Why is a circle 1-dimensional?

Solution 1:

Suppose we're talking about a unit circle. We could specify any point on it as: $$(\sin(\theta),\cos(\theta))$$ which uses only one parameter. We could also notice that there are only $2$ points with a given $x$ coordinate: $$(x,\pm\sqrt{1-x^2})$$ and we would generally not consider having to specify a sign as being an additional parameter, since it is discrete, whereas we consider only continuous parameters for dimension.

That said, a Hilbert curve or Z-order curve parameterizes a square in just one parameter, but we would certainly not say a square is one dimensional. The definition of dimension that you were given is kind of sloppy - really, the fact that the circle is of dimension one can be taken more to mean "If you zoom in really close to a circle, it looks basically like a line" and this happens, more or less, to mean that it can be paramaterized in one variable.

Solution 2:

Continuing ploosu2, the circle can be parameterized with one parameter (even for those who have not studied trig functions)... $$ x = \frac{2t}{1+t^2},\qquad y=\frac{1-t^2}{1+t^2} $$

Solution 3:

Officially, it's one-dimensional because at any point, the tangent space is a one-dimensional vector space.

Unofficially, it's one-dimensional because if you zoom in enough on a tiny little piece of it, it is indistinguishable from a segment of the real line.

Sorry, I have a bad habit of answering a question title and not exactly answering the question itself. You say you need an $x$ and $y$ value to identify a circle. So you are identifying its center. Even though you've just named a point with two numbers in its address, you have only identified a $0$-dimensional thing at this point. It's just a point, there is no direction to travel in.

To identify the circle, you'd need a third number: the radius. And the circle will consist not of the point you had earlier identified, but rather of all points that are that radius from your identified center.

This collection of points can be parametrized using only one independent variable, as a few other answers have shown.

Solution 4:

When the book says, "a dimension is the number of independent parameters needed to specify a point" it's talking about the parameters needed to specify a point on the circle. Only one is needed to describe which point on a certain circle.

That's different from what's needed to describe a circle itself. A circle spans two dimensions, but a single point on it can be described with a single value.