2021年10月4日 · A 3-sphere cannot fit into 3-dimensional euclidean space, just as a 2-sphere cannot be cut from a piece of paper. So while you can't just "look at it", here are two nice visualizations of what it would look like being inside the 3-sphere: Spherical Geometry Is Stranger Than Hyperbolic - Hyperbolica Devlog #2

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: Isn't that what it means to have a 4th dimension? Just add time? Apparently not. This image is from ...

2014年6月4日 · Could you recommend a good introductory book(s) or web sites (besides Wikipedia, of which I am aware ) on the subject of the 3-sphere. My goal is to build intuition about the 3-sphere by starting with simple analogies, examples etc. I am not too familiar with group theory, would like to approach the 3-sphere without group notation, if possible.

2014年4月22日 · The Wikipedia article on the subject shows the hypermeridians, but their mathematical formulation is not given. I am new to the 3-sphere, and would like to do as little guessing as possible. There are other resources on the 3-sphere, but they use group theory formulation, with which I am unfamiliar. Thanks!

The 3-sphere can be represented by the set $\{(z,w) \in \mathbb C^2 : |z|^2+ |w|^2 =2\}$. This contains the subset $|z|=|w|=1$, which is the product of two circles, that is, a torus. This torus is the common boundary of the subsets where $|z|\le 1\le |w|$ and $|w|\le 1\le|z|$.

If there was a way to parametrize a 3-sphere with just 2 parameters, every little neighbourhood of it would be homeomorphic to an open set of $\mathbb{R}^2$. This would imply that we could establish an homeomorphism from an open set of $\mathbb{R}^3$ to an open set of $\mathbb{R}^2$ , which is absurd.

2015年10月28日 · A 3-sphere is a sphere in 4D space. A 3-sphere -- or any other object in 4space with non-zero volume -- can have two planes of rotation. These planes are always at right angles to one another, and may have entirely independent periods of rotation. To make this more down to Earth I like to imagine being on the surface of a rotating 3-sphere planet.

2012年1月9日 · What you have at the top is not the metric of a 3-sphere, but simply the metric of R^3 in spherical polar coordinates. A 3-sphere metric will have r fixed and 3 angles to describe where you are (a sphere is hollow remember), just as a 2-sphere has r fixed and 2 angles to describe where you are.

It is a 2-sphere embedded in the 3-sphere. Reasoning as you did in the lower-dimensional case, you could say that the set of rays from the projection point in the 3-sphere to the plane in $\mathbb{R}^3$ form a half-hyperplane intersecting the 3-sphere at the projection point. The intersection between the 3-sphere and a hyperplane is a 2-sphere.

Moreover, since the mapping class group of the 2-sphere is trivial, this is really the only thing you could obtain: whatever gluing you choose for your 3-balls you always end with a 3-sphere.