S^5 Sphere: Visualizing and Understanding

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In summary, an S^5 sphere is a subset of five-dimensional points in R^6 that satisfy a specific equation, while the "Ball" subset is a set of points that satisfy a similar equation but with a less strict inequality. It is important to note that S^5 is five-dimensional, not four-dimensional as previously mentioned. To visualize it, one can compute the volume of the "Ball" subset and use that to deduce the "area" of the S^5 sphere.
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What does S^5 sphere mean? How can I imagine it?
Thanks
 
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http://en.wikipedia.org/wiki/Hypersphere" [Broken]
 
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  • #3
It is the subset of R5 of points [itex](x_1, x_2, x_3, x_4, x_5)[/itex] such that
[tex]x_1^2+ x_2^2+ x_3^2+ x_4^2+ x_5^2= 1[/tex]

(The "Ball", B5, is the set of points [itex](x_1, x_2, x_3, x_4, x_5)[/itex] such that
[tex]x_1^2+ x_2^2+ x_3^2+ x_4^2+ x_5^2<= 1[/tex])
 
  • #4
HallsofIvy said:
It is the subset of R5 of points [itex](x_1, x_2, x_3, x_4, x_5)[/itex] such that
[tex]x_1^2+ x_2^2+ x_3^2+ x_4^2+ x_5^2= 1[/tex]

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

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.
 
  • #5
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.
 

1. What is a S^5 sphere?

A S^5 sphere is a five-dimensional hypersphere, also known as a 5-sphere, which is a mathematical construct used in geometry and topology to visualize and understand spaces with five dimensions.

2. How is a S^5 sphere visualized?

A S^5 sphere can be visualized as a three-dimensional object in the same way a 2-sphere (a regular sphere) can be visualized in two dimensions. This is often done using projections or stereographic techniques.

3. What is the significance of a S^5 sphere?

A S^5 sphere is important in mathematics and physics as it helps us understand and study higher dimensional spaces, which have important applications in fields such as string theory and cosmology.

4. Can the S^5 sphere be visualized in the real world?

No, the S^5 sphere cannot be visualized in the physical world as it exists in five dimensions, which is beyond our perception and understanding. It can only be visualized using mathematical constructs and techniques.

5. How is the S^5 sphere used in research?

The S^5 sphere is used in various research fields, such as topology, geometry, and physics, to understand and study higher dimensional spaces and their properties. It also has practical applications in data analysis and computer graphics.

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