Dimension of the circle in the plane is 1

[LikeMath]

I am becoming confused when I read in Wiki that the dimension of the circle in the plane is 1!
It is said that the dimension of circle is 2 (in general )!
I do not get it!

 

[DonAntonio]

I am becoming confused when I read in Wiki that the dimension of the circle in the plane is 2!
It is said that the dimension of circle is 2 (in general )!
I do not get it!

Where exactly in Wiki, or wherever, is written such a thing and in what context? Please do write down a link.

DonAntonio

 

[LikeMath]

http://en.wikipedia.org/wiki/Dimension


In mathematics, the dimension of an object is an intrinsic property, independent of the space in which the object may happen to be embedded. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate (the polar coordinate angle), so the circle is 1-dimensional even though it exists in the 2-dimensional plane. This intrinsic notion of dimension is one of the chief ways in which the mathematical notion of dimension differs from its common usages.

 

[LikeMath]

Please note that I have changed the first post (2 becomes 1)

 

[micromass]

The dimension should be seen as a local property. That is, we should look at fictitious inhabitants of the circle and ask them what the dimension is.
If we ask them, then they will say us that they can only go forwards or backwards, and thus they will think that they live on some sort of line. This suggests that the dimension of the circle should be 1.

 

[HallsofIvy]

I am becoming confused when I read in Wiki that the dimension of the circle in the plane is 1!
It is said that the dimension of circle is 2 (in general )!

Where is that said? It is certainly wrong.

I do not get it!

A circle is a one dimensional closed curve imbedded in a set of dimension at least two.

I suspect you are thinking "circle" when you mean "disk".

A disk (the circle and the points inside the circle) is a two dimensional object.

 

[DonAntonio]

Please note that I have changed the first post (2 becomes 1)

You, and anybody else, shouldn't do that, as it confuses things. If you want to correct an old post add a NEW post with the correction, do

not alter the original one.

Anyway, with the correction nothing's wrong...under certain assumptions and definitions, of course.

DonAntonio

 

[LikeMath]

You, and anybody else, shouldn't do that, as it confuses things. If you want to correct an old post add a NEW post with the correction, do

not alter the original one.

Anyway, with the correction nothing's wrong...under certain assumptions and definitions, of course.

DonAntonio

Thank you. I am Sorry.

 

[LikeMath]

Where is that said? It is certainly wrong. A circle is a one dimensional closed curve imbedded in a set of dimension at least two.

I suspect you are thinking "circle" when you mean "disk".

A disk (the circle and the points inside the circle) is a two dimensional object.

So the dimension of the sphere is 2, but the dimension of the solid sphere is 3. Thank you all. I got it!

 

[HallsofIvy]

The "solid sphere" is, mathematically, a "ball". A sphere is the surface of a ball.

 

[Arian.D]

There are many other people on this forum that know much more than me, I'm not informed about mathematics as much as they are but my guess is this:

You could parametrize a circle by the polar equations:
x = rcos(\theta) + x_0
y = rsin(\theta) + y_0

So you'll need only one parameter to describe a circle. Even though the result will be in the real plane because for any value of theta that function gives a point (x,y) on the circle, but one variable is enough to describe the circle in \mathbb{R}^2. Therefore its dimension is one in the real plane. I'm just saying that intuitively, not based on any particular definitions.