The document discusses different possible shapes of Möbius bands based on a discrete lattice model that accounts for both in-plane stretching and out-of-plane bending. The model predicts four characteristic shapes depending on the dimensionless parameter k ̃, which represents the ratio of in-plane stretchability to out-of-plane bendability. Very low stretchability leads to developable bands, while higher stretchability results in more complex twisted shapes with expanded circular centerlines or self-intersecting forms. The shapes transition continuously from developable to highly stretchable as k ̃ increases.

1. Bend, twist, and stretch:
The many shapes of Möbius bands
David M. Kleiman†∗
, Denis F. Hinz†
, Eliot Fried†
†
Mathematical Soft Matter Unit, Okinawa Institute of Science and Technology, Okinawa, Japan 904-0495;
∗
E-mail: david.kleiman@oist.jp
Significance
Most existing investigations of Möbius bands assume developability, leading to one characteristic shape
of such developable bands [3, 4, 5]. However, on the nano and micro-scale, in-plane stretching becomes im-
portant and the assumption of developability is no longer expected to hold. Here, we formulate a discrete
model for the surface of a Möbius band that accounts for in-plane stretching and show that Möbius bands
adopt four characteristic shapes depending on the in-plane stretchability of the material. Our results afford
insights regarding the mechanical properties of topological micro- and nano colloids and can be applied
for future design and manufacturing of novel materials.
Questions
• Is it possible to achieve different equilibrium
shapes of Möbius bands?
• What is the effect of stretching on the equilib-
rium shape of a Möbius band?
The lattice model
To model the surface of a Möbius band, we con-
sider a lattice of equilateral triangles with neigh-
boring points connected through linear and angu-
lar harmonic spring potentials given by
Ul =
1
2
kl(r − r0)2
and Uθ =
1
2
kθ(θ − θ0)2
,
where kl and kθ are spring constants, r is the cur-
rent length of a linear spring, r0 is the equilib-
rium length of a linear spring, θ is the current an-
gle of an angular spring, and θ0 = π is the equi-
librium angle of an angular spring. Nondimen-
sionalization yields a dimensionless spring ratio
˜k =
klr2
0
kθ
that characterizes the in-plane stretch-
ability versus the out of plane bendability of the
lattice. Whereas small ˜k correspond to stretch-
able bands, large ˜k correspond to the limiting
case of developable bands. We generate the lat-
tice surface of the Möbius band through an iso-
metric mapping. The equilibrium shapes are
Figure 1: The lattice model.
obtained through a
conjugate-gradient
minimization algo-
rithm that minimizes
the spring potentials
subject to the geometric
constraints. We con-
duct simulations for a
large range of ˜k and
aspect ratios a = L/w.
Different shapes of Möbius bands
Figure 2: Bands for ˜k = 100
to ˜k = 106
˜k
π 2π 4π 6π 8π
2
2 × 106
2 × 105
2 × 104
2 × 103
2 × 102
2 × 101
a
Figure 2: Möbius bands show four characteristic shapes
depending on the spring ratio: (1) Bands in blue boxes
vary continuously from the developable bands (colored in
red) to more stretchable bands. (2) The band in the red box
is a self intersecting achiral shape with expanded circular
centerline. (3) Bands in the yellow boxes have expanded
circular centerlines. (4) The band in the cyan box has a
non-expanded circular centerline.
10
2
10
4
10
60
100
200
300
˜k
ET/(Akθ)
10
2
10
4
10
60
100
200
300
˜k
EB/(Akθ)
10
2
10
4
10
60
10
20
30
˜k
ES/(Akθ)
a = π
a = 2π
a = 4π
a = 6π
a = 8π
π 2π 4π 6π 8π
10
2
10
4
10
6
a
˜k
0.05
0.1
0.15
0.2
ES/EB
Figure 3: The nondimensional bending and stretching
energies per unit area of a Möbius band depend on its
aspect ratio: Lower aspect ratios have higher bending and
stretching energy per unit area. The shape of Möbius
bands is characterized by the ratio of stretching energy
to bending energy: The colors of the markers in the con-
tour plot correspond to the characteristic shapes observed
in Figure 2.
