## Abstract

A Möbius band can be formed by bending a sufficiently long rectangular unstretchable material sheet and joining the two short ends after twisting by 180^{°}. This process can be modelled by an isometric mapping from a rectangular region to a developable surface in three-dimensional Euclidean space. Attempts have been made to determine the equilibrium shape of a Möbius band by minimizing the bending energy in the class of mappings from the rectangular region to the collection of developable surfaces. In this work, we show that, although a surface obtained from an isometric mapping of a prescribed planar region must be developable, a mapping from a prescribed planar region to a developable surface is not necessarily isometric. Based on this, we demonstrate that the notion of a rectifying developable cannot be used to describe a pure bending of a rectangular region into a Möbius band or a generic ribbon, as has been erroneously done in many publications. Specifically, our analysis shows that the mapping from a prescribed planar region to a rectifying developable surface is isometric only if that surface is cylindrical with the midline being the generator. Towards providing solutions to this issue, we discuss several alternative modelling strategies that respect the distinction between the physical constraint of unstretchability and the geometrical notion of developability.

## 1. Introduction

In mechanics, an unstretchable material sheet can sustain only deformations corresponding to isometric mappings. Although every surface obtained by bending a flat unstretchable material sheet is developable, a mapping of a prescribed planar region into a developable surface is not necessarily isometric. In spite of this, non-isometric mappings of planar regions into developable surfaces have frequently been used in misguided attempts to describe equilibrium configurations of two-dimensional bodies made from allegedly unstretchable material sheets. The primary purpose of this paper is to clarify the salient issues. Although we focus on Möbius bands, our conclusions apply equally well to orientable twisted strips.

This paper is organized as follows. Some background on the modelling of Möbius bands, as initiated by Sadowsky [1–6], is provided in §2. An expanded discussion of the issues related to the modelling of unstretchable material sheets and the flaws resulting from restricting attention to non-isometric mappings that generate developable surfaces is contained in §3. Basic ideas associated with the parametrization of a surface and the associated notion of a mapping from a planar region to a surface in three-dimensional Euclidean point space are reviewed in §4. The concepts of stretch and curvature, as they relate to the in-plane deformation and out-of-plane bending of a flat material sheet identified with a planar region, are discussed in §5. Developable surfaces and isometric mappings from planar regions to surfaces in space are reviewed in §6. Our main result is established in §7. There we show that, while an isometric mapping always deforms a planar region to a developable surface, a mapping from a planar region to a developable surface need not be isometric. To demonstrate this point, we explicitly work with a class of mappings that have been used frequently to describe deformations of rectangular regions into ribbons and Möbius bands. Any such mapping describes a ruled surface that lies on the rectifying developable of its midline. Most importantly, we establish that the mappings in question are not isometric—except in the trivial case, wherein the mapped surface is cylindrical with the midline being the generator. A concrete illustration of our result is presented in §8. In that illustration, a rectangular region is mapped to a helical ribbon that lies on the surface of a cylinder. In addition to showing that the mapping in question is not isometric, we identify an isometric mapping that takes a parallelogram to the same helical ribbon. The latter mapping is directly relevant to Sadowsky’s [1,2] construction showing that it is possible to bend a rectangular region into a Möbius band without stretching. In §9, we show that a non-isometric mapping from a planar region to a developable surface of the type discussed in §7 can be expressed as the composition of a non-isometric mapping between two planar regions and an isometric mapping to a surface. This is achieved by a straightforward change of independent variables. Finally, a few alternative modelling strategies that respect the distinction between the physical constraint of unstretchability and the geometric notion of developability are discussed in §10.

## 2. Background

In 1930, Sadowsky [1,2] published an appealingly simple constructive proof showing that a rectangular region can be bent, without stretching or tearing, into a Möbius band. In essence, his proof amounts to smoothly joining three helical ribbons bent from parallelograms, with three isosceles trapezoids (two or more of which may degenerate to isosceles triangles), to form a surface with the requisite one-sided spatial connectivity. Recognizing that a strip of paper bent to adopt the shape of his construction would change shape in the absence of externally applied loads, Sadowsky also proposed a variational strategy for determining stable equilibrium shapes of Möbius bands made from unstretchable sheets. That strategy begins by identifying a Möbius band with a developable surface *H* and d*a* denote the mean curvature and element of surface area on *η* is the ratio of the torsion *τ* and the curvature *κ* of *s* is the element of arclength along

