Applied Geometry: Foldings and Unfoldings

Hellmuth Stachel

[email protected] — http://www.geometrie.tuwien.ac.at/stachel

ISTEC 2016, International Science and Technology Conference July 13–15, Vienna University of Technology Table of contents

1. Unfolding and folding 2. Flexible quad-meshes 3. Curved folding, Example 1 4. Curved folding, Example 2

Funding source: Partly supported by the Austrian Academy of Sciences

July 13: ISTEC 2016, Vienna University of Technology 1/57 1. Unfolding and folding

unfold

fold

There are standard procedures provided for the construction of the unfolding (development, net) of polyhedra or developable surfaces. The result is unique, apart from the placement of the diﬀerent components, and it shows the intrinsic metric of the spatial structure.

July 13: ISTEC 2016, Vienna University of Technology 2/57 E.g., the unfolding of the Oloid

2 3

Technische Universitaet Wien Technische Universitaet Wien Technische Universitaet Wien

Oloid

Oloid Oloid Oloid

Institut fuer Geometrie fuer Institut Institut fuer Geometrie fuer Institut 1

Developable surfaces are ruled surfaces with vanishing Gaussian curvature

July 13: ISTEC 2016, Vienna University of Technology 3/57

4/57 Technology of University Vienna 2016, ISTEC 13: July

cos + 1 cos + 1 9 s s

+ ln = ) ( . s y

1+7cos 7 + 11 2 3 s √

cos + 1 s −

) cos + (1 9 s

√

− = ) ( arccos s x

cos 2 3 2 s ) cos 2 + )(1 cos 2(1 s s √ √

nodn ftecircles: the of unfolding 1

r-eghprmtiaino the of parametrization arc-length

Oloid: Institut fuer Geometrie Institut fuer Geometrie

Oloid Oloid Oloid Oloid

Technische Universitaet Wien Universitaet Technische Wien Universitaet Technische Technische Universitaet Wien Universitaet Technische

2 3 1. Unfolding and folding

unfold

fold

The inverse problem, i.e., the determination of a folded structure from a given unfolding is more complex. In the smooth case we obtain a continuum of bent poses. In the polyhedral case the computation leads to a system of algebraic equations. Also here the corresponding spatial object needs not be unique.

July 13: ISTEC 2016, Vienna University of Technology 5/57 1. Unfolding and folding

Only if the polyhedron bounds a convex solid then the result is unique, due to Aleksandr Danilovich Alexandrov (1941).

In this case, for each vertex the sum of intrinsic angles for all adjacent surfaces is < 360◦ (= convex intrinsic metric).

Theorem: [Uniqueness Theorem] For any convex intrinsic metric there is a unique convex polyhedron.

July 13: ISTEC 2016, Vienna University of Technology 6/57 1. Unfolding and folding

If convexity is not required the unfolding of a polyhedron needs not deﬁne its spatial shape uniquely !

Deﬁnition 1: A polyhedron is called globally rigid if its intrinsic metric deﬁnes its spatial form uniquely — up to movements in space.

July 13: ISTEC 2016, Vienna University of Technology 7/57

1. Unfolding and folding

If convexity is not required the unfolding 7

6 f

f f f 4 5 6 of a polyhedron needs not deﬁne its f spatial shape uniquely !

f2 f2 f 1 f 1 Definition 1: A polyhedron is called globally rigid if its intrinsic metric defines its spatial form uniquely — up to movements in space. A flipping (or snapping) polyhedron admits two sufficiently close realizations e.g., a tetrahedron – by applying a slight force.

July 13: ISTEC 2016, Vienna University of Technology 7/57 1. Unfolding and folding

Definition 2: A polyhedron is called (continuously) flexible if there is a continuous family of mutually incongruent polyhedra sharing the intrinsic metric. Each member of this family is called a flexion.

July 13: ISTEC 2016, Vienna University of Technology 8/57 1. Unfolding and folding

Deﬁnition 2: A polyhedron is called (continuously) ﬂexible if there is a continuous family of mutually incongruent polyhedra sharing the intrinsic metric. Each member of

this family is called a ﬂexion.

8 f

f f

f 5 8 f 6 2

f 1 f f 1 f 1

Even a regular octahedron is ﬂexible — after being re-assembled. The regular pose on the left hand side is called locally rigid.

