## Dimensions! Clarification needed.

1) Why is it that a sphere is a 2-dimensional object even though we need 3-dimensions to actually *draw* the sphere.

2) If the Cartesian plane is a 2-dimensional *space*, then the number-line should be a 1-dimensional *space*, right? (not sure if space is the appropriate term here)

3) Would a point be considered a 1-dimensional or a 0-dimensional entity?

Thanks.
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 It's the surface of the sphere that is 2D, not the sphere itself. Yes, a line or axis is only one dimension. A point has no dimensions.
 Recognitions: Science Advisor But a 2 dimensial space cannot connect to the ends. In 2 dimensions up and down gives no meaning! it's only 2 dimensions, back and forth, and left and right. up and down is preserved to the third dimension. So if you want a ball you must add the third dimension even though you only think if the surface of it. Am I not right?

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## Dimensions! Clarification needed.

No. Look up on non-Euclidean geometries.
Although such geometries can readily be regarded as geometries-on-curved-surfaces i.e, within a 3-D world), their postulates (in particular their alternative to Euclid's fifth postulate) contains no mention of a third dimension.
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 Quote by cesiumfrog It's the surface of the sphere that is 2D, not the sphere itself.
I see. So would you still call sphere a 3-dimensional object?

Also, what does it mean to say that a particular function lives in a certain dimension. For example, say $$f$$ lives in $$\mathbb{R}^3$$

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 Quote by Swapnil I see. So would you still call sphere a 3-dimensional object? Also, what does it mean to say that a particular function lives in a certain dimension. For example, say $$f$$ lives in $$\mathbb{R}^3$$
Technically, a sphere is a 2D-object [a closed surface] whereas a [tacitly assumed solid] ball is a 3D-object.

Do you mean the range instead of the domain?

I mean, you can have a function f that maps every point on the number line to a set of 3 points. Then would say that the function lives in 1-dimension because of its domain? Or would you say that the function lives in 3-dimension because of its range?
 Blog Entries: 47 Recognitions: Gold Member Homework Help Science Advisor well... clearly "lives" is not a precise enough term to be used seriously. Function may be somewhat vague as well... hence terms like real-valued function and vector-valued function. Maybe "lives" should be replaced by a more technical notation like the mapping $$f: M \rightarrow N$$ and discuss the dimensionality [if it exists] of the domain M and of the range N.

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 Quote by Swapnil 1) Why is it that a sphere is a 2-dimensional object even though we need 3-dimensions to actually *draw* the sphere.
Here's the general approach: ask yourself how many numbers you need to specify to completely define a point on a sphere. That number tells you the dimensionality (crudely, but rule-of-thumbishly speaking - see link) of the sphere(edited from "object" to "sphere").

 2) If the Cartesian plane is a 2-dimensional *space*, then the number-line should be a 1-dimensional *space*, right? (not sure if space is the appropriate term here)
Both statements are true, and space is a correct usage. In particular, the set of Reals is a vector space (defined with the usual addition and mutiplication of reals).

 3) Would a point be considered a 1-dimensional or a 0-dimensional entity?
Apply the test described above.

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 Quote by Swapnil Also, what does it mean to say that a particular function lives in a certain dimension. For example, say $$f$$ lives in $$\mathbb{R}^3$$
As robphy stated, the notation is somewhat imprecise. But I would expect it to mean that f is a function into R³.

Informally, people sometimes think of such an f as a vector that varies with time, or with position (depending on how we think of f's domain), which explains why one might think of it as "living" in R³. It also provides a convenient way to be sloppy with notation. But it's probably not best to actually think of f in that way, unless you're willing to swallow a hefty dose of formal logic before doing so.

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 Quote by Swapnil 1) Why is it that a sphere is a 2-dimensional object even though we need 3-dimensions to actually *draw* the sphere.
Because that is the definition of dimension we have chosen - it is intrinsic to the object, not any embedding of it in an ambient space (which is clearly not a well defined number anyway). The dimension refers to the 'local dimension' (non-standard term) of that object: locally, i.e. in any small patch about a point, it is homeomprhic to the open disc or R^2. Since this is true for every point on the surface, we say it is 2-dimensional. Some things don't have a well defined dimension because they are not everywhere locally homeomorphic to R^n. There is another definition of dimension. The sphere is defined, in R^3, by the equation f:=x^2+y^2+z^2-1=0. The (algebraic) functions from the sphere to R are then R[x,y,z]/f, and this ring has krull dimension 1. Which is just saying that f is irreducible.

 2) If the Cartesian plane is a 2-dimensional *space*, then the number-line should be a 1-dimensional *space*, right? (not sure if space is the appropriate term here)
The real line is a 1-dimensional vector space, and a 1-dimensional manifold, yes.

 3) Would a point be considered a 1-dimensional or a 0-dimensional entity? Thanks.
zero dimensional.

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