeterPerimeter HalvingHalving

v4 x v3 v5

v2 y

v1 Edge-to-EdgeEdge-to-Edge GluingsGluings

 Restricts gluing of whole edges to whole edges.

[Lubiw & O’Rourke, 1996]

VideoVideo

[Demaine, Demaine, Lubiw, JOR, Pashchenko (Symp. Computational Geometry, 1999)] GluingGluing TreesTrees

v4

1/3

v5 v3

e4 e3 e e e5 4 3 e2

v4 v5 v3 2/3

v1 v2 v1 e1 v2 1 1 e5 e2 (a) (b) (c) 1 e1 FoldingFolding ofof nonconvexnonconvex pentagonpentagon

v4

v5 v3

e e e5 4 3 e2 e e 4 3 v e1 4

v1 e1 v2

e5 (a) e2 v 2 v5 v1 v3 ExponentialExponential NumberNumber ofof GluingGluing TreesTrees

α

β

x (a) y

β αβαβ αβ αβαβα (b) x ... y αβαβ αβ αβαβαβ

β β

β α α βαα α β βα x ... (c) α α y αβαβαβ βα α

β β ExponentialExponential NumberNumber ofof GluingGluing TreesTrees GeneralGeneral GluingGluing AlgorithmAlgorithm

 No edge-to-edge assumption.  Implementations: Anna Lubiw, Koichi Hirata (independently)  Exponential-time, dynamic programming flavor. Open:Open: Polynomial-timePolynomial-time FoldingFolding DecisionDecision AlgorithmAlgorithm

Given a polygon P of n vertices, determine in time polynomial in n if P has an Aleksandrov folding, and so can fold to some convex polyhedron. TwoTwo CaseCase StudiesStudies

 The Latin Cross  The Square FoldingFolding thethe LatinLatin CrossCross

 85 distinct gluings 6 5 6  7 4 6 4 7 4 Reconstruct shapes 8 3 7 3 8 3 2 F43/P7 F54/P8 8 2 7 2 8 F19/P9 2 by ad hoc 7 1 1 6 1 1 7 1 techniques 6 2 5 2 6 1 5 8 5 3 4 3 5 8 5  23 incongruent 4 3 4

convex polyhedra 5 44 4 5 4 5 3 43

4 3 6 3 3 5 7 3 F53/P10 2 F3/P11 2 5 F12/P12 2 7 6 1 1 8 1 6 1

7 2 8 1 7 1

8 6 6 2 7 2 8 7 5 6 TheThe 2323 convexconvex polyhedrapolyhedra foldablefoldable fromfrom thethe LatinLatin crosscross

Sasha Berkoff, Caitlin Brady, Erik Demaine, Martin Demaine, Koichi Hirata, Anna Lubiw, Sonya Nikolova, Joseph O’Rourke CubeCube ++ FlatFlat QuadrilateralsQuadrilaterals

1x1

1

2

3 LatinLatin crosscross TetrahedraTetrahedra

1x1

4 5 6 7 8 9 10 LatinLatin crosscross PentahedraPentahedra

1x1

11

12

13 LatinLatin crosscross HexahedraHexahedra

1x1

14

15 16

17 LatinLatin crosscross OctahedraOctahedra

1x1

18 19 20 21 22 23 2323 LatinLatin CrossCross PolyhedraPolyhedra ReconstructionReconstruction ofof TetrahedronTetrahedron

p3 a2 p2

p0 q0 p1=q1

a0 a1 (a) (b)

a 2 p2

q2

p3

p0 q0 p1=q1 a1

(c) a0 LatinLatin crosscross HexahedraHexahedra

1x1

14

15 16

17 OctahedronOctahedron ReconstructionReconstruction

(a) (b) FoldingsFoldings ofof aa SquareSquare

 Infinite continuum of polyhedra.  Connected space CreasesCreases

A AA AA AA AA

A = 0 A = 1/2

As A varies in [0,½], the polyhedra vary between a flat triangle and a symmetric tetrahedron. NNineine CCombinatoriaombinatoriall ClClassesasses ofof PolyhedraPolyhedra foldablefoldable fromfrom aa SquareSquare • Five nondegenerate polyhedra: – Tetrahedra. – Pentahedra: 5 vertices and a single quadrilateral face, – Pentahedra : 6 vertices and three quadrilateral faces (and all other faces triangles). – Hexahedra: 5-vertex, 6-triangle polyhedra with vertex degrees (3,3,4,4,4). – Octahedra: 6-vertex, 8-triangle polyhedra with all vertices of degree 4. • Four flat polyhedra: –A right triangle. –A square. –A 1 P ½ rectangle. –A pentagon with a line of symmetry. Dynamic Web page

A = 0 0 c

3/2 A = 0 = A

1 3 1/2 = A 1 1 c c 0 c 1/2 3/2 2 3 c c 1/2 1 c 1/2 2 c 1/2 1 1 0 c 1 c 0 3/2 2 c 3

A = 1/4

c

1/2

A A A 1 1 A A 3 1 c 1/2 1 0 3 c c

0 3 1/2 0 2 1 0 1 c c c c

3/2 1/2

1 1 A A = A c c

1

A = 1/2 = A A 2 1 2 1 c 1 c c 1/2 A = 1 1 AA 3 c A = 0 to A = .25A = 0 to A = .5 A = .25 to 1/2 3 0 c c 1 1 1 1 0 1 B 2 2

c c

c 1/2

A = 0 = A A = 1 = A

AA 0 1/2 = A 1 A = 1/8 to A = .219A = 1/8 to A = 1 A = .219 to c 1 0 1 c

1

AA A = 0 = A A = 1/8 A = 0 to A = 0 = A

A = 1/8

A = 1/4 = A

A = 1/2 = A 1

A A = 1/2 = A

1 A = 0 = A 0 c c 1 1 0 0 3 2 c c C

1 A = 1/4 = A

D A = 1/4 = A

1 3 1/4 = A c A A 1/2 0 c 1 0 1 c 1 2 0 3 c c c 1/2 1 1 0 2 1 c c A 1 1 MaxMax VolumeVolume PolyhedronPolyhedron

Question due to Joseph Malkevitch , Feb 2002 Open:Open: Fold/RefoldFold/Refold DissectionsDissections

Can a cube be cut open and unfolded to a

polygon that may be 1

refolded to a regular 1/2 tetrahedron?

3 -1/2 [M. Demaine 98] 1/2