Short Möbius Bands in R³

Reference: B. Halperin and C. Weaver, "Inverting a Cylinder through Isometric Immersions and Isometric Imbeddings," Trans.Amer. Math. Soc. 230 (1977), 41-70.

   Question . How short can a flat Möbius band in R³ be?

Denote by L the length and by W the width (height) of an abstract Möbius band. For an isometric copy of it in R³, L and W are the measurements of its abstract original. Only proportions matter here, so we call a band short if its length per unit width (that is, L/W) is small.

It seems obvious that an abstract Möbius band with, say, L=W, is too short to be isometrically imbedded in R³--there just isn't enough length to accomplish the half-twist. But we have seen that such an imbedding is possible for a longer band. So the question above becomes:  Where is the cut?

One way to shorten the example given in the page above is to simply increase W, leaving L alone. This example had L=310=30 and W=3.4, hence L/W about 8.8. A top of view of this band is shown at the left below. To its right is the corresponding band with W increased to 8, hence L/W=3.75.

 

This shorter band is still imbedded, since its planar regions are in three different parallel planes. To shorten it further, we will soon have to reduce the radii of the cylinders used to bend the band. It should be clear that in limitinga case (not an imbedding) the model will look like an equilateral triangle--covered three times.

A model near the limit can be produced from a piece of paper as follows:

Fold the outer right triangles inward, then make the final fold--the arrowed edges now match, so the result represents a Möbius band.

From the figure, L=3 and W=3, so L/W=3 in the limit. Thus any L/W>3 can be realized by an isometric imbedding.

It's not certain who first found a flat Möbius band in R³, but Halperin and Weaver in the referenced paper proved this remarkable result

    Theorem . A flat Möbius band in R³ cannot have L/W less than /2~1.57.

Looking at the figure above, they conjectured that the cutoff is actually at 3~1.73.

BACK to Flat Möbius Bands in R³.

BACK to Drawing Surfaces

HOME