Is it mathematically correct to call a sphere the 3-d equivalent of a circle?

[Hypercube]

I realized something the other day. Almost every three-dimensional shape we know of - cube, cone, cylinder, prisms, etc - can be formed simply by extending their two-dimensional equivalents - square, circle, trapezoid, etc. - into the third dimension; in other words by being given depth. There are several that this doesn't really apply for such as the square based pyramid, but that is a combination of the square and triangular-faced cone.

 

The sphere however is different; because technically it has no two dimensional equivalent, most would say that the circle is the 2d sphere, but in reality the circle is a 2d cylinder, since if you give a circle depth, it becomes a cylinder. It is easy to see why people would consider a sphere to be a 3d circle, but technically it isn't; the sphere might be a shape in of itself.

[ecoli]

I'm not sure if that's true. If you rotate a half-circle around an axis (thereby giving it volume) don't you get a sphere?

[waitforufo]

Are both a sphere and a circle round? People say the earth is round.

[timo]

A unit-sphere and a unit-circle are the respective objects of [math] \{ \vec x \, : \, \| \vec x \| = 1 \} [/math] for [math] \vec x \, \in R^3 [/math] and [math] \vec x \, \in R^2 [/math], that's a similarity but not "the equivalence".

 

There is no universal mathematical definition of the term "equivalence", therefore the answer to the thread-title is "depends on what 'equivalence' means". I do not agree that "almost every three-dimensional shape we know of [...examples...] can be formed simply by extending their two-dimensional equivalents": Looking around I spontaneously see exactly no 3D-shape for which an obviously similar 2D shape exists. Examples: Telephone, car, computer, bottle of water, cup of coffee.

[Fuzzwood]

If you rotate a square, you get a cylinder. The OP speaks of extending, and extending, e.g. give depth to a circle, forms a cylinder

[ajb]

Hypercube,

 

The circle [math]S^{1}[/math] is 1-dimensional. The sphere [math]S^{2}[/math] is 2-dimensional.

 

Anyway, what Atheist says is correct. I think you could call them "related" or something as they are both defined by the conditions Atheist stated. (well, not quite true as Atheist has define one possible embedding in [math] R^{3}[/math] for the two manifolds )

 

I don't think that all manifolds of a given dimension can be decomposed into a product of lower dimensional ones, apart from locally.

[geoguy]

No, not mathematically. But mathematics isn't really a 'science' but a tool to pursue science. Being a tool it can be remachined and words like 'equivalence' redefined. However, as state above by a poster, 'equivalency' in the current common usage is not the term to use.

A circle is not equivalent to a sphere anymore than a 'point' in the circle is the equivalent to a circle.

 

Your observation is an interesting one. The type that is too often lacking in teaching math. It's the fuel that can make math an exciting subject rather than a stress-filled chore.

[the tree]

Both a circle and a sphere are sets of all points of equal distance from one focus. One exists in two dimensional space and the other in three dimensional space.

 

They aren't equivalent, but only because that word is already used for something else.

[Hypercube]

How is a sphere 2-dimensional? If it was 2-dimensional it wouldhave to be depthless, and hence, couldn't physically exist.

[Royston]

How is a sphere 2-dimensional? If it was 2-dimensional it wouldhave to be depthless, and hence, couldn't physically exist.

 

You're not thinking in the right terms...only the surface is defined on a 2 dimensional sphere, there's no volume (2-manifold)...where as a 3-sphere requires the co-ordinates in 4d Euclidean space (3-manifold.)

 

Think of it this way, if you start with an x,y plane, and a point is defined by the relationship between x and y, that is one dimensional, as depth (z) isn't included. The point isn't 2 dimensional, because all it is a reference of y with respect to x. I think you're slipping up, because you're thinking of lines and points as solely physical objects i.e you draw a line, the ink from your pen is 3 dimensional in a 4 dimensional space. A line or point doesn't require higher dimensions to be defined. The same with the surface of a sphere. Does that make sense ?

[the tree]

*smacks forehead*

 

It is correct to say a sphere is the higher dimensional analogue of a circle.

 

That was pretty much the OP's original point, just in standard terminology.