Absent from the discussion leading to (2.2) but present in two contemporaneous papers by Sadowsky [3–6] is the crucial provision that *K* of ** γ** of

*l*>0 and half-width

*b*>0 to

**and**

*γ***must satisfy closure conditions**

*t**s*

_{i}∈[0,

*l*], namely points

*s*

_{i}at which

*tη*

^{′}(

*s*)>0 for each (

*s*,

*t*) belonging to

*H*and area element

*da*of

*κ*(1+

*η*

^{2}(

*s*))/2 for the mean curvature, which is simply the restriction to

In what is perhaps the first published paper that recognizes Sadowsky’s contributions to the mechanics of Möbius bands, Wunderlich [7,8] noticed that (2.9) can be used without approximation to yield
*b* and length *l* admits an isometric immersion as a Möbius band in three-dimensional Euclidean point space if and only if *πb*<*l* and, moreover, conjectured that such a strip can be isometrically embedded as a Möbius band in three-dimensional Euclidean point space only if the more restrictive inequality

## 3. Critique

It appears to have gone unnoticed that an unstretchable flat rectangular sheet identified with a region *cannot* in general sustain deformations described by mappings of the form (2.6). In §6 of the present paper, we show that a mapping *τ*=0 at each point along the midline *τ*=0 on a closed curve ** b** of the Frenet frame of

The oversight described above might stem from a misinterpretation of the commonly encountered characterization of a developable surface, which states that any such surface can be mapped isometrically to a planar region. Any surface parametrized in accordance with (2.6) is indeed developable and can, therefore, be mapped isometrically to a planar region. However, setting aside the degenerate case where *τ* vanishes at each point along the midline

This error inevitably undermines any effort to determine stable equilibrium shapes of Möbius bands by minimizing Wunderlich’s functional (2.10). As the isometric flattenings of two surfaces parametrized by different mappings of the form (2.6) for a given region *b* and length *l* but rather to juxtaposing the energies of two differently shaped flat regions mapped into developable surfaces. Hence, such a comparison cannot generally yield useful information towards determining the equilibrium shape of the Möbius band made from an unstretchable sheet of given prescribed shape. Moreover, as using

Our conclusions regarding the family of mappings defined in (2.6) are independent of whether the condition (2.8) needed to ensure that *et al.* [23]. Even so, there is no reason to believe, *a priori*, that the class of mappings (2.6) is rich enough to include all possible equilibrium shapes of such a sheet.

## 4. Parametrization of a surface: deformations and associated gradients

Consider a surface ** x** denotes a generic point on

*s*and

*t*are parameters. A parametrization of a surface is not unique and need not be tied to a physical process. If, however, the surface represents a configuration occupied by a material sheet, then the situation changes markedly. Under such circumstances, each ordered pair (

*s*,

*t*) serves to label a unique material particle of the sheet in some (possibly flat) reference configuration and

In mechanics, the mapping ** r** to denote the independent variable on which the mapping

*s*and

*t*being interpreted as the components of

**with respect to an orthonormal basis. This appears to be consistent with the formulation of many authors who treat the parameters**

*r**s*and

*t*as the coordinates of a point in the planar region relative to an orthogonal Cartesian coordinate system. We denote the vectorial translation spaces associated with

Introducing a planar region ** F**, is called the deformation gradient. Each value of

**is a linear transformation that maps**

*F***, which can be viewed as a linear transformation from**

*G***as the ‘second gradient’.**

*G*## 5. Stretch and curvature

The stretch associated with a deformation ** F** and the second gradient

**.**

*G*### (a) Stretch

We employ the usual Euclidean norms and the corresponding inner products in the domain ** F**, where

