S^5 Sphere: Visualizing and Understanding

[yola]

What does S^5 sphere mean? How can I imagine it?
Thanks

 

[torquil]

http://en.wikipedia.org/wiki/Hypersphere"

 

[HallsofIvy]

It is the subset of R⁵ of points $(x_1, x_2, x_3, x_4, x_5)$ such that
$$x_1^2+ x_2^2+ x_3^2+ x_4^2+ x_5^2= 1$$

(The "Ball", B⁵, is the set of points $(x_1, x_2, x_3, x_4, x_5)$ such that
$$x_1^2+ x_2^2+ x_3^2+ x_4^2+ x_5^2<= 1$$)

 

[torquil]

It is the subset of R⁵ of points $(x_1, x_2, x_3, x_4, x_5)$ such that
$$x_1^2+ x_2^2+ x_3^2+ x_4^2+ x_5^2= 1$$

(The "Ball", B⁵, is the set of points $(x_1, x_2, x_3, x_4, x_5)$ such that
$$x_1^2+ x_2^2+ x_3^2+ x_4^2+ x_5^2<= 1$$)

No, S^5 is the unit sphere in R^6, not of R^5. It should be five-dimensional, not four-dimensional. Your B^5 is correct, though.

 

[mathwonk]

you might try computing the volume of B^5 to get a first idea of what it "looks" like. and then maybe you can deduce the "area" of S^4.