# Construction of Unknotted and Knotted Symmetric Developable Bands

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## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Construction

#### 2.1.1. Scaffolds

#### 2.1.2. Connecting Curve of a Scaffold

#### 2.1.3. Maximum Tilt Angle

#### 2.1.4. Planar Development and Maximum Width of a Band

#### 2.2. Elastic Bending Energy of a Band

#### 2.3. Force Needed to Tighten an Unknotted Band on Its Supporting Scaffold

## 3. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Sequence of intermediate steps in isometrically threading a flat rectangular strip (top) into a Möbius band (also shown in Figure 8) through a scaffold of three cylinders. The properties that a scaffold must have to allow for the construction of a band are the primary focus of this paper. Starting from the flat configuration, the first cylinder is placed on the strip making contact at the line where parallelogram 1 (yellow and denoted by “1” on the strip) begins. The band is next wrapped around the first cylinder until parallelogram 1 conforms fully to the cylinder. The second cylinder is next placed on the band, making contact along the line where parallelogram 2 begins and the band is wrapped around the second cylinder, until parallelogram 2 conforms fully to the cylinder. The third cylinder is finally put into place and parallelogram 3 is wrapped over it in the same fashion as parallelograms 1 and 2 were wrapped around the first and second cylinders. This aligns the vertical edge of the strip to the right of parallelogram 3 with the edge of the strip to the left of parallelogram 1 and allows for the strip to be smoothly joined into a band, thereby creating a flat trapezoid from the portions of the strip to the left and right of parallelograms 1 and 3.

**Figure 2.**(Top) N line segments (here shown for the case of $N=3$) equidistantly distributed on a base circle of radius r. These lines serve as the axes of the cylinders once they have been rotated around radial axis of the base circle C by the angle $\alpha $ such that the shortest distance between associated lines equals $2R$. (Bottom) The basic construction after cylinders have been added for $N=3$, 4, 5 (scaffolds $\left(i\right)$, $\left(ii\right)$, $\left(iii\right)$, respectively) and $k=1$ and for $N=5$ cylinders (scaffold $(iv)$) and $k=2$. The indicated angle $\omega $ depends on the number N of cylinders as well as k and is given by $\omega =2\pi k/N$. The integer $k\ge 1$ specifies that a cylinder has a common point (in red) with each of the two “k-th” neighboring cylinders. For $k=1$, we obtain convex scaffolds (scaffolds $\left(i\right)$, $\left(ii\right)$, $\left(iii\right)$). For a nonconvex scaffold ($k>1$) to be possible (scaffold $\left(iv\right)$) two conditions must be satisfied: $k<N/2$ and k is not a divisor of N.

**Figure 3.**Side view of the basic construction for $N=3$ displaying the tilt angle $\alpha $, the length L of the cylinders (granted that they are extended until their common endpoints, shown in red, meet), and the radius R of the cylinders.

**Figure 4.**The connecting curve $\Gamma $ (in blue) for $N=3$. Also shown is the incidence angle $\gamma $ at which $\Gamma $ meets the cylinder; $\gamma $ also appears as half of the angle between the axes of two cylinders which have a common point (in red). Also depicted is the angle $\varphi $ for which $\Gamma $ is covering each cylinder and the length $\Lambda $ for which a helical part of $\Gamma $ traverses along the cylinder axis.

**Figure 5.**(

**Left**) Illustration of the length a of the shortest path between a contact point of two cylinders (in red) and the connecting curve $\Gamma $ (in blue) for the case $N=3$. The length a bounds the possible half-width w of the band as discussed in Section 2.1.4. (

**Right**) For the maximum tilt angle ${\alpha}_{m}$ of the scaffold, the length a vanishes and the curve $\Gamma $ intersects all contact points.

**Figure 6.**The maximum tilt angle ${\alpha}_{m}$ (see Figure 3) for which the connecting curve $\Gamma $ can be constructed as a function of $\omega (N,k)$ resulting from the solution of the transcendental Equation (12). Indicated are all scaffolds ${N}_{k}$ with $N\le 9$ and the highest possible value for ${\alpha}_{m}$ occuring at ${\omega}^{*}\approx 124.{4393}^{\circ}$ (with ${\alpha}_{m}^{*}\approx 24.{9894}^{\circ}$) which is close to ${\omega}^{{3}_{1}}={120}^{\circ}$ (with ${\alpha}_{m}^{{3}_{1}}\approx 24.{9443}^{\circ}$). Considering nonconvex scaffolds, it is easy to find a case with ${\alpha}_{m}>{\alpha}_{m}^{{3}_{1}}$, for example, ${14}_{5}$ (with ${\alpha}_{m}^{{14}_{5}}\approx 24.{9486}^{\circ}$). Since every real number can be approximated arbitrarily close by a rational number, one can find a nonconvex scaffold with $\omega $ being arbitrarily close to ${\omega}^{*}$.

