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  1. Copilot Answer
    Overview

    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 interio… See more

    Definition

    In coordinates, a 3-sphere with center (C0, C1, C2, C3) and radius r is the set of all points (x0, x1, x2, x3) in real, 4-dimensional space (R ) such that
    The 3-sphere centered at the origin with radius 1 is … See more

    Properties

    The 3-dimensional surface volume of a 3-sphere of radius r is
    while the 4-dimensional hypervolume (the content of the 4-dimensional region, or ball, bounded by the 3-sphere) is
    Every non-… See more

    Topological construction

    There are several well-known constructions of the three-sphere. Here we describe gluing a pair of three-balls and then the one-point compactification.
    A 3-sphere can be constructed topologically by … See more

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