**is a linear transformation from**

*R***is a linear transformation (called the stretch tensor) from**

*U*The Cauchy–Green deformation tensor ** C** is defined by

**. It follows from (5.2) that**

*F***is positive-definite and therefore has a square root**

*C***is called the stretch tensor. We now define tensor**

*U***and**

*a***of**

*b***preserves the length of any line segment in**

*R***can be thought of as a linear transformation that rotates vectors in**

*R*A mapping ** a** in

**, and hence the stretch tensor**

*C***, must be the identity tensor. The material sheet identified with**

*U***=**

*C***or, equivalently,**

*I***=**

*U***.**

*I*### (b) Curvature

Given linearly independent elements ** c** and

**of**

*d***and**

*n***by**

*G*

*n***is defined so that**

*G***⋅((**

*a*

*n***)**

*G***)=**

*b***⋅**

*n***[**

*G***⊗**

*a***]=**

*b***⋅(**

*G***⊗**

*n***⊗**

*a***) for all choices of**

*b***and**

*a***in**

*b*The curvatures of the surface ** a** of

*κ*of the surface

*F***is defined by**

*a**κ*

_{1}and

*κ*

_{2}of

*κ*for all

**in**

*a**dκ*/

*d*

**=**

*a***0**, which leads to

*κ*

_{1}and

*κ*

_{2}are the eigenvalues of

*C*^{−1}

**. Additionally, the mean and Gaussian curvatures**

*D**H*and

*K*of

*κ*

_{1}and

*κ*

_{2}by

**by**

*D***of**

*v***=**

*v*

*F***, we find that**

*a**κ*

_{1}and

*κ*

_{2}are also the eigenvalues of

**.**

*L*## 6. Developable surfaces and isometric mappings

A surface is said to be developable if its Gaussian curvature vanishes everywhere. It is common knowledge that the image of a planar region under an isometric mapping must be a developable surface. As a basis for the ensuing discussion, we provide a simple proof of this fact.

Let ** F** must satisfy

**must satisfy**

*F***and**

*a***in**

*b***,**

*a***and**

*b***in**

*c***in**

*a***in the space**

*A***and**

*a***of**

*b***|=|**

*a***|=1 and**

*b***⋅**

*a***=0) and defining elements**

*b***ℓ**and

**of**

*m***ℓ**=

*F***and**

*a***=**

*m*

*F***, we find, as a consequence of (5.11) and (6.5), that the second gradient admits a representation of the form**

*b***,**

*a***,**

*b***and**

*c***in**

*d***being the third-order deformation gradient. Taking advantage of the various symmetries of**

*H***and**

*G***, we find from (6.8) that**

*H***in (6.9), we obtain**

*G***and**

*a***are orthonormal, we recognize the left-hand side of (6.10) as the determinant of**

*b***and, with reference to definition (5.16) of the Gaussian curvature, we conclude that**

*D*We have shown that if a mapping *not* generally isometric, and, therefore, are unsuitable for modelling unstretchable material sheets.

## 7. Mappings from planar regions to rectifying developable surfaces are typically not isometric

Consider a surface *r*_{1}=** r**⋅

*e*_{1}and

*r*

_{2}=

**⋅**

*r*

*e*_{2}are the components of

**with respect to a positively oriented orthonormal basis {**

*r*

*e*_{1},

*e*_{2}} for

**represents a unit speed curve**

*γ**κ*=|

*t*^{′}| and

*τ*=

**⋅(**

*t***×**

*p*

*p*^{′}) are the curvature and torsion of

*η*is the ratio

The deformation gradient ** F** of

*t*^{′}=

*κ*

**,**

*p*

*p*^{′}=−

*κ*

**+**

*t**τ*

**and**

*b*

*b*^{′}=−

*τ*

**. Importantly, by invoking (7.4), we have tacitly assumed that the curvature**

*p**κ*of

*κ*of

*l*], then the unit normal

**and unit binormal**

*p***of the Frenet frame**

*b*The second gradient ** G** of

**to the surface**

*n***and**

*c***being replaced by**

*d*

*e*_{1}and

*e*_{2}, respectively. Assuming, consistent with the previously stipulated invertibility of

*r*

_{2}

*η*

^{′}(

*r*

_{1})>0 for all

**in**

*r*Although the image of a mapping *not* isometric unless it satisfies some highly restrictive conditions that rule out the intended utility of (7.1). To verify the foregoing assertion, we first use (7.5) in (5.2) to give
** C** corresponding to a mapping

**if and only if**

*I**η*vanishes at each point of the midline

*η*, if and only if the torsion

*τ*vanishes at each point of

*κ*.

