Circles and lines are normally regarded as one-dimensional objects. However, when embedded in two-space, they require two coordinates $(x,y)$ to specify a point within them. Are they still considered one-dimensional, and why? This is somewhat related to a separate but very similar question of why a sphere embedded in three-space is (to my knowledge) considered two-dimensional, even though it has a height, a width and a length. To laymen, spheres are generally considered 3D objects. This is where a potential misunderstanding may lie. Therefore, I am also curious of what difference, if any, there is between a mathematician's conception of what a "dimension" is versus a layman's?
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1You probably confuse the dimension of the space and the number of parameters describing the curve. A line, a circle, an ellipse can all three be defined in 2D space, where they are determined by 2, 3 and 5 parameters respectively, but they are 1D entities as they can be mapped to the real line (you can graduate them with a single number). – May 25 '15 at 10:29
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Related: "Why the unit circle in $R^2$ has one dimension?", "Confused about dimension of circle", "Why is a circle one-dimensional?". – Blue May 25 '15 at 10:30
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2Along with all of the answers here and in the links, which all more or less state that a circle is locally equivalent to a line, I'll add another thing. It has been proven that a line is not locally equivalent to a plane, space, etc. nor are two $n$ and $m$-dimensional euclidean spaces locally equivalent, unless $n=m$. This isn't trival (although in the case of a line it's much simpler to prove). It's important to say this because otherwise the concept of dimension is meaningless! – GPerez May 25 '15 at 14:31
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Roughly speaking, dimension refers to the degree of freedom of the geometric shape, regardless of the ambient space where that shape happens to embed in. So, lines and circles are one-dimensional, intuitively, since you have one degree of freedom at each point (you can only move along one direction). The sphere in three dimensions, much like the earth, is two dimensional since at each point you can move in precisely two independent directions. This should not be confused with the intrinsic geometry of the shape. Things like the intrinsic curvature of the shape have little to do with the dimension of the shape itself (though they can be loosely related to the dimension of the ambient space). For instance, a circle of large radius has smaller intrinsic curvature than a circle of smaller radius. Finally, particularities of how precisely a shape sits in the ambient space is a different story altogether. An important story, but different. Differential geometry is the area of study of these concepts where all of the above becomes rigorous with the help of proper definitions. The differences between how a mathematician interprets things and how a lay-person does only stem from syntactical inaccuracies.
Ittay Weiss
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The idea is that you can call objects "n-dimensional" if you can parameterise them using n distinct coordinates.
For example, the line is the image of the function $f(t)=(t,at+b)$, and the circle is the image of the function $f(t)=(a+R\cos(t),b+R\sin(t))$. The sphere is two-dimensional because it can be parameterised by $f(\theta,\varphi)=(R\sin\varphi\cos\theta,R\sin\varphi\sin\theta,R\cos\varphi)$.
john
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Roi Blumberg
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Any point on a line (or a circle, or any shape) can be described in terms of a single coordinate - how far along the line the point is. The area circumscribed by the shape is a two-dimensional object, the points requiring two coordinates to describe their locations. Similarly, the surface of a sphere can be described with two coordinates (longitude and latitude), but the volume of the sphere is three-dimensional. I think the misunderstanding arises because talking about the perimeter/surface area of a polygon/solid is a little less intuitive than talking about its area/volume.
Nuclear Hoagie
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You need to define some point on the line to be "$0$" before it makes sense to talk about how far along any other point is. But apart from that detail you're right. – Henrik supports the community May 25 '15 at 10:29
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Look it like this: If i give you a box and put something inside it a layman will think that the object inside it has to be related to the box and that is the reason why it is inside it. But a mathematician's view would be something else he would first try to define what is the purpose of the box then he would look into an object and then try to classify why is that object inside the box and what are the properties of that object instead of directly relating it to the box. To give you an example you can think of $\Bbb{Q}$ and $\Bbb{R}$ How are they related? First thought they are subsets of each other. But if you just look at it you will see most of the reals are there in rational and some manipulations will give you other reals to, So a layman will think it as $\Bbb{Q}$ being the box and $\Bbb{R}$ being the object inside it but actually its the inverse for a mathematician.
So now extending it to your question the $x-y$ plane which is made from the $x,y$ axis which are itself lines. So to think it in a straight forward way we have a box($x-y$ plane) where i am putting another box(line) . Now for a line and try to deform it that is try joining the two ends(here we are considering a line segment) we get a circle and since i am taking a one dimensional object and deforming it without adding anything else it still remains one dimensional.
For a sphere we have to add stuff to the line. We have to stack lines on top of each other that then just glue the ends we get a sphere!! Now since we added something to the line which will change its dimension and we now cant keep it in the earlier box ($x-y$ plane) SO we know make a new box where we put this so called sphere. Now since lines were 1-dimesional this object that we get has to 2-dimensional.
Technically: As given in one of the you can use parameters to show dimension . Therefore the number of parameters will tell you the dimension. SO as we know the formula of a line is $at+b$ here the parameter is just $t$ and hence it is one dimensional. A sphere of radius a can be described by$$x=a\cos\theta\cdot\sin\phi$$ $$y=a\sin\theta\cdot\sin\phi$$ $$z=a\cos\phi$$ So here there are two parameters and hence it is two dimensional .
There are also other complicated ways to define a dimension which you can find in dimension theory.