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    3-sphere - Wikipedia

    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,
    The … 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

    Coordinate systems on the 3-sphere

    The four Euclidean coordinates for S are redundant since they are subject to the condition that x0 + x1 + x2 + x3 = 1. As a 3-dimensional manifold one should be able to parameterize S by three coordinates, just as one can … See more

    Group structure

    When considered as the set of unit quaternions, S inherits an important structure, namely that of quaternionic multiplication. Because the set of unit quaternions is closed under multiplication, S takes on … See more

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