July 13: ISTEC 2016, Vienna University of Technology 8/57 4 1. Unfolding and folding

B A′ Two particular examples of ﬂexible octahedra C′ C′ where two faces are omitted. Both have an B′ A axial symmetry (types 1 B and 2) A C A′ Below: Nets of the two octahedra. B′ C C B C A′ A′ A′ A′ B

B C′ A B′ C′ ′ C C′ A ′

July 13: ISTEC 2016, Vienna University of Technology 9/57 8 1. Unfolding and folding

B′

According to R. Bricard (1897) t2 there are three types of flexible t1 C′ octahedra (four-sided double- A′ N pyramids). The first two have axis or plane of C symmetry. Those of type 3 admit two flat poses. In each such pose, A B the pairs (A, A′), (B, B′), and (C, C′) of opposite vertices are associated points of a strophoid . S S

P July 13: ISTEC 2016, Vienna University of Technology 10/57 1. Unfolding and folding

B′

According to Bricard’s construction, all bisectors must pass C through the midpoint N of the A N concentric circles.

C′ B′′ The two ﬂat poses of a type-3

B A′ ﬂexible octahedron, when ABC remains ﬁxed.

C′′ A′′

July 13: ISTEC 2016, Vienna University of Technology 11/57 1. Unfolding and folding

This polyhedron called

06 “Vierhorn” is locally

05 01 03 05 01 05 02 rigid, but can ﬂip 08 08 02 02 between its spatial shape and two ﬂat

13 realizations in the

09 09 09 16 12 16 12 planes of symmetry 10 15 16 (W. Wunderlich, C. Schwabe).

At the science exposition “Phänomena” 1984 in Zürich this polyhedron was exposed and falsely stated that this polyhedron is ﬂexible.

July 13: ISTEC 2016, Vienna University of Technology 12/57 1. Unfolding and folding

05 01 06 01 08 05 04 02 07 08 02 03

12 09 16 14 12 09 13 15 11 10 16

the “Vierhorn” and its unfolding

Wolfram MathWorld: A flexible polyhedron which flexes from one totally flat configuration to another, passing through intermediate configurations of positive volume.

July 13: ISTEC 2016, Vienna University of Technology 13/57 2. Flexible quad-meshes

A polygonal mesh is a simply connected subset of a polyhedral surface (sphere-like or with boundary) consisting of (not necessarily planar) polygons, edges and vertices in the Euclidean 3- space. The edges are either internal when they are shared by two faces, or they belong to the boundary of the mesh.

When all polygons are quadrangles, then it is called a quadrilateral surface.

July 13: ISTEC 2016, Vienna University of Technology 14/57 2. Flexible quad-meshes

In modern architecture, most freeform surfaces are designed as polyhedral surfaces – like the Capital Gate in Abu Dhabi, built by the Austrian company Waagner Biro (160 m high, 18◦ inclination)

July 13: ISTEC 2016, Vienna University of Technology 15/57 2. Flexible quad-meshes

f 3

a3 V3 V2 f 4 a4 Under which conditions is this 3 3 f 0 a2 f 2 mesh continuously ﬂexible ? × V4 V1 a1

f 1

July 13: ISTEC 2016, Vienna University of Technology 16/57 2. Flexible quad-meshes

f 3 V3 β2 γ2 I30 1 B1 2 α2 γ 2 δ A1 B γ 2

α A2 1 β1 f 0 f 2 β2 I20 α2 ϕ1 ϕ3 ϕ2 I10 V4 2 I30 I10 δ1 I20 δ δ1 β1 α1 γ1 f 1

Transmission from f 1 to f 3 via the quadrangle f 2 Composition of two spherical four-bar linkages

July 13: ISTEC 2016, Vienna University of Technology 17/57 2. Flexible quad-meshes

A Kokotsakis mesh is continuously ﬂexible f ⇐⇒ 3 f f the transmission from 1 to 3 can in two I30 ways be decomposed into two spherical V3 V2 f four-bar mechanisms, one via V1 and V2, 4 I40 f I20 f the other via V4 and V3. 0 2 V4 V1 I10 The internal edges can be arranged in two f 1 ‘horizontal’ (blue) and two ‘vertical’ edge folds (red).

