1. Background Our Model Results Conclusions
Bend, twist, and stretch: Shape transitions of
non-developable Möbius bands
David M. Kleiman1,2, Denis F. Hinz2, Eliot Fried2
1Department of Mathematics and Statistics, McGill University, Montréal, QC H3A 2K6,
Canada
2Mathematical Soft Matter Unit, Okinawa Institute of Science and Technology, Okinawa,
Japan
david.kleiman@oist.jp
August 11, 2014
David M. Kleiman (McGill) Möbius band equilibria August 11, 2014 1 / 7

2. Background Our Model Results Conclusions
Introduction
What is a Möbius band?
The Möbius band is the canonical example of a one-sided (or
non-orientable) surface.
The band has long been studied by mathematicians, but also by
artists like M.C. Escher.
Recently, the Möbius band has found applications in chemistry and
materials science.
brief communications
ngle crystals
u m can b e coaxe d into a novel topology.
almost equal to zero because the edge magnetism directly
ses from the localized edge states while no edge states are
med when the width of the ribbon is as small as 2. With
increase in the ribbon width, edge states are developed
the decrease in the TMM after some critical width while it
never decreases with the width in the ZGNRs. Such mag-
netic properties that are different from the ZGNRs may make
the Möbius strips graphene building blocks in spintronic de-
FIG. 4. ͑Color online͒ The AMMs of
edge carbons are shown in ͑a͒. The
spin polarized density of states and
spin density for Möbius molecule
formed by graphene nanoribbons ͑with
Nz =8͒ are shown in ͑b͒ and ͑c͒,
respectively.
103-3 Wang et al. Appl. Phys. Lett. 97, 123103 ͑2010͒
ba
d
c
200nm 100nm
Trace
50nm
LETTERS NATURE NANOTECHNOLOGY DOI: 10.1038
Tanda et al. 2002 Wang et al. 2010 Han et al. 2010
David M. Kleiman (McGill) Möbius band equilibria August 11, 2014 2 / 7

3. Background Our Model Results Conclusions
The shape problem
What is the shape of a Möbius band?
The problem of finding the shape of a Möbius band was first
formulated by Sadowsky in 1930.
Mahadevan and Keller (1993) and then Starostin and van der Heijden
(2007) found shapes of unstretchable Möbius bands numerically.
Little is known about the effect of in-plane stretching on the shapes
of these peculiar objects.ETTERS
a a
b
c
d
b
LETTERS
a a
b
c
d
e
b
c d
e f
LETTERS
a a
b
c
d
e
b
c d
Starostin & van der Heijden 2007
David M. Kleiman (McGill) Möbius band equilibria August 11, 2014 3 / 7

4. Background Our Model Results Conclusions
Our approach
A discrete model
We use a discrete model that allows us to vary the in-plane
stretchability of the bands.
(width, stretchability) → shape
David M. Kleiman (McGill) Möbius band equilibria August 11, 2014 4 / 7

5. Background Our Model Results Conclusions
Shapes of stretchable Möbius bands
a = ⇡ a = 2⇡ a = 4⇡
←−
More
Stretchable
−→
Narrower
π 2π 4π 6π 8π
10
−6
10
−4
10
−2
10
0
a
k
0.05
0.1
0.15
0.2
0.25
ΨS/ΨB
David M. Kleiman (McGill) Möbius band equilibria August 11, 2014 5 / 7

6. Background Our Model Results Conclusions
Conclusions
Synthesis of Möbius bands
Möbius bands of stretchable material are easier to make.
If the material is too stretchable, then Möbius bands could be
impossible to synthesize.
Möbius bands could be useful to tune the bulk properties of
suspensions.
Current work
Simulating dense suspensions of Möbius bands.
Simulating sedimenting Möbius bands.
Sedimentation experiments using 3D printed Möbius bands.
David M. Kleiman, Denis F. Hinz, Eliot Fried. Bend, twist, and stretch: Shape transitions of non-developable Möbius bands.
To be submitted.
David M. Kleiman (McGill) Möbius band equilibria August 11, 2014 6 / 7

7. Background Our Model Results Conclusions
Conclusions
Thanks for your attention! Questions?
David M. Kleiman (McGill) Möbius band equilibria August 11, 2014 7 / 7