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  2. 3-sphere - Wikipedia

    en.wikipedia.org/wiki/3-sphere

    Direct projection of 3-sphere into 3D space and covered with surface grid, showing structure as stack of 3D spheres (2-spheres) 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.

  3. Hopf fibration - Wikipedia

    en.wikipedia.org/wiki/Hopf_fibration

    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.

  4. Homotopy groups of spheres - Wikipedia

    en.wikipedia.org/wiki/Homotopy_groups_of_spheres

    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 1 ...

  5. n-sphere - Wikipedia

    en.wikipedia.org/wiki/N-sphere

    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 ...

  6. Klein bottle - Wikipedia

    en.wikipedia.org/wiki/Klein_bottle

    A two-dimensional representation of the Klein bottle immersed in three-dimensional space. In mathematics, the Klein bottle (/ ˈ k l aɪ n /) is an example of a non-orientable surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down.

  7. 3-manifold - Wikipedia

    en.wikipedia.org/wiki/3-manifold

    The Poincaré homology sphere (also known as Poincaré dodecahedral space) is a particular example of a homology sphere. Being a spherical 3-manifold, it is the only homology 3-sphere (besides the 3-sphere itself) with a finite fundamental group. Its fundamental group is known as the binary icosahedral group and has order 120.

  8. Poincaré conjecture - Wikipedia

    en.wikipedia.org/wiki/Poincaré_conjecture

    The two-dimensional analogue of the Poincaré conjecture says that any two-dimensional topological manifold which is closed and connected but non-homeomorphic to the two-dimensional sphere must possess a loop which cannot be continuously contracted to a point. (This is illustrated by the example of the torus, as above.)

  9. Three-dimensional space - Wikipedia

    en.wikipedia.org/wiki/Three-dimensional_space

    A perspective projection of a sphere onto two dimensions. A sphere in 3-space (also called a 2-sphere because it is a 2-dimensional object) consists of the set of all points in 3-space at a fixed distance r from a central point P. The solid enclosed by the sphere is called a ball (or, more precisely a 3-ball). The volume of the ball is given by