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Is a circle just a line (therefore 1 dimension) or is it a 2-dimensional object because it occupies some surface?

Thanks in advance!

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Åre you mixing the circle (1d) with the disk (2d)? –  lhf May 5 '11 at 19:50
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Expanding on the previous comment: mathematicians (when they're being careful) use the word "circle" for what you might call the circumference, and the word "disk" for the filled-in circle. Non-mathematicians use the word "circle" for both concepts. So if your question is about terminology, rather than mathematics, the answer is, 1-dimensional if a mathematician is involved, depends on context if not. –  Gerry Myerson May 6 '11 at 0:29
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up vote 12 down vote accepted

A circle is a one-dimensional object, although one can embed it into a two-dimensional object. More precisely, it is a one-dimensional manifold. Manifolds admit an abstract description which is independent of a choice of embedding: for example, if you believe string theorists, there is a $10$- or $11$- or $26$-dimensional manifold that describes spacetime and a few extra dimensions, and we can study this manifold without embedding it into some larger $\mathbb{R}^n$.

Incidentally, you might guess that $n$-dimensional things ought to be embeddable into $\mathbb{R}^{n+1}$ ($n+1$-dimensional space). Actually, this is false: there are intrinsically $n$-dimensional things, such as the Klein bottle (which is $2$-dimensional) which can't be embedded into $\mathbb{R}^{n+1}$. The Klein bottle does admit an embedding into $\mathbb{R}^4$. More generally, the Whitney embedding theorem tells you that smooth $n$-dimensional manifolds can be embedded into $\mathbb{R}^{2n}$ for $n > 0$, and for topological manifolds see this MO question.

There are also other notions of dimension more general than that for manifolds: see the Wikipedia article.

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