July 13: ISTEC 2016, Vienna University of Technology 18/57 2. Flexible quad-meshes

I. Planar-symmetric type (Kokotsakis 1932):

The reﬂection in the plane of symmetry of V1 and V4 V3 V2 maps each horizontal fold onto itself while the two vertical folds are exchanged. V1 V4

II. Translational type: V3 There is a translation V1 V4 and V2 V3 mapping a4 V2 the three faces on the right→ hand side→ onto the triple on the left hand side. V V4 1

July 13: ISTEC 2016, Vienna University of Technology 19/57 6 2. Flexible quad-meshes

III: Isogonal type (Kokotsakis 1932):

A Kokotsakis mesh is ﬂexible when at each vertex Vi opposite angles are either equal or complementary, i.e., αi = βi , γi = δi or Vi α = π β , γ = π δ and (n = 4) i − i i − i δi sin α1 sin γ1 sin α2 sin γ2 βi ± ± α γ sin(α1 γ1) sin(α2 γ2) i i − − sin β3 sin γ3 sin β4 sin γ4 = ± ± sin(β3 γ3) sin(β4 γ4) − −

A quad mesh where all 3 3 complexes are of this type is continuously ﬂexible and called Voss surface (Kokotsakis,× Graf, Sauer)

July 13: ISTEC 2016, Vienna University of Technology 20/57 2. Flexible quad-meshes

These are Voss surfaces:

Nadja Posselt: Synthese von zwangläuﬁg beweglichen 9-gliedrigen Vierecksﬂachen Diploma thesis, TU Dresden 2010

July 13: ISTEC 2016, Vienna University of Technology 21/57 2. Flexible quad-meshes

IIIa. Generalized isogonal type: A. Kokotsakis (1932): At all vertices opposite angles are congruent or complementary. G. Nawratil (2010): At least at two of the four pyramids opposite angles are congruent.

IV. Orthogonal type (Graf, Sauer 1931): Here the horizontal folds are located in parallel (say: V4 V3 horizontal) planes, the vertical folds in vertical planes V1 V2 (T-ﬂat).

July 13: ISTEC 2016, Vienna University of Technology 22/57 2. Flexible quad-meshes

V. Line-symmetric type (H.S. 2009): f3 a3 A line-reﬂection maps the pyramid at V1 V3 ρ1 V2 onto that of V4; another one exchanges the f 4 a4 2 3 f pyramids at V and V . 0 a2 f2 V4 ρ2 V1 a1 This includes Kokotsakis’ example of a f1 ﬂexible tessellation.

July 13: ISTEC 2016, Vienna University of Technology 23/57

2. Flexible quad-meshes folding Miura-Ori 1) Miura-ori is a Japanese folding technique (1970?) named after Prof. Koryo Miura, The University of Tokyo (military secret in Russia).

mountain folds full valley folds dashed

It is used for solar panels because it can be unfolded into its rectangular

folding Miura-Ori shape by pulling on one corner only. Unfolded miura-ori; On the other hand it is used as kernel dashs are valley folds, to stiﬀen sandwich structures. full lines are mountain folds

July 13: ISTEC 2016, Vienna University of Technology 24/57 replacements 2. Flexible quad-meshes

180◦

we start with two parallelograms sharing one edge . . .

July 13: ISTEC 2016, Vienna University of Technology 25/57 replacements 2. Flexible quad-meshes

and rotate the right one against the left one through the angle 2δ.

ε1 The lower sides span a plane ε1, 2δ the upper sides a plane ε2 parallel ε1 .

July 13: ISTEC 2016, Vienna University of Technology 26/57 2. Flexible quad-meshes

ε2

ε1

By translations we generate a zig-zag strip of parallelograms between the two parallel planes ε1 and ε2 .

July 13: ISTEC 2016, Vienna University of Technology 27/57 2. Flexible quad-meshes

ε2

ε1

By reﬂection in ε2 we generate a second zig-zag strip of parallelograms sharing the border line in ε2 with the initial strip — and we iterate . . .

July 13: ISTEC 2016, Vienna University of Technology 28/57 2. Flexible quad-meshes

ε2

ε1

By reﬂection in ε2 we generate a second zig-zag strip of parallelograms sharing the border line in ε2 with the initial strip — and we iterate . . .