If *κ* and *τ* obey *κ*>0 and *τ*=0 at each point of ** b** of the Frenet frame of

It is noteworthy that the requirement necessary to ensure that a mapping *τ* vanishes identically, is also necessary to preserve the lengths of the lines of constant *r*_{1} in the region ** b**+

*η*

**is of magnitude**

*t**η*(

*s*)≠0 at some

*r*

_{1}in [0,

*l*], then the corresponding line in the planar region determined by the isometric flattening of

*e*_{2}unless

*τ*≡0.

Randrup & Røgen [12] and Sabitov [14] considered an alternative to the parametrization (7.1) for a surface *r*_{1} in the rectangular region *η*=0 (or equivalently *τ*=0) at each point of *β*(*ατ*−*βκ*)=0, and that mapping (7.12) is isometric if and only if *α*=1,*β*=0 and ** b** is constant. This is in complete agreement with the conclusion drawn above.

In an effort to establish the existence of an embedding of a Möbius band in three-dimensional Euclidean point space, Chicone & Kalton [25] considered a class of developable surfaces with parametrization
** ω** is a unit vector-valued function. The parametrization (7.13) encompasses (7.12), and therefore (7.1) and (7.11), as special cases. It is easy to show that the parametrization (7.13) is isometric if and only if

**is constant and everywhere orthogonal to**

*ω***=**

*t*

*γ*^{′}. This is again in agreement with the central conclusion of the present work.

Because the calculations performed in this section rely on representing vector and tensor fields in terms of their components relative to a fixed rectangular Cartesian basis, it is perhaps natural to wonder whether the results remain true if a curvilinear basis is used instead. Granted that the rectangular region

## 8. A relevant example

### (a) General helical ribbons

We now provide an explicit example of a non-isometric mapping of the form (7.1) that takes the rectangular region *e*_{1},*e*_{2},*e*_{3}} for *e*_{1} and *e*_{2} need not be the base vectors previously used for *e*_{3}, radius *ρ*, pitch angle *θ*, and arclength *l*>0. Invoking the well-known identities
*η*, we find that
** o** denotes the point at the origin. Moreover, using (2.4), (8.2) and (8.3), we find that the combination

**+**

*b**η*

**is constant and given by**

*t**l*>0, we require in (8.5) that

*e*_{3}and radius

*ρ*.

Since a mapping *τ* (and, thus, *η*) vanishes, we conclude from (8.2) that the particular mapping *not* isometric. Indeed, by (7.10) and (8.2), the Cauchy–Green tensor ** C** corresponding to (8.5) has the representation

**for any non-zero pitch angle**

*I**θ*.

The mapping *not* rectangular.

The planar region ** U** in the composition is the gradient of a mapping that non-isometrically deforms

*D*to another planar region, which is identified as

**in the composition is the gradient of a mapping that isometrically deforms the planar region**

*R*

*R*^{⊤}of the second element

**of the composition is also isometric and maps**

*R***corresponding to the mapping**

*U***in (8.7) and has the representation**

*C**r*

_{1},

*r*

_{2})-coordinates of the vertices of

*r*

_{1}in the isometric flattening of

Moreover, the tensor (*Q*** U**)

^{−1}takes

**. The differences between the expression for**

*C**not*an isometric mapping of the rectangular region

### (b) Specific helical ribbon

For illustrative purposes, we take
*b* to be of unit length, in which case it follows that

The mapping ** o** and basis {

*e*_{1},

*e*_{2},

*e*_{3}}, it, respectively, maps the corners (0,−1), (

*π*,−1), (

*π*,1) and (0,1) of

The transformation tensor *Q*** U** defined in (8.10) specializes to

*r*

_{1},

*r*

_{2})-coordinates of the vertices of

*π*−1,−1), (

*π*+1,1) and (1,1). For the particular example (8.15), material fibres oriented along

*e*_{2}are elongated by a factor of

Moreover, the isometric mapping ** o** and basis {

*e*_{1},

*e*_{2},

*e*_{3}},

*π*−1,−1), (

*π*+1,1) and (1,1) of

The procedure of constructing the mappings

## 9. Inducing isometry by a change of variables

For a mapping ** r** defined by a mapping

**and invoking (7.10), we find that the gradient**

*r***of**

*J**η*=

*τ*/

*κ*involving the curvature

*κ*and torsion

*τ*of the midline

*τ*vanishes identically. The discussion of helical ribbons in §8 provides an example of this difference in the particularly simple case where

*η*is constant. Any strategy that seeks to determine a mapping

## 10. Alternative strategies

We have shown that the class of mappings of the form (7.1), which has been used extensively in the literature to model bands and ribbons, is not suitable when the bands and ribbons are made of unstretchable material sheets. We now offer three possible strategies for avoiding the drawbacks of working with such mappings. One of these strategies involves relinquishing the constraint of unstretchability. The other two mimic approaches that are familiar from treatments of internally constrained three-dimensional bodies.

