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In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point. The interior of a 3-sphere is a 4-ball. It is called a 3-sphere because topologically, the surface itself is 3-dimensional, even ...
2 = 1; This is the set of points in 3-dimensional Euclidean space found exactly one unit away from the origin. It is called the 2-sphere, S 2, for reasons given below. The same idea applies for any dimension n; the equation x 2 0 + x 2 1 + ⋯ + x 2 n = 1 produces the n-sphere as a geometric object in (n + 1)-dimensional space. For example, the ...
In mathematics, an n-sphere or hypersphere is an -dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The circle is considered 1-dimensional, and the sphere 2-dimensional, because the surfaces themselves are 1- and 2-dimensional respectively, not because they ...
Technically, Hopf found a many-to-one continuous function (or "map") from the 3-sphere onto the 2-sphere such that each distinct point of the 2-sphere is mapped from a distinct great circle of the 3-sphere . [1] Thus the 3-sphere is composed of fibers, where each fiber is a circle — one for each point of the 2-sphere.
T d is isomorphic to S 4, the symmetric group on 4 letters, because there is a 1-to-1 correspondence between the elements of T d and the 24 permutations of the four 3-fold axes. An object of C 3v symmetry under one of the 3-fold axes gives rise under the action of T d to an orbit consisting of four such objects, and T d corresponds to the set ...
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.
In differential topology, sphere eversion is a theoretical process of turning a sphere inside out in a three-dimensional space (the word eversion means "turning inside out"). It is possible to smoothly and continuously turn a sphere inside out in this way (allowing self-intersections of the sphere's surface) without cutting or tearing it or ...
If n is 1 mod 4 it has order 1 or 2; in particular it has order 1 if n is 1, 5, 13, 29, or 61, and William Browder proved that it has order 2 if = mod 4 is not of the form . It follows from the now almost completely resolved Kervaire invariant problem that it has order 2 for all n bigger than 126; the case n = 126 {\displaystyle n=126} is still ...