Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

Thanks in advance!

share|cite|improve this question
Åre you mixing the circle (1d) with the disk (2d)? – lhf May 5 '11 at 19:50
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
up vote 14 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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.