July 13: ISTEC 2016, Vienna University of Technology 28/57 2. Flexible quad-meshes

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Miura-ori, a Japanese folding technique

July 13: ISTEC 2016, Vienna University of Technology 29/57 2. Flexible quad-meshes

z P There is a hidden local ∗3 symmetry at each P 1 vertex V : P3 ψ The parallelograms P1, P2 with angle α V α ϕ ϕ and the elogations y P3∗, P3∗ of those with ε2 x ψ P angle 180◦ α form 2 − P∗4 a pyramid symmetric with respect to the P4 ﬁxed planes.

July 13: ISTEC 2016, Vienna University of Technology 30/57 2. Flexible quad-meshes

2) Any arbitrary plane quadrangle is a tile for a regular tessellation of the plane. It is obtained from the initial f quadrangle 4 f 2 by iterated 180 -rotations about the • ◦ f 3 C midpoints of the sides — or f 1 by iterated translations of centrally • symmetric hexagon.

A. Kokotsakis, 1932 For a convex f 1 this polyhedral surface Athens is continuously ﬂexible.

July 13: ISTEC 2016, Vienna University of Technology 31/57 2. Flexible quad-meshes

2) Any arbitrary plane quadrangle is a tile for a regular tessellation of the plane. It is obtained from the initial f quadrangle 3 f 2 by iterated 180 -rotations about the • ◦ f 4 C midpoints of the sides — or f 1 r l by iterated translations of centrally • symmetric hexagon.

A. Kokotsakis, 1932 For a convex f 1 this polyhedral surface Athens is continuously ﬂexible.

July 13: ISTEC 2016, Vienna University of Technology 31/57 2. Flexible quad-meshes

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At each ﬂexion all vertices are located on a right circular cylinder.

July 13: ISTEC 2016, Vienna University of Technology 32/57 2 2. Flexible quad-meshes

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July 13: ISTEC 2016, Vienna University of Technology *Kokotsakis*tessellation*Kokotsakis* 33/57

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*Kokotsakis*tessellation*Kokotsakis* 2. Flexible quad-meshes

3 3 2 3 2 4 4 5 5 6 l r l r ρ4 l r lr ρ4 lr r ρ4 r 2 2 2 3 3 4 4 5 1 5 l r ρ4 l r lr ρ4 lr r ρ4 r l r ρ4 − Theorem: l2r 2 2 lr 3 3 r 4 1 4 l 1r 5 lr ρ4 r ρ4 l− r ρ4 − lr lr 2 2 r 3 1 3 l 1r 4 2 4 i j i+j ρ4 r ρ4 l r ρ4 − l− r ρ4 −2 1 − l r 2 − (f 11) lr rρ r 2 l 1r 2 l 1r 3 l 2r 3 l 2r 4 4 − ρ4 − − ρ4 − for i + j 0 (mod 2) f  r 1 1 2 2 2 2 3 3 3 ij = ≡ ρ4 l rρ l r l r ρ l r l r ρ  i j 1 i+j 3 − 4 − − 4 − − 4  − −  l 2 r 2− ρ4(f 11) 1 2 3  f11 l 1 l r l 2r l r 2 l 3r 2 l r 3  − ρ4 − − ρ4 − − ρ4 − for i + j 1 (mod 2)  ≡   ( r = ρ2 ρ1 and l = ρ4 ρ1) This scheme of a 7 7 tesselation mesh indicates ◦ ◦ × which product of helical motions l, r and 180◦- rotations ρ4 maps f11 onto f ij.

July 13: ISTEC 2016, Vienna University of Technology 34/57 2. Flexible quad-meshes

Coaxial helical motions commute = 3 3 2 3 2 4 4 5 5 6 l r l r ρ4 l r lr ρ4 lr r ρ4 r ⇒ 2 2 l2r 3 3 lr 4 4 r 5 1 5 l r ρ4 lr ρ4 r ρ4 l− r ρ4 l2r 2 2 lr 3 3 r 4 1 4 l 1r 5 lr ρ4 r ρ4 l− r ρ4 − Theorem: lr lr 2 r 2 r 3 l 1r 3 l 1r 4 l 2r 4 A ﬂexion of a m n tesselation ρ4 ρ4 − ρ4 − − ρ4 × 2 1 3 2 4 mesh closes with a vertical shift lr rρ4 r l 1r 2ρ l r l 2r 3ρ l r − 4 − − 4 − of k faces r 1 2 2 3 ρ4 l 1rρ l r l 2r 2ρ l r l 3r 3ρ ⇐⇒ − 4 − − 4 − − 4 there exist a, b Z with 2 3 f11 l 1 l 1r l 2r l r 2 l 3r 2 l r 3 ∈ − ρ4 − − ρ4 − − ρ4 − a b l r = d2 , k = a b. π − −

