# Flexible Polyhedral Surfaces with Two Flat Poses

## Abstract

**:**

## 1. Introduction

- Firstly, we analyze the geometry behind Miura-ori.
- Secondly, we study Kokotsakis’ flexible tessellation of convex quadrangles and a particular case with a second flat pose. These two examples include flexible 3 × 3-complexes of quadrangles, so-called quadrangular Kokotsakis meshes [3].
- Finally, we recall the well-known Bricard octahedra of Type 3. They can be characterized by having no global symmetry, and they admit two flat poses. We will emphasize the role of a particular cubic curve, the strophoid.

**Lemma 1.**Let a flexible polyhedral surface be given that passes through two flat poses. Then, at each vertex, where four faces are meeting, the reduced vertex figure has either a plane of symmetry, at least after replacing one or two vertices by their antipodes, or the sides coincide in pairs.

**Proof.**The side lengths of the reduced vertex figure satisfy 0 < α,…,δ < π (see Figure 2). The spherical quadrangle admits two aligned poses if two of the following four equations hold:

- α+β+γ+δ=2π,
- α+β−γ−δ=0,
- α−β+γ−δ=0,
- α−β−γ+δ=0

_{0}B

_{0}, equals the sum of the other three, since this spherical quadrangle admits only the aligned position. In any non-aligned pose of the polygon A

_{0}ABB

_{0}with side lengths β, γ and δ, the distance between the initial point and the endpoint would be shorter than α. For a similar reason, a spherical quadrangle with β+γ+δ−α = 2π cannot be continuously flexible.

**P**

_{3}and

**P**

_{4}in Figure 6).

## 2. Miura-ori

_{1}, and the upper sides span a plane E

_{2}parallel to E

_{1}. We continue and extend the two parallelograms to a zig-zag strip by adding parallelograms, which are translatory congruent alternately to the left-hand or right-hand parallelogram of the initial pair. After this, the complete strip has its lower zig-zag boundary still placed in E

_{1}and the upper one in E

_{2}(see Figure 4c). This remains valid when we fix the plane E

_{1}, but vary the bending angle φ.

_{2}parallel to E

_{1}(Figure 4d) or after translations orthogonal to E

_{1}, the complete Miura-ori flexion is obtained as depicted in (Figure 5). When finally the two initial parallelograms become coplanar due to a bending angle of 180°, Miura-ori reaches a totally folded pose, which is again flat.

_{1}and placed in horizontal planes. They are aligned in the stretched initial pose as depicted in Figure 3. The other folds are located in mutually parallel vertical planes, since their edges are obtained by iterated reflections in planes parallel to E

_{1}. We call these folds “vertical”. While the edges of each horizontal fold alternate between valley and mountain fold, all edges of a vertical fold are of the same type, either valley or mountain folds.

_{1}can vary without restricting the flexibility; analogously, the distances between the planes through the horizontal folds need not be the same everywhere).

**P**

_{1},…,

**P**

_{4}be the four parallelograms meeting at V. We start with the motion of

**P**

_{1}: the two edges of

**P**

_{1}with the common endpoint V rotate within the respective fixed planes, such that the included interior angle α < 90° remains constant (compare with Figure 4a). The reflection in the horizontal plane E

_{2}maps

**P**

_{1}onto the second parallelogram

**P**

_{2}. It has the same interior angle α at V and moves like

**P**

_{1}.

**P**

_{3}and

**P**

_{4}(with interior angle 180°−α) beyond the fixed vertical plane by the unit length. This gives two additional parallelograms ${\mathbf{P}}_{3}^{*}$ and ${\mathbf{P}}_{4}^{*}$ with the interior angle α at V. Each shares a “vertical” edge with

**P**

_{1}or

**P**

_{2}, respectively. Hence, the reflection in the fixed vertical plane must map

**P**

_{1}onto ${\mathbf{P}}_{3}^{*}$ and P

_{2}onto ${\mathbf{P}}_{4}^{*}$.

**P**

_{3},

**P**

_{4}with congruent interior angles at V are replaced by their “horizontal elongations” ${\mathbf{P}}_{3}^{*}$ and ${\mathbf{P}}_{4}^{*}$, respectively, we obtain a pyramid of four congruent parallelograms

**P**

_{1},

**P**

_{2}, ${\mathbf{P}}_{3}^{*}$, ${\mathbf{P}}_{4}^{*}$ with apex V. This pyramid flexes, such that it remains symmetrical with respect to the planes of the horizontal fold and the vertical fold passing through V (compare with Lemma 1).

_{2}and with the [yz]-plane spanned by the vertical fold passing through V (see Figure 6). Let 2φ and 2ψ be the bending angles between consecutive segments of the horizontal and vertical folds, respectively. Thus, the sides of

**P**

_{1}have the direction vectors:

**P**

_{1}is located in the vertical [yz]-plane; the corresponding (half) bending angles are φ = 0° and ψ = 90° − α. The other limit is the totally folded position with

**P**

_{1}in the [xy]-plane, φ = α and ψ = 90°.