[Dave]

How is a sphere 2-dimensional? If it was 2-dimensional it wouldhave to be depthless, and hence, couldn't physically exist.

 

Not sure whether Snail's post will clarify so I'm adding my own comment here.

 

The simplest way of thinking about the dimensionality of [math]S^2[/math] is to consider a 'small' square on the surface of the sphere. Because the section is small, to all intents and purposes it would look like a square in the unit plane (which is of two dimensions). It is this property which makes [math]S^2[/math] a 2-dimensional (smooth) manifold.

 

Indeed, we can construct an explicit homeomorphism [math]f:[0,2\pi) \times [0,\pi] \to S^2[/math] by [math]f(\theta, \varphi) = (\cos\theta \sin\varphi, \sin\theta\sin\varphi, \cos\varphi)[/math]. If one considers the domain as a vector space, then it is clearly of dimension two and, f being an isomorphism, so is [math]S^2[/math].

[Xerxes]

The sphere however is different; because technically it has no two dimensional equivalent, most would say that the circle is the 2d sphere,

This is, of course, nonsense. A circle is a 1-sphere, as I told you a few days ago

but in reality the circle is a 2d cylinder,

and this is worse. You really believe this?

There is no universal mathematical definition of the term "equivalence",

There most certainly is, where did you get that idea from?

whereas a 3-sphere requires the co-ordinates in 4d Euclidean space (3-manifold.)

Why? Local coordinates work just fine. Why do you think you "require" coordinates on the ambient space for a 3-manifold, but not for a 2-manifold? What, for example, do you think the ambient coordinates might be for the 4-manifold? (which I can assure you does exist - it's called spacetime, by the way)

[timo]

There most certainly is [a universal definition of the term "equivalence"], where did you get that idea from [that it wasn't the case]?

From the mathematical term "equivalence relation" and the resulting equivalence classes. I can call/define two vectors or R² equivalent if they have the same x-coordinate. But I can as well call them equivalent if they have the same y-coordiante or if they have the same length. That's already three different meanings of equivalence for one very specific class of objects.

[Bluenoise]

I'd say a sphere is closer to being a 3d circle than a cylinder because if you take a 2d slice out of a sphere any way you want you get a circle but there is only one angle you can take a 2d slice out of a cylinder and still get a circle.

[Royston]

Why? Local coordinates work just fine. Why do you think you "require" coordinates on the ambient space for a 3-manifold, but not for a 2-manifold? What, for example, do you think the ambient coordinates might be for the 4-manifold? (which I can assure you does exist - it's called spacetime, by the way)

 

Sorry, I didn't make that very clear, I was just talking about a sphere in a 'physically real' sense, in the context Hypercube was talking i.e how can you have a 2 dimensional sphere...mathematically you can, I was just explaining how a sphere would exist as Hypercube was describing.

 

Judging by the question, I was trying to explain in Layman terms (clearly failed) ho hum.

[Dave]

From the mathematical term "equivalence relation" and the resulting equivalence classes. I can call/define two vectors or R² equivalent if they have the same x-coordinate. But I can as well call them equivalent if they have the same y-coordiante or if they have the same length. That's already three different meanings of equivalence for one very specific class of objects.

 

I was going to point out the very same fact, but you appear to have gotten here first

 

Additionally - since we are talking about topology here - two spaces are generally topologically 'equivalent' if you can construct a homeomorphism between them. Or maybe you could define equivalence by constructing a homology, or perhaps doing some clever tricks with identification spaces. There's lots of ways in which you might wish to define equivalence.

[Xerxes]

The claim is that there's no precise definition of equivalence. I claim otherwise:

 

A member of an arbitrary partition of any space is called an "equivalence class" iff, with respect to the relation ~ the following conditions are met:

 

x~x (reflexivity)

if x~y, then y~x (symmetry)

if x~y and y~z, then x~z (transitivity)

 

for all x, y and z in [x].

 

Under these circumstances the relation ~ is called "an equivalence relation". Merely because the partition is arbitrary doesn't mean there can be no precise definition of when that partition induces an equivalence relation.