**Figure 7.**Example of a nonconvex scaffold with a knotted, non-self-intersecting band. The scaffold is identical to that in Figure 2 ($iv$) with $N=5$ and $k=2$. The corresponding ${5}_{2}$ band is a $(5,2)$ torus knot and a nonorientable Möbius band.

**Figure 8.**A ${3}_{1}$ band (

**left**) and its flat rectangular development (

**right**). The trapezoids (grey) in the development remain flat in the deformed state while the parallelograms (yellow) lie on the curved cylinder surfaces. The midline lengths of the trapezoids (${l}_{\mathrm{line}}$) and parallelograms (${l}_{\mathrm{helix}}$) and the half-width w of the band are indicated. The opening cut of the band (emphasized by a dashed line) is located at the middle of one of the trapezoids.

**Figure 9.**A ${7}_{2}$ knotted band with maximal band width for the prescribed $\alpha $, extending the edge out to the common points. The band clearly self-intersects along a closed curve.

**Figure 10.**(

**Left**) Illustrating the length b of the shortest path between the center of the construction and the connecting curve $\Gamma $ (in blue) for the case $N=3$. The half-width w of the band must not exceed the length b otherwise the band self-intersects. (

**Right**) For the special angle $\alpha ={\alpha}_{c}$ the case $a=b$ occurs and the band can extend from the center out to the contact point, see Section 2.1.3 for the definition of a.

**Figure 11.**The relevant forces on a cylinder for a ${3}_{1}$ band construction if vertical loads $\pm \mathit{f}/3$ are applied at the ends of the cylinder through a heavy platform and the floor (contacting at the position of the cyan balls) leading to a force couple. This results in a force couple of magnitude ${\mathit{f}}_{\phantom{\rule{-1.25pt}{0ex}}c\phantom{\rule{0.97214pt}{0ex}}}$ at the common points of the cylinder with the other two cylinders (red balls). The force ${\mathit{f}}_{\phantom{\rule{-1.25pt}{0ex}}c\phantom{\rule{0.97214pt}{0ex}}}$ is then split up into normal (${\mathit{f}}_{{n}_{1,2}}$) and tangential (${\mathit{f}}_{{t}_{1,2}}$) components to the cylinder surfaces at the common points. The sum of the forces ${\mathit{f}}_{{t}_{1,2}}$ leads to a radial force ${\mathit{f}}_{r}$ acting on the cylinder which needs to be balanced by the band.

**Figure 12.**A band wrapped around a cylinder. The magnitude of the pulling forces ${\mathit{f}}_{{p}_{1}}$ and ${\mathit{f}}_{{p}_{2}}$ necessary to tighten the band is given in (29).

**Figure 13.**Photographs of a table made from three aluminum cylinders (with 3D-printed spherical white caps), an acrylic plate, and a ${3}_{1}$ band (a synthetic printing canvas). (Top left) The disassembled parts. (Top right) The assembled table. (Bottom) The table in relation to the surrounding chairs. Each cylinder has a radius of $R=50$ mm, a total length (without the spherical caps) of ${L}_{\mathrm{tot}}=800$ mm, and a mass (including the spherical caps) of 3.7 kg. The band has a length of $l=976$ mm, a half-width of $w=50$ mm, and a thickness of 0.28 mm. The short ends of the synthetic band were connected using an ultrasonic welder. The acrylic plate has a diameter of 1 m, a thickness of 18 mm, and a mass of 17 kg. The values for R and l are chosen to yield the value $\alpha ={18}^{\circ}$ for the tilt angle. The total height of the table is 366 mm.

${\mathit{N}}_{\mathit{k}}$ | ${3}_{1}$ | ${4}_{1}$ | ${5}_{1}$ | ${5}_{2}$ | ${6}_{1}$ | ${7}_{1}$ | ${7}_{2}$ | ${7}_{3}$ | ${8}_{1}$ | ${8}_{3}$ | ${9}_{1}$ | ${9}_{2}$ | ${9}_{4}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\alpha}_{m}{(}^{\circ})$ | 24.94 | 22.56 | 19.63 | 23.97 | 17.13 | 15.11 | 24.00 | 22.38 | 13.46 | 24.71 | 12.12 | 21.06 | 21.01 |

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**MDPI and ACS Style**

Schönke, J.; Grunwald, M.; Fried, E.
Construction of Unknotted and Knotted Symmetric Developable Bands. *Symmetry* **2021**, *13*, 431.
https://doi.org/10.3390/sym13030431

**AMA Style**

Schönke J, Grunwald M, Fried E.
Construction of Unknotted and Knotted Symmetric Developable Bands. *Symmetry*. 2021; 13(3):431.
https://doi.org/10.3390/sym13030431

**Chicago/Turabian Style**

Schönke, Johannes, Michael Grunwald, and Eliot Fried.
2021. "Construction of Unknotted and Knotted Symmetric Developable Bands" *Symmetry* 13, no. 3: 431.
https://doi.org/10.3390/sym13030431