Inspired by the rational numbers between 0 an 1 thread, where it was shown that all points on two concentric circles can be matched by drawing lines from the larger circle to the center.
Same can be done probably with a cylinder too.
So what about about two “concentric” Möbius strips, with the same geometry except one is larger in “diameter”. Is there a function that can map all points from one to the others ?
Yes, through much the same argument. Let’s suppose that the larger Möbius strip is twice the size of the smaller one. Take some point in the middle of the smaller strip as fixed. For each point A on the smaller Möbius strip, draw a ray from the fixed point through it. Then construct a “matching” point A’ that is twice as far from the fixed point as A is. If you do this for each point on the smaller strip, the constructed points will form a Möbius strip that’s twice as large in every dimension.
Thanks MikeS. Wouldn’t the ray intersect the smaller strip at two points or more for some of the locations ?
No, why would it? These are rays drawn on the surface of the Möbius strip. If you’d like, you can cut the strip open and lay it out flat before you draw the rays: It’s still the same number.
Not sure I understand this either, Chronos. If the rays are perpendicular to the surface, then at the places the strip curves, the rays will be emanating in a diverging sense and not will not reach the outer strip from the inner strip. Or maybe I am getting this wrong.
They’re not perpendicular to the surface. They’re on the surface. Like, make the strip out of paper, and draw on it with a pencil.
There’s no difference between the Möbius strip and the cylinder you mentioned. The two “concentric” strips can be paired point for point by rays exactly as two “concentric” cylinders can. Don’t let the twist confuse you. The outer strip always matches the inner strip point for point everywhere along their surfaces. You just look at the two at that point and move over one point at a time. That eliminates the curvature that’s throwing you off. So would cutting the strips and lying them flat, which makes it obvious that you’re comparing a plane to a plane.
Remember, connecting rays is just a way to visualize the problem. If it’s not helping understanding, toss it.