**P**

_{1}:

**Theorem 2.**During the self-motion of Miura-ori, the horizontal folds remain in mutually-parallel planes. The same holds for vertical folds, and their planes are orthogonal to the planes spanned by the horizontal folds. The dihedral bending angles 2γ along the edges of the horizontal folds are everywhere the same, as well as the angles 2γ for the vertical folds. They satisfy Equation (2), while φ and ψ are related by Equation (1).

## 3. Kokotsakis’ Flexible Tessellation

**P**

_{1}. By iterated rotations about the midpoints of the sides, we obtain a well-known regular tessellation of the plane (Figure 7). We can generate the same tessellation also in another way: when we glue together two adjacent quadrangles, we obtain a central-symmetric hexagon (blue shaded in Figure 7). Now, the same tessellation arises when this hexagon undergoes iterated translations, as indicated in Figure 7 by the red arrows.

**P**

_{1},…,

**P**

_{4}be four pairwise congruent faces with the common vertex V

_{1}(shaded area in Figure 8). They form a four-sided pyramid with apex V

_{1}, and the four interior angles at V

_{1}are respectively congruent to the four interior angles of a quadrangle. Our pyramid is flexible, provided the fundamental quadrangle is convex; otherwise, one interior angle at V

_{1}would be greater than the sum of the other three interior angles, so that the only realization is flat.

_{1}, ρ

_{2}, ρ

_{3}or ρ

_{4}, which exchanges the two faces. Therefore, e.g.,

**P**

_{2}= ρ

_{1}(

**P**

_{1}). The axis of ρ

_{1}is perpendicular to the common edge V

_{1}V

_{2}and located in a plane that bisects the dihedral angle between

**P**

_{1}and

**P**

_{2}.

**P**

_{1}, this is mapped via

**P**

_{2},

**P**

_{3}and

**P**

_{4}onto itself. This implies:

_{1}exchanges not only

**P**

_{1}with

**P**

_{2}, but transforms the pyramid with apex V

_{1}onto a congruent copy with apex V

_{2}sharing two faces with its preimage. This is the area hatched in green in Figure 8, right. Analogously, ρ

_{4}generates a pyramid with apex V

_{4}sharing the faces

**P**

_{1}and

**P**

_{4}with the initial pyramid. Finally, there are two ways to generate a pyramid with apex V

_{3}. Either we transform ρ

_{2}by ρ

_{1}and apply ρ

_{1}○ ρ

_{2}○ ρ

_{1}, which exchanges V

_{2}with V

_{3}, or we proceed with ρ

_{4}○ ρ

_{3}○ ρ

_{4}, which exchanges V

_{4}with V

_{3}. Thus, we obtain two mappings, ρ

_{1}○ ρ

_{2}and ρ

_{4}○ ρ

_{3}, with V

_{1}↦ V

_{3}, which are equal by virtue of Equation (3). Hence, each flexion of the initial pyramid with apex V

_{1}is compatible with a flexion of the 3 × 3 complex of quadrangles. This can be extended to the complete tessellation (in [6], it is shown that this tessellation is a particular case of the line-symmetric Type 5 of flexible Kokotsakis meshes).

_{1}is not flat, then the axes of the 180°-rotations ρ

_{1},…,ρ

_{4}are pairwise skewed; the common perpendicular of any two of them is unique. Equation (3) implies that the axes of the four rotations have a common perpendicular a. Hence, the motions ρ

_{1}○ ρ

_{2}and ρ

_{3}○ ρ

_{2}= ρ

_{4}○ ρ

_{1}are helical motions with a common axis a. All vertices of the flexion have the same distance to a; they lie on a cylinder Ψ or revolution (in [11], T. Tarnai addresses another relation between planar tessellations and cylindrical shapes; he shows how semiregular tessellations can be folded into cylindrical shells).

**P**

_{3}and

**P**

_{4}are glued together, we obtain a line-symmetric skewed hexagon, one half of our initial pyramid with apex V

_{1}; the half-turn ρ

_{3}maps this hexagon onto itself (Figure 8). We can generate the complete flexion by applying iterated helical motions ρ

_{1}○ ρ

_{2}and ρ

_{3}○ ρ

_{2}on this hexagon.

**Theorem 3.**Any flexion of the Kokotsakis tessellation is obtained from the line-symmetric hexagon consisting of the two planar quadrangles by applying the discrete group of coaxial helical motions generated by ρ

_{1}○ ρ

_{2}and ρ

_{3}○ ρ

_{2}. In the initial flat pose, these generating motions are the translations applied to a centrally symmetric hexagon, thus generating the regular tessellation in the plane.

_{1}admits a bifurcation between two continuous motions of the pyramid. Hence, the complete quad mesh admits two self-motions (Figure 9). When starting with a trapezoid

**P**

_{1}(compare with Figure 14 in [12]), one of these self-motions is trivial: it results from mutually-independent bending along aligned edges of the initial pose.

