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The set of all spheres satisfying this equation is called a pencil of spheres determined by the original two spheres. In this definition a sphere is allowed to be a plane (infinite radius, center at infinity) and if both the original spheres are