July 13: ISTEC 2016, Vienna University of Technology 35/57 2. Flexible quad-meshes

*Kokotsakis*tessellation*mesh* 6 Closing ﬂexion*Kokotsakis*tessellation*mesh* *Kokotsakis*tessellation*mesh* of a 7 9 tesselation mesh with l − r = d2π, k =5. *Kokotsakis*tessellation*mesh* *Kokotsakis*tessellation*mesh* *Kokotsakis*tessellation*mesh* *Kokotsakis*tessellation*mesh* *Kokotsakis*tessellation*mesh* × *Kokotsakis*tessellation*mesh* *Kokotsakis*tessellation*mesh* *Kokotsakis*tessellation*mesh* *Kokotsakis*tessellation*mesh* *Kokotsakis*tessellation*mesh* *Kokotsakis*tessellation*mesh* How obtainable ? *Kokotsakis*tessellation*mesh**Kokotsakis*tessellation*mesh* *Kokotsakis*tessellation*mesh* *Kokotsakis*tessellation*mesh* *Kokotsakis*tessellation*mesh* *Kokotsakis*tessellation*mesh* numerically: *Kokotsakis*tessellation*mesh* *Kokotsakis*tessellation*mesh* • minimize a distance, or *Kokotsakis*tessellation*mesh* f 11

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57 *Kokotsakis*tessellation*mesh*

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July 13: ISTEC 2016, Vienna University of Technology 36/57

*Kokotsakis*tessellation*mesh* *Kokotsakis*tessellation*mesh*

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Flexible quad-meshes*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* The analogon*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* of a Schwarz boot with trapezoids; a closing ﬂexion of a 17 16 *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* 10 6 *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* × tesselation mesh*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* with l − r = d2π and k =4. *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* 4,1 *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* 4,0 f 16 *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* 5,0 *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* 5,1 *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* f f 62 f 42 f 22 *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* f 142 f 122 f 102 82 f f f 27 *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* f 92 f 72 f 52 32 12 f 17 *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*f 172 6,1 *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*6,0 *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* f f 31 f 11 *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* f f 91 f 71 51 f 18 *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* f 151 f 131 111 f *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* f 81 f 61 f 41 21 f 28 *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* f 141 f 121 f 101 *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*f 161 7,0 *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*7,1 *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*

f 115 f 210 9,0 9,1

10,1

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12,0 12,1 11,0 11,1

July 13: ISTEC 2016, Vienna University of Technology 37/57

*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*

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f f 32

1616 110

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118 *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation 310

*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation11

318

117

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317 f *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation

*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation Theorem:

*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation

Each closing ﬂexion of a m n tesselation*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation mesh is inﬁnitesimally rigid. *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation×

July 13: ISTEC 2016, Vienna University of Technology 38/57 *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation

*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation

*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation f 3. Curved folding, Example 1

A common way of producing small boxes is to push up appropriate planar cardbord forms Φ0 with prepared creases. Below the case of creases along circular arcs c0.

c0 c

Φ Φ0

planar version with circular creases corresponding box with planar creases

July 13: ISTEC 2016, Vienna University of Technology 39/57 f 3. Curved folding, Example 1

As proved by W. Wunderlich (1958), the spatial creases c are again planar and known as meridians of surfaces of revolution with constant Gaussian curvature.

c0 c

Φ Φ0

planar version with circular creases corresponding box with planar creases

July 13: ISTEC 2016, Vienna University of Technology 40/57 3. Curved folding, Example 1

