Moebius Strips: How Many Sides and Surfaces?
Date: 10/18/2001 at 17:53:13 From: Melissa Bercume Subject: Moebius strips I have looked at almost every Web site there is, and I still cannot find the difference between sides and surfaces. Can you help? Thank you so much. Melissa
Date: 10/19/2001 at 17:00:30 From: Doctor Peterson Subject: Re: Moebius strips Hi, Melissa. In simple terms, a surface is a set of points, while a side adds an orientation to those points. We look not only at where we are, but at which way we are facing. Take a sphere. To show that it is a single surface, imagine making one point glow blue, and let it wander all over the sphere. You will see that it can move freely anywhere on the sphere, so the whole sphere is one contiguous surface: ooooooooooo ooooo ooooo oooo oooo oo oo o o oo * oo o o o o o o o o o o o o o o o o o o oo oo o o oo oo oooo oooo ooooo ooooo ooooooooooo To see how many sides this surface has, imagine replacing that point with something like a tack; its flat head rests on the surface with the point sticking up: ooooooooooo ooooo ooooo oooo oooo/ oo ..../oo o . / . o oo . * . oo o ..... o o o o o o o o o o o o o o o o o oo oo o o oo oo oooo oooo ooooo ooooo ooooooooooo Now let this wander around, and you will see that, although it will be able to get to any spot on the surface, the "point" of the tack will always be pointing the same direction whenever it gets to that point. You would need to put a red tack on the inside of the sphere in order to point the other way. There are two distinct sides to the sphere. But on a Moebius strip, you will find that the single blue tack will not only be able to go everywhere on the surface, it will be able to point in both directions at every point, just by sliding around. You won't need that second red tack. So this surface is one-sided. Now notice that you don't really have to have the point on the tack. Just slide around a circle with an arrow going clockwise around it. On a Moebius strip, you will be able to slide that circle around and back to the same location, but with the arrow going the other way! This allows us to define orientation without reference to the space outside the surface. ---->---- ----<---- / \ / \ / \ / \ | | | | | | | | | up | | down | | | | | | | | | \ / \ / \ / \ / --------- --------- You are asking for a "technical" definition of a "popular" concept, so if you're looking for a technical definition, this won't satisfy you; but it gives the general idea. Mathematicians don't need to define "side," because we don't call surfaces like the Moebius strip "one-sided." We call them "non-orientable"; that is, you can't distinguish two orientations. What I gave you is a definition of orientability - still pictorial, but closely related to what we would say in an introductory textbook. So don't worry if you can't say what a "side" is; I've told you what "one-sided" means. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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