S^5 Sphere: Visualizing and Understanding

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Discussion Overview

The discussion revolves around the concept of the S^5 sphere, its mathematical definition, and how to visualize it. Participants explore its properties in relation to higher-dimensional spaces, specifically R^5 and R^6, and consider implications for understanding its geometry and volume.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant asks for clarification on the meaning of the S^5 sphere and how to visualize it.
  • A link to a Wikipedia article on hyperspheres is provided for additional context.
  • Another participant defines S^5 as the subset of R^5 consisting of points satisfying the equation x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 = 1, and describes the corresponding "Ball" B^5.
  • A subsequent post corrects the dimensionality, asserting that S^5 is actually the unit sphere in R^6, not R^5, while agreeing that the definition of B^5 is accurate.
  • One participant suggests computing the volume of B^5 to gain insights into its geometric properties and potentially deducing the area of S^4.

Areas of Agreement / Disagreement

There is disagreement regarding the dimensionality of S^5, with some participants asserting it is in R^6 while others initially reference R^5. The discussion remains unresolved regarding the visualization and understanding of higher-dimensional spheres.

Contextual Notes

Participants express uncertainty about visualizing higher-dimensional objects and the implications of their definitions. There are also unresolved aspects concerning the calculations of volumes and areas related to these spheres.

Who May Find This Useful

This discussion may be useful for individuals interested in higher-dimensional geometry, mathematical definitions of spheres, and those exploring concepts in topology and related fields.

yola
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What does S^5 sphere mean? How can I imagine it?
Thanks
 
Physics news on Phys.org
http://en.wikipedia.org/wiki/Hypersphere"
 
Last edited by a moderator:
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)
 
HallsofIvy said:
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.
 
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.
 

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