ρ1c′′

c′ y P α c

ρ 2 The Gaussian curvature is deﬁned M2 x

1 as K = κ1κ2 . Hence, ρ K = const. ⇐⇒ y + Ky =0, x = 1 y 2. M1 ′′ ′ − ′ provided that cos α =0. On surfaces of revolution the meridians and parallel circles are the principal curvature lines. Therefore, the signed principal curvatures are

y ′′ cos α κ1 = , κ2 = . −cos α y

July 13: ISTEC 2016, Vienna University of Technology 41/57 3. Curved folding, Example 1

The general solution of y ′′ + Ky = 0 with constant K =0 is for K > 0 : y = a cos s√K + b sin s√K , for K < 0 : y = a cosh s√ K + b sinh s√ K − − with constants a, b R, and ∈ x = p1 y 2 ds. − ′ we can restrict to six cases, up to similarities (Gauß, Minding). Pseudosphere (tractroid)

July 13: ISTEC 2016, Vienna University of Technology 42/57 3. Curved folding, Example 1

1.y 2. y 3. y

x x x

a = 0.78 a = 1.00 a = 1.18 4. y 6. y 5. tractrix M1 x y x

b = 0.3 a = 0.8

1 M2 x

There are six types of meridians to distinguish at the surfaces of revolution with constant Gaussian curvature K =0.

July 13: ISTEC 2016, Vienna University of Technology 43/57 2 3. Curved folding, Example 1

z Let c0 satisfy y0′′+Ky0 =0 and bound a cylindrical patch Φ0 with generators Φ orthogonal to the x-axis a0.

y0 y Theorem: If at a cylindrically bent β pose Φ of Φ0 the boundary c lies in P a plane ε, then it satisﬁes the same c diﬀerential equation as c0. a Proof: T ε y0(s) = y(s) cos β with β < π/2 being the (constant) angle of inclination of the cylinder. x

The axis of c is the meet of ε and the plane of the orthogonal section a, which is the bent counterpart of the original axis a0 of c0.

July 13: ISTEC 2016, Vienna University of Technology 44/57 3. Curved folding, Example 1

Theorem:

Let Φ0 be a planar ‘ruled surface’ with a transversal curve (crease) c0, which separates Φ0 into two patches Φ10 and Φ20. Φ10 Suppose the generators of the ruling remain straight at the bent pose Φ1, Φ2 with a curved edge c between. c0 Then c must be a planar curve.

If all generators of Φ1 and Φ2 are extended to inﬁnity, Φ20 we obtain two torses, which are symmetric with respect to the plane of c.

E.g., take a cone of revolution with a parabolic section c and reﬂect the part opposite to the apex in the plane of c. In Origami this is called reﬂection operation.

July 13: ISTEC 2016, Vienna University of Technology 45/57 3. Curved folding, Example 1

Theorem:

Let Φ0 be a planar ‘ruled surface’ with a transversal curve (crease) c0, which separates Φ0 into two patches 1/κ Φ10 and Φ20. g Φ10 Suppose the generators of the ruling remain straight at α the bent pose Φ1, Φ2 with a curved edge c between. c0 Then c must be a planar curve.

If all generators of Φ1 and Φ2 are extended to inﬁnity, Φ20 we obtain two torses, which are symmetric with respect to the plane of c.

E.g., take a cone of revolution with a parabolic section c and reﬂect the part opposite to the apex in the plane of c. In Origami this is called reﬂection operation.

July 13: ISTEC 2016, Vienna University of Technology 45/57 3. Curved folding, Example 1

Sketch of the Proof: Let κ(s) and τ(s) denote the curvature and torsion of c. In terms of the angle γ1(s) between the osculating plane of c and the tangent plane of the torse Φ1, the geodesic curvature of c w.r.t. Φ1 is

κg = κ cos γ1.

The geodesic curvature κ must be the same w.r.t. Φ2 = γ2 = γ1. g ⇒ −

The angle α between the tangent of c and the generator of Φ1 satisﬁes

cos α : sin α = (τ γ ) : κ sin γ1. − 1′ − The angle α must be the same w.r.t. Φ2 = (τ γ ) = (τ + γ ), hence τ =0. ⇒ − 1′ − 1′

July 13: ISTEC 2016, Vienna University of Technology 46/57 4. Curved folding, Example 2

c0 P0

Unfolding and corresponding spatial form (photos: G. Glaeser)

The spatial form Φ is obtained by gluing together the semicircles with the straight segments. How to model the resulting convex body ?

