Information About 3-sphere

	CATEGORIES ABOUT 3-SPHERE
	4-dimensional geometry
	algebraic topology
	elementary geometry
	geometric topology
	mathematical analysis
	quaternions
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In an entirely analogous manner one can define higher-dimensional spheres called Hypersphere s or ''n''-spheres. Such objects are ''n''-dimensional Manifold s. Some people refer to a 3-sphere as a glome from the Latin word ''glomus'' meaning ''ball''. Roughly speaking, a glome is to a sphere as a sphere is to a circle. DEFINITION In Coordinates , a 3-sphere with center (''x''0, ''y''0, ''z''0, ''w''0) and radius ''r'' is the set of all points (''x'',''y'',''z'',w) in R4 such that :$(\; x\; -\; x\_0\; )^2\; +\; (\; y\; -\; y\_0\; )^2\; +\; (\; z\; -\; z\_0\; )^2\; +\; (\; w\; -\; w\_0\; )^2\; =\; r^2.\; \backslash ,$ The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted ''S''3. It can be described as a subset of either '''R'''4, '''C'''2, or '''H''' (the Quaternion s): :$S^3\; =\; \backslash left\backslash \{(x\_1,x\_2,x\_3,x\_4)\backslash in\backslash mathbb\{R\}^4\backslash mid\; x\_1^2\; +\; x\_2^2\; +\; x\_3^2\; +\; x\_4^2\; =\; 1\; ight\backslash \}$
	:<math>S^3	\left\{q\in\mathbb{H}\mid q = 1 ight\}</math>
	{ Class	"wikitable"

  Where '''u''' (''u''₁, ''u''₂, ''u''₃) is a vector in '''R'''³ and ''u''² = ''u''₁² + ''u''₂² + ''u''₃² In the second equality above we have identified ''p'' with a unit quaternion and '''u''' = ''u''₁ ''i'' + ''u''₂ ''j'' + ''u''₃ ''k'' with a pure quaternion (Note that the division here is well-defined even though quaternionic multiplication is generally noncommutative) The inverse of this map takes ''p'' = (''x''₀, ''x''₁, ''x''₂, ''x''₃) in ''S''³ to


  :<math>\mathbf{v} rac{1}{\u\^2}\mathbf{u}</math>