Representation and determination of a 3-sphere

[HarleQuin]

I was wondering about the construction,ideally from first principles of 1 to 2 to a 3-sphere. What criteria are having to be met for a 3-sphere,and at what point or what are the boundaries of a 4-sphere? ;)

[HallsofIvy]

I was wondering about the construction,ideally from first principles of 1 to 2 to a 3-sphere. What criteria are having to be met for a 3-sphere,and at what point or what are the boundaries of a 4-sphere? ;)

I'm not sure what you mean by "construction" or "representation" or "determination" of a 3 sphere but I can answer your last question. The 4-sphere, exactly like the 3-sphere, 2-sphere, and 1-sphere, has NO boundary.
(Did you mean "ball" rather than sphere? The boundary of a 4-ball is a 4-sphere, again just like the boundary of a 3-ball is a 3-sphere, the boundary of a 2-ball is a 2-sphere, and the boundary of a 1-ball is a 1-sphere.