A Flat Band

This work was inspired by a flat mobius band which I saw on the web site of the [now (2 October 2016) vanished] Geometry Center [which was] located at the University of Minnisota [@ the mid 1990s]. The work was also inspired by a conversation with a mathematics professor at Brown University (September 1995). Although I was told that it is possible to create a developable mobius band, I did not really believe it, and I did not consider the problem again until I saw the above mentioned flat mobius band on the web.

figure 1:

The curve goes around the torus and ends directly below its starting point.

figure 2:

The red quadrilaterals are sections of planes which are tangent to the torus at the white points along the curve. The blue lines are the generators for the envelope of tangent planes along the curve: As white points are added so that the distance between them shrinks to zero, the red quadrilaterals shrink down to the blue lines. The surface thus produced is the developable surface circumscribed to the torus along the curve.

figure 3:

w = 1.0000 (equations are below)

figure 4:

w = 0.5000

figure 5:

w = 0.2500

figure 6:

w = 0.1000

figure 7:

w = 0.0001
This surface is intrinsically flat and seems to be a mobius band.

figure 8:

side view

Initial Band:

The limit process: Let w go to zero.

As it stands, the curvilinear coordinate system (Initial Torus) above is left handed. We can make the coordinate system right handed by inserting a minus sign in the equation for z. (To see the difference between the right and left handed cases, click here.) For the right handed case, the figure below shows the coordinate surfaces: u, v, and w are constant on the surfaces which have respectively red, green, and blue grid lines.