July 13: ISTEC 2016, Vienna University of Technology 47/57 4. Curved folding, Example 2

Unfolding and corresponding spatial form (photos: G. Glaeser)

The crucial point is here that the ruling is unknown.

M. Kilian, S. Flöry, Z. Chen, N.J. Mitra, A. Sheﬀer, H. Pottmann: Curved Folding. ACM Trans. Graphics 27/3 (2008), Proc. SIGGRAPH 2008.

July 13: ISTEC 2016, Vienna University of Technology 48/57 y 4. Curved folding, Example 2

The helix-like curve c = c1 D0 c10′ A0 • ∪ c2 is a proper edge of Φ; the resulting solid is the convex hull c10 of c. r M0 The semicircular disks are Φ0 • c20 bent to cones with apices A and C. Hence, Φ is a C1-

C0 c20′ B0 compound of two cones and a torse between. A physical model shows: The body has an axis a of • The spatial body with its developable boundary Φ symmetry which connects the is• convex and uniquely deﬁned. midpoint M with the remaining transition point B = D on c.

July 13: ISTEC 2016, Vienna University of Technology 49/57 4. Curved folding, Example 2

B0′ =D0 E10′ c10′ A0 At A and C the surface Φ •can be approximated by a right c10 cone with apex angle 60 . E20 g0 ◦

r The tangent tA to c1 at A M0 • Φ0 is a generator, the osculating c20 E10 plane of c1 a tangent plane of this cone; the rectifying plane C0 c20′ E20′ B0 passes through the cone’s axis.

When g0 meets both straight Consequences: • sides of c0, then g meets c1 Because of the straight segments of c10, the • and c2 at points with parallel developable surface on the left hand side of c1 tangents = coinciding belongs to the rectifying torse of c1. tangent indicatrices.⇒

July 13: ISTEC 2016, Vienna University of Technology 50/57 4. Curved folding, Example 2

E c The tangent at the analogue B0′ =D0 10′ 10′ A0 • point E1 c1 is parallel to the gM ∈ c10 ﬁnal tangent tC of c2. E20 ψ r The subcurves AE1 c1 M0 • ⊂ and E2C c2 have conciding Φ0 ⊂ c20 E10 tangent indicatrices.

C0 c20′ E20′ B0 At a ﬁrst approximation the cone with apex A is speciﬁed as The tangent at the point E2 c2 of transition right cone with apex angle 60 ; •between the cone with apex A and∈ the torse must ◦ c1 is a geodesic circle on this be parallel to t . A cone.

July 13: ISTEC 2016, Vienna University of Technology 51/57 4. Curved folding, Example 2

A0 A′′ A′′′

c0 c1′′′ tA′′′ c1′′

tA′′ E1′′′ E1′′ Approximation 1: B′′ B′′′ c1 is an algebraic curve. a′′′ tA is parallel to the tangent at E2′′′ E2 c2. Analogously, tC is parallel t′ ∈ A to the tangent at E1 c1. This ∈ A′ deﬁnes the axis a of symmetry.

E1′ We notice a contradiction since c′′′ 2 the osculating plane of c1 at B is c1′ C′′′ not orthogonal to BC. B′ July 13: ISTEC 2016, Vienna University of Technology 52/57 2 4. Curved folding, Example 2

Approximation 2 is deﬁ- ned by alined side views of the tangent indicatrices (right) = tB ⇒ a tB a the subcurve AE1 c1 is •a curve of constant slope⊂ . tC =tE1 tC =tE 1 the central torse is a t =t E2 A tE2 =tA cylinder• ,

a translation maps AE1 Left: Tangent indicatrices of c1 and c2 for the ﬁrst • onto the subcurve E2C approximation; no coinciding subcurves! ⊂ c2.

July 13: ISTEC 2016, Vienna University of Technology 53/57 A′′ A′′′ c1′′′

a1′′′ F1′′′ F1′′

E1′′ E1′′′

c1′′ Approximation 2: M′′ a′′′ M′′′ B′′′ B′′ c2′′ The product of the translation A E2 c′′′ 2 and the half-rotation about a maps→ the E2′′ E2′′′ subcurve AE1 onto itself, but in reverse

F2′′ order. F2′′′ a2′′′ Therefore this portion AE1 has an axis a1 of symmetry passing through the C′′ C′′′ F2′ M′ F1′ midpoint F1.

