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March 20th, 2015, 04:17 PM   #1
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2D shape with 1 side?

A mobius strip like the one you make out of paper is a 2D shape with 1 side.

How is it possible for a 2D shape to have 1 side, or really any n-dimensional shape to have 1 side?

The closest that I see to 1 side is a single straight line and even then it is 0 sides.

A circle or its n-dimensional analog has ∞ sides and so is nowhere close to 1 side.

Last edited by caters; March 20th, 2015 at 04:22 PM.
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March 20th, 2015, 04:40 PM   #2
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Here is a three dimensional visible shape with one side, a Klein bottle. It would require four dimensions to create one, however, since the object appears to pass through itself in our three dimensions.

Klein bottle - Wikipedia, the free encyclopedia

Klein Bottle -- from Wolfram MathWorld

Last edited by Compendium; March 20th, 2015 at 04:56 PM.
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March 20th, 2015, 05:33 PM   #3
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Quote:
Originally Posted by caters View Post
The closest that I see to 1 side is a single straight line and even then it is 0 sides.

A circle or its n-dimensional analog has ∞ sides and so is nowhere close to 1 side.
I think that we have a confusion here about what is meant by side. Perhaps using the terminology "edge" and "face" would be clearer.

You can argue that an edge should be straight and thereby get a circle to have an infinite number of edges. Personally, I would say that an edge is smooth but not necessarily straight. So a circle has one edge, just as a line does.

However, when we talk about a möbius strip having one side, I usually take this to mean that it has one face.

A cube obviously has six faces. A two dimensional shape in three dimensional space has two faces, like a coin has heads and tails (yes, it also has an edge - but then it's a three dimensional object). It is in this sense that a möbius strip has only one face (or side).

Having said that, in the same sense that a circle has only one edge, a möbius strip also has only one edge.

In both cases (the face and the edge) the way to see this is to put a pen onto the face (or edge) and draw a continuous line along the face (or edge) until you reach the point you started from. In both cases you will see that what at first glance appears to be "the other" face (or edge) has pen marks on it, and thus must be part of the same face (or edge) that you started from.
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