### (a) Removing the constraint of unstretchability

Since mappings of the form (7.1) are not generally isometric and are hence inadequate for the purpose of modelling pure bendings of unstretchable flat material sheets, one possible remedy would involve dropping the isometry requirement in favour of considering stretchable flat material sheets. Among other things, this would require a modification of the elastic energy function to include the change of elastic energy induced by in-plane stretching. The Sadowsky and Wunderlich functionals (2.2) and (2.10) incorporate bending only. As a consequence, these functionals are insensitive to the energy that is required to deform, for example, the rectangular strip ** U** (or, if the flat material sheet is assumed to be isotropic, the principal stretches of

**), in addition to the curvature of the surface. Associated models are more complicated, both kinematically and in regard to constitutive relations. There is, however, another more fundamental and seemingly inescapable problem with this strategy. The class of mappings of the form (7.1) has been used previously because of the belief that it describes the deformations of unstretchable flat material sheets. As we have demonstrated that this belief is unfounded, a justification would therefore be needed to support continued use of such mappings in a context where stretching energy is properly incorporated.**

*U*### (b) Using strictly isometric mappings

Another possible strategy would be to replace (7.1) with the correct and complete class of isometric parametrizations. This strategy is consistent with the spirit of the literature concerning Möbius bands made from unstretchable flat material sheets, as the constraint (5.9) serves as a good approximation for a large class of two-dimensional materials, including those often used to construct model Möbius bands. It has the obvious advantage of leading to a description of great simplicity, in which the energy function depends on the mean curvature only. The task of characterizing the class of three-times continuously differentiable isometric mappings was recently addressed by the present authors and Fosdick [27]. That class neither contains nor is contained in the class (7.1), albeit there exists an intersection consisting of precisely those mappings in (7.1) with zero torsion *τ*, namely the degenerate case where the midline is planar and the conditions needed to describe a Möbius band cannot be met. Moreover, in contrast to the position of a point on a surface

### (c) Using a theory with properly imposed constraints

The third strategy, which is perhaps more familiar to workers in mechanics and which we are currently pursuing independent of the work reported here, would be to develop a theory for internally constrained flat material sheets. In such a theory, unstretchability is treated as a constraint on the class of admissible mappings used to parametrize surfaces. The theory can be developed on general grounds for a class of constraints that includes the constraint (5.9) of unstretchability as a special case. Having derived the general theory, an energy functional that incorporates bending only, and is therefore compatible with the constraint of unstretchability, can be used. The partial-differential equations of equilibrium and the complete set of edge conditions can be derived by computing the first variation of that energy functional subject to the constraint (5.9). The resulting boundary-value problem, which includes reactive forces due to the constraint, needs to be solved in conjunction with the constraint. By contrast, the approach described in the previous subsection amounts to satisfying the constraint *a priori* and substituting the result in the objective functional.

## Data accessibility

The research reported here is purely analytical and thus does not have any associated experimental data.

## Authors' contributions

The authors contributed equally to the research and to writing the manuscript. They both gave final approval for submission.

## Competing interests

Neither of the authors has competing interests.

## Funding

The work of Eliot Fried was supported by the Okinawa Institute of Science and Technology Graduate University with subsidy funding from the Cabinet Office, Government of Japan.

## Acknowledgements

The authors thank Alain Goriely, Ian Murdoch, Roger Fosdick, Giulio Giusteri and Johannes Schönke for reading previous versions of this paper and offering many insightful and constructive suggestions. Yi-chao Chen thanks the Okinawa Institute of Science and Technology for hospitality and generous support during a sabbatical and several subsequent visits.

- Received June 9, 2016.
- Accepted August 2, 2016.

- © 2015 The Authors.

Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.