5 10 15 20 25 30 35 40 45 50 55 60
0.5 1 1.5 2 2.5 3 3.5 4 4.5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.05 0.1 0.15 0.2 0.25 0.3
0.02 0.04 0.06 0.08 0.1 0.12 0.14
0.5 1 1.5 2 2.5
x 10
8
5 10 15 20 25 30 35 40
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.05 0.1 0.15 0.2 0.25
0.02 0.04 0.06 0.08 0.1 0.12
−16000 −14000 −12000 −10000 −8000 −6000 −4000 −2000 0
−8 −7 −6 −5 −4 −3 −2 −1 0
−0.5 −0.4 −0.3 −0.2 −0.1 0
−0.22 −0.2 −0.18 −0.16 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02
−0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0
H2, ˜k = 2 × 106 H2, ˜k = 2 K, ˜k = 2
Developable Bands Stretchable Bands
π
2π
4π
6π
8π
a
Figure 4: Developable bands have two nearly flat trape-
zoidal regions of low bending energy bounded by regions
of high bending energy with higher aspect ratios having
a more evenly distributed bending energy. Stretchable
bands have near zero mean curvature, so bending energy
contributions are due almost entirely to the Gaussian cur-
vature.
Continuum theory
The Canham–Helfrich bending energy functional
for biomembranes is given by
Ebend =
1
2
µ
S
H2
dA + µ
S
K dA, (1)
where µ and µ are the splay and saddle-splay mod-
uli of the membrane and H and K are the mean
and Gaussian curvatures, respectively [1]. Further,
the stretching energy of a membrane is given by
Estretch =
1
2
µs
S
(J − 1)2
dA, (2)
where µs is the stretching modulus and J is the de-
terminant of the deformation gradient.
Connection between continuum and lattice theory
Through a series of numerical experiments and ana-
lytical verification we estimate the relation between
µ, µ, and µs and kl and kθ. Having determined the
moduli, we compute (1) and (2) and relate them to
the discrete energy of the spring potentials. We find
that
EB =
S
(
2
√
3
kθ
3
j=1
κ2
gj +
1
2
6
√
3kθH2
−
√
3kθK)dA,
where κgj is the geodesic curvature at a point with j
identifying the direction of the curve with the direc-
tion of one of the three lattice lines. Similarly, for the
stretching energy
ES ≈
1
2
√
3
2
kl
S
(J − 1)2
dA.
Comparison with (1) and (2) yields µ = 6
√
3kθ,
µ = −
√
3kθ, and µs ≈
√
3
2 kl.
Conclusions
• Our results show that the shape of a Möbius band can be characterized by the ratio of stretching
energy to bending energy related to a single dimensionless parameter ˜k. This parameter can further
be related to continuum elastic moduli [2], allowing for the prediction of the shape of micro- and nano
colloids with Möbius topology based solely on ˜k.
• In a different scenario, it is also possible to infer the value of ˜k on the basis of the characteristic shape
a material adopts when twisted into a Möbius band.
• Our results demonstrate that for sufficiently low values of the aspect ratio a and the dimensionless
parameter ¯k, the Möbius topology is unstable with the model favoring a self-intersecting shape. This
shape is achiral, indicating that chirality is a source of higher deformation energy states in topological
materials.
• Our full results [2] provide insights regarding the connection between topology and surface deforma-
tion energy, allowing for the prediction of cracks and other defects.
References
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possible experiments. Zeitschrift fur Naturforschung.Teil C:
Biochemie, Biophysik, Biologie, Virologie, 28(11):693–703, 1973.
[2] D. M. Kleiman, D. F. Hinz, E. Fried. In preparation. 2014.
[3] L. Mahadevan, J. B. Keller. The Shape of a Mobius Band.
Proceedings of the Royal Society A: Mathematical, Physical and
Engineering Sciences, 440(1908):149–162, 1993.
[4] E. L. Starostin, G. H. M. van der Heijden. The equilibrium
shape of an elastic developable Möbius strip. Proceedings in
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