I'd say a sphere is closer to being a 3d circle

If by 3d you mean 3-dimensional, you are wrong. The sphere (note the definite article here) is defined to be a 2-dimensional object. You may call it the 2-sphere if you want (but most people don't), just as the 0-sphere is a line, the 1-sphere is a circle, the 3-sphere is a ball etc.

 

Anyway, who said we were doing topology, I never saw that stipulated.

[Sisyphus]

You guys are getting way too caught up with semantics here. It doesn't matter what definitions for "equivalence" are currently being tossed around English-speaking math departments. And, there's more than one way to define circles and spheres. the tree gave the best answer, IMO:

 

The simplest way in which circles and spheres are equivalent (or analogous, or whatever) in their respective dimensionalities is that they are both simply the locus of points equidistant from one point. A circle is that locus limited to a Euclidian plane, a sphere is this locus limited to a Euclidian volume.

 

The way hypercube was using "equivalent" in the OP was a little different, i.e. just "adding depth" uniformly to a plane figure to make a solid. A sphere doesn't have any 2d "equivalent" in this sense, obviously, but then, only prisms would.

[Xerxes]

You guys are getting way too caught up with semantics here. It doesn't matter what definitions for "equivalence" are currently being tossed around English-speaking math departments.

"Semantics"?? So if I tell you the thing I drive to work in each morning is a "table" and the thing I take for walks, goes woof and chases rabbits is a "telephone", that's just semantics? You may think so, I don't; it's a matter of agreeing definitions, which is what a lot of the language of mathematics is about. I gave the agreed definition of equivalence - notice the word definition

 

The simplest way in which circles and spheres are equivalent (or analogous, or whatever)

I can do no more - I gave the precise definition of equivalence. What does "analogous" or, god help us, "or whatever" mean? Give us definitions.

A sphere doesn't have any 2d "equivalent" in this sense,

I have no idea what this means. Do you?

[Sisyphus]

"Semantics"?? So if I tell you the thing I drive to work in each morning is a "table" and the thing I take for walks, goes woof and chases rabbits is a "telephone", that's just semantics?

 

No, but it is if I ask you where you walked your dog this morning, and you have no idea what I'm talking about because you actually jogged. Or it is if somebody asks a coherent question in perfectly understandable English, but you feign ignorance because one of the words he uses you've learned can mean other things in different contexts.

 

I gave the agreed definition of equivalence

 

Which, sadly, is not what the OP question was about.

 

I can do no more - I gave the precise definition of equivalence. What does "analogous" or, god help us, "or whatever" mean? Give us definitions.

I have no idea what this means. Do you?

 

What does analogous mean? You don't know what an analogy is?

[Xerxes]

Or it is if somebody asks a coherent question in perfectly understandable English, but you feign ignorance because one of the words he uses you've learned can mean other things in different contexts.

Er, um... the context here is mathematics, isn't it? Or have I come to the wrong party?

 

Which {that is, definition of equivalence}, sadly, is not what the OP question was about.

Then you are too self-effacing, my friend, I was talking directly to you. You made some assertions, specifically that equivalence was not well defined, with which I disagreed, what's wrong with that? You may disagree with my presentation, I'm cool with that, but why might you think it was misplaced?

What does analogous mean? You don't know what an analogy is?

Sure I do, but it's not in the mathematician's lexicon, as far as I know.

Is this a mathematics forum, or what? If it is, am I not allowed to give the mathematician's definition of terms, irrespective of what others might say here?

[Sisyphus]

It is a mathematics forum, yes, and it's also an English-language forum. Obviously we have different perspectives on this. Your training is (I assume) in mathematics or some subject that utilizes mathematics; mine is in the history and philosophy of mathematics. I'm not as interested in whatever the current minutiae of conventions are, because I know they're recent, non-universal, and soon to change. It's being understood that matters. I can afford this broader perspective because no bridges are going to collapse if I speak ambiguously to get across an unconventional idea...

[Xerxes]

Ok.

[shadowacct]

I would call a sphere not a real extension of a circle. A cylinder (with open end caps) could be called more of an extension, it simply is the product [math]S^1 \times L[/math], where [math]L[/math] is a line segment. Another extension could be a torus, being [math]S^1 \times S^1[/math].