Wave propagation on a Mobius strip

A fun way to visualize the solution to the inaugural "Fiddler on the Proof"

This week's Fiddler is all about Mobius strips. Specifically, starting from a point on the Mobius strip, what point(s) have the greatest shortest distance from the starting point? Put another way, if you sent a circular ripple out at constant from the starting point, what would be the last point(s) to feel the ripple, and how far would the ripple have gone?

I solved the wave equation on the Mobius strip and then propagated a pulse out from the central point.

I found it useful to visualize this by "unfolding" the ribbon by first detaching the tape, marking the corners to keep track of where they were connected, then peeling the "back" off and laying it down next to the "front" on either side.

It turns out the quickest way that the last point is reached is by traveling $(0,y)$ one way and $(W, L-y)$ the other way around, where $L$ and $W$ are the length and width of the original paper, and $y<L/2$. These are the same point because of how the Mobius strip edges "wrap around". By setting the distances equal, we find $y=(L^2-W^2)/2L$.

With $L=10$ inches and $W=1$ inch, as in the Fiddler, the points at the maximum shortest distance are at $y=\pm 5.05$ inches from the starting point traveling along the centerline, equivalent to traveling $\Delta y = \pm 4.95$ inches and $\Delta x= \pm 1.0$ inches. There are two such unique points.

The animation shows the case $L=128$ and $W=32$. In this case the farthest points are 68 units away along the centerline, or $\pm 60$ units in $y$ and 32 units in $x$.

The waves are shown on the same grid as in the second figure above: point “O” is in the center and at the 4 corners of the animated plot.

For you partial differential equation lovers out there, here’s part of the solution to the wave equation PDE: