Search results
Results from the WOW.Com Content Network
The 3-sphere is homeomorphic to the one-point compactification of R 3. In general, any topological space that is homeomorphic to the 3-sphere is called a topological 3-sphere. The homology groups of the 3-sphere are as follows: H 0 (S 3, Z) and H 3 (S 3, Z) are both infinite cyclic, while H i (S 3, Z) = {} for all other indices i.
If is a linear space with a real quadratic form:, then {: =} may be called the unit sphere [3] [4] or unit quasi-sphere of . For example, the quadratic form x 2 − y 2 {\displaystyle x^{2}-y^{2}} , when set equal to one, produces the unit hyperbola , which plays the role of the "unit circle" in the plane of split-complex numbers .
b each has a radius of 1 / √ 2 , their Clifford torus product will fit perfectly within the unit 3-sphere S 3, which is a 3-dimensional submanifold of R 4. When mathematically convenient, the Clifford torus can be viewed as residing inside the complex coordinate space C 2, since C 2 is topologically equivalent to R 4.
For example, one sphere that is described in Cartesian coordinates with the equation x 2 + y 2 + z 2 = c 2 can be described in spherical coordinates by the simple equation r = c. (In this system—shown here in the mathematics convention—the sphere is adapted as a unit sphere, where the radius is set to unity and then can generally be ignored ...
S 2: a 2-sphere is an ordinary sphere; S 3: a 3-sphere is a sphere in 4-dimensional Euclidean space. Spheres for n > 2 are sometimes called hyperspheres. The n-sphere of unit radius centered at the origin is denoted S n and is often referred to as "the" n-sphere. The ordinary sphere is a 2-sphere, because it is a 2-dimensional ...
The 3-sphere is the boundary of a -ball in four-dimensional space. The -sphere is the boundary of an -ball. Given a Cartesian coordinate system, the unit -sphere of radius can be defined as:
In the mathematical field of geometric topology, the Poincaré conjecture (UK: / ˈ p w æ̃ k ær eɪ /, [2] US: / ˌ p w æ̃ k ɑː ˈ r eɪ /, [3] [4] French: [pwɛ̃kaʁe]) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.
The first component is a complex number, whereas the second component is real. Any point on the 3-sphere must have the property that |z 0 | 2 + |z 1 | 2 = 1. If that is so, then p(z 0, z 1) lies on the unit 2-sphere in C × R, as may be shown by adding the squares of the absolute values of the complex and real components of p