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a 2-sphere is a normal sphere. A 3-sphere is

$$ x^2 + y^2 + z^2 + w^2 = 1 $$

My first question is, why isn't the w coordinate just time? I can plot a 4-d sphere in a symbolic math program and animate the w parameter, as w goes from .1 to .9:

4d surface

Isn't that what it means to have a 4th dimension? Just add time?

Apparently not.

This image is from wikipedia,

3-sphere

The caption says that this is a "Stereographic projection of the hypersphere's parallels (red), meridians (blue) and hypermeridians (green)". I don't get that at all. What is a parallel, meridian, hypermeridian? Why can't we just

There is an article here which talks about the 3-sphere in terms of Poincare's Conjecture.

Here there is an image of the "Hopf fibration of the 3-sphere".

hopf

This looks very cool and there are formulas that break down the Hopf fibration into understandable algebra, but what does this mean, at a high level?

Edit: I am looking at the dimensions videos and they are actually very good.

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    Time is one way to add another dimension, but not always the most interesting way to visualize it (since you can't see everything at once). That's why certain projections are sometimes used, as in this case.2012-05-21
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    You could view the unit circle $\Bbb S$ as two dots on the $x$-axis splitting from the origin to the endpoints $\pm 1$ and then coming back, using time as a dimension. But that is hardly "the" way to visualize a circle. Also, you could view a cube as a square standing still for an interval in time. Hardly enlightening, and if you rotate the cube the corresponding "animation" (constructed with cross-sections as frames) will be drastically harder to piece together into a solid figure. "Time" is merely a crutch we stand on to help us visualize.2012-05-21
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    The square of the space-time "distance" is $x^2+y^2+z^2-c^2t^2$, so it is not directly related to the distance function on $\mathbb{R}^4$.2012-05-21
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    I think you're missing the fundamental point that a four-dimensional space doesn't actually come equipped with four distinct "dimensions". Being four-dimensional just means that you can always find four linearly independent directions, but there isn't a fixed set of directions that are canonically labeled as "the first dimension", "the second dimension", and so on. So thinking that time is "*the* fourth dimension" is quite misleading.2012-05-21
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    Time isn't **the** fourth dimension, it's **a** fourth dimension.2012-05-21

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