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I am really confused about what is the minimum length for a dimension to exist (I don't know the exact word). Like I mean, we call the paper to be a 2 dimensional object but if we magnify its edges, we have the third dimension as well.

So my question is, for example, what should be the minimum breadth of the edge of the paper so that we can say that the paper is only 2 dimensional?

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The paper should have $0$ breadth. I suggest you to post this question on math.SE for good answer. I've given a trivial answer to a similar question here. –  user31782 Dec 24 '13 at 14:22
    
This would be helpful:math.stackexchange.com/questions/465818/dimension-of-an-object –  user31782 Dec 24 '13 at 17:33
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The mathematicians have a careful definition of dimension which would say that every physical object is three dimensional. In physics and engineering it is often useful to consider something to have a different dimensionality, but that will be an approximation. Your example of a piece of paper is a good one. When I am drawing on it, the useful concept is that it is two dimensional. If I look at it under a microscope, the surface is no longer flat and it is clearly three dimensional. Which view is useful depends on the problem. An example that is used a lot is a garden hose. When you are close, it is clearly three dimensional. Step back a ways and it looks like a line, 1 dimension. Flow problems will often use this approximation. Step back further and it is a mass point, 0 dimensions. If you put it in orbit around the earth, this will be useful.

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It depends on the context. In some cases a planet can be considered a point-particle (0-D), while an atom can be considered 3-D.

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