_{1}…V

_{4}intersects the circumcylinder Ψ along an ellipse E. Given a convex quadrangle V

_{1}…V

_{4}, there is a pencil of conics passing through the four vertices. This pencil includes an infinite set of ellipses, which is bounded by parabolas or pairs of parallel lines. Hence, we can specify an ellipse ε out of this pencil and choose one of the two right cylinders passing through ε.

_{1}and ρ

_{4}; their axes pass through the midpoints of the sides V

_{1}V

_{2}and V

_{1}V

_{4}, respectively, and intersect the cylinder’s axis a perpendicularly (see Figure 10). The common normal of the lines V

_{2}V

_{3}and a is the axis of ρ

_{1}○ρ

_{2}○ρ

_{1}; that between V

_{3}V

_{4}and a is the axis of ρ

_{4}○ρ

_{3}○ρ

_{4}. Iterations of these half-turns transform our initial face

**P**

_{1}into all faces of the flexion.

**Theorem 4.**When the basic quadrangle of the flexible Kokotsakis’ tessellation is cyclic, i.e., has a circumcircle, it admits a second flat pose: all quadrangles are packed into a single circumcircle. However, this fails for real-world models, because of self-intersections.

**Proof.**In the case of a cyclic quadrangle, the pencil of circumscribed conics includes one circle. There is a unique right cylinder passing through this circle, and the corresponding axes of half-turns ρ

_{1},…,ρ

_{4}are coplanar. The complete flexion is flat; the circumcircles of all faces coincide (compare with Figure 12). Similar to Miura-ori, this tessellation mesh can be “packed” into a very small size, such that it finds a place in the circumcircle of one quadrangle, at least “theoretically”, i.e., without paying attention to self-intersections. However, a real-world model consisting of at least 3 × 3 quadrangles cannot approach such a pose, since the faces have to be placed one upon the other. Then, there is a collision between the “edges” spanning over distant levels and the vertices of quadrangles placed between them. □

## 4. Bricards’ Octahedra of Type 3

_{1}and t

_{2}. Furthermore, the strophoid S is the locus of points X with the property that one angle bisector of the lines XP and XP′ passes through the node N. This holds simultaneously for all pairs (P, P′) of associated points of S.

**Theorem 5.**For any strophoid S, each triple of associated pairs of points (A, A′), (B, B′) and (C, C′) defines a flat pose of a flexible octahedron. This octahedron has a second flat pose.

**Proof.**In order to obtain the second flat pose, let us keep the face ABC fixed, while the opposite triangle A′B′C′ is replaced by a congruent copy A″B″C″. Both flat poses must show respectively equal distances AB′ = AB″, AC′ = AC″,…, CB′ = CB″. We show that the reflection in the line AB maps C′ onto C″, the reflection in the line AC maps B′ onto B″ and the reflection in the line BC maps A′ onto A″ (see Figure 14). Firstly, these reflections preserve the distances to the points of the fixed triangle. Secondly, since both pairs of lines, (AB, AB′) and (AC, AC′), are symmetric with respect to AM, the angles ∢B′AB″ = 2 ∢B′AC and ∢C′AC″ = 2 ∢C′AB are equal. Hence, there is a rotation about A, which sends the side B′C′ onto B″C″ and preserves the length. The same holds for the other sides A′B′ and A′C′. □

## 5. Conclusions

## Conflicts of Interest

## References and Notes

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**Figure 1.**This polyhedron called “Vierhorn” is locally rigid, but snaps between its spatial pose (

**b**) and the two flat realizations (

**a,c**). Dashes in the unfolding (

**d**) indicate valley folds.

**Figure 2.**The interior angles at the vertex V serve as side lengths of the spherical four-bar A

_{0}ABB

_{0}, which controls the dihedral angles φ

_{1}and φ

_{2}during the flexion.

**Figure 6.**The two-fold local symmetry at each vertex V becomes visible when the edge between P

_{3}and P

_{4}is extended beyond V.

**Figure 8.**The flexion of a four-sided pyramid can be extended to a flexion of the complete tessellation.

**Figure 10.**There is an infinite set of ellipses passing through the vertices of a convex quadrangle.

**Figure 11.**This is an infinitesimally rigid pose of a flexible tessellation mesh [12].

**Figure 13.**In the flat pose, the pairs of opposite points of Bricard’s Type-3 flexible octahedron are associated with a strophoid S.

© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Stachel, H.
Flexible Polyhedral Surfaces with Two Flat Poses. *Symmetry* **2015**, *7*, 774-787.
https://doi.org/10.3390/sym7020774

**AMA Style**

Stachel H.
Flexible Polyhedral Surfaces with Two Flat Poses. *Symmetry*. 2015; 7(2):774-787.
https://doi.org/10.3390/sym7020774

**Chicago/Turabian Style**

Stachel, Hellmuth.
2015. "Flexible Polyhedral Surfaces with Two Flat Poses" *Symmetry* 7, no. 2: 774-787.
https://doi.org/10.3390/sym7020774