  In 1993 Mahadevan and Keller used the Kirchhoff rod theory to predict the
shape of a M\"obius band. Starting from the solution for a square cross-section
(isotropic), they employ numerical continuation in the cross-sectional aspect
ratio in order to approach the solution for a thin strip. Certain smoothly
varying configurations are obtained. More recently in 2007, Starostin and van
der Heijden pointed out that an actual M\"obius band "localizes" into a nearly
flat triangular configuration as the ratio of the strip width to center-line
circumference is no longer small. Accordingly they return to the developable,
thin-plate model of Sadowsky and Wunderlich, obtaining such localized shapes.
In this work we strike a middle ground between these two approaches. We employ
the standard two-director (special) Cosserat model, and we also use the
cross-sectional aspect ratio as a numerical continuation parameter.
Nonetheless, we are able to capture both the smoothly varying shapes and the
localized, nearly triangular configurations. Our point of departure is that we
do not always employ the strength-of-materials formula for the torsional
stiffness - a standard feature of the Kirchhoff theory. Our key observation
comes from the usual Cosserat ansatz in light of developability, suggesting an
increasing torsional stiffness ratio (as well as one increasing bending
stiffness ratio) as the thickness becomes small. We demonstrate that the
closed-loop configurations we obtain are stable with respect to small
perturbations.