E2′ A′ C′ E1′

a c2′ ′ c1′ B′

July 13: ISTEC 2016, Vienna University of Technology 54/57 A c1

F1 a1

Approximation 2 shows an excellent a accordance with the physical model. B c20 . . . but there remains a contradiction. E2

F2 a2

July 13: ISTEC 2016, Vienna University of Technology 55/57 C 4. Curved folding, Example 2 A

E10′ F10′ A0

F10

ψ N a1 N F1 M0 E10

E1 E20′ B0

Due to the symmetry w.r.t. a1, the midpoint N of AE1 lies on a1. The distances A0F10 and A0E10 are preserved, the triangle ANF1 is congruent to its counterpart A0N0F10 in the unfolding. But NF1 is not (exactly) orthogonal to the tangent of c1 at F1.

July 13: ISTEC 2016, Vienna University of Technology 56/57 Thank you for your attention!

July 13: ISTEC 2016, Vienna University of Technology 57/57 References

[1] A.D. Alexandrov: Convex Polyhedra. Springer Monographs in Mathematics, Springer 2005, ISBN 9783540263401, (ﬁrst Russian ed. 1950).

[2] A.I. Bobenko, I. Izmestiev: Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes. Annales de l’Institut Fourier 58/(2), 447–505 (2008), arXiv:math.DG/0609447.

[3] R. Bricard: Mémoire sur la théorie de l’octaèdre articulé. J. math. pur. appl., Liouville 3, 113–148 (1897).

[4] R. Bricard: Leçon de Cinématique, Tome II, de Cinématique appliqueé. Gauthier- Villars et Cie, Paris 1927.

July 13: ISTEC 2016, Vienna University of Technology 58/57 [5] E.D. Demaine, J. O’Rourke: Geometric folding algorithms: linkages, origami, polyhedra. Cambridge University Press 2007.

[6] H. Dirnboeck, H. Stachel: The Development of the Oloid. J. Geom. Graph. 1, 105–118 (1997).

[7] M. Kilian, S. Flöry, Z. Chen, N.J. Mitra, A. Sheﬀer, H. Pottmann: Curved Folding. ACM Trans. Graphics 27/3 (2008), Proc. SIGGRAPH 2008.

[8] A. Kokotsakis: Über bewegliche Polyeder. Math. Ann. 107, 627–647, 1932.

[9] G. Nawratil, H. Stachel: Composition of spherical four-bar-mechanisms. In D. Pisla et al. (eds.): New Trends in Mechanism Science, Springer 2010, pp. 99–106.

[10] G. Nawratil: Flexible octahedra in the projective extension of the Euclidean 3-space. J. Geom. 14/2, 147–169 (2010).

July 13: ISTEC 2016, Vienna University of Technology 59/57 [11] G. Nawratil: Reducible compositions of spherical four-bar linkages with a spherical coupler component. Mech. Mach. Theory 46/5, 725–742 (2011).

[12] H. Stachel: What lies between the ﬂexibility and rigidity of structures. Serbian Architectural J. 3/2, 102–115 (2011).

[13] H. Stachel: On the Rigidity of Polygonal Meshes. South Bohemia Math. Letters 19/1, 6–17 (2011).

[14] H. Stachel: A Flexible Planar Tessellation with a Flexion Tiling a Cylinder of Revolution. J. Geometry Graphics 16/2, 153–170 (2012).

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[16] H. Stachel: Flexible Polyhedral Surfaces With Two Flat Poses. In Special Issue, B. Schulze (ed.): Rigidity and Symmetry, Symmetry 7(2), 774–787 (2015).

July 13: ISTEC 2016, Vienna University of Technology 60/57 [17] H. Stachel: Two examples of curved foldings. Proceedings of the 17th Internat. Conf. on Geometry and Graphics, Beijing 2016 (to appear).

[18] W. Whiteley: Rigidity and scene analysis. In J.E. Goodman, J. O’Rourke (eds.): Handbook of Discrete and Computational Geometry, 2nd ed., Chapman & Hall/CRC, Boca Raton 2004, pp. 1327–1354.

[19] W. Wunderlich: Aufgabe 300. El. Math. 22, p. 89 (1957); author’s solution: El. Math. 23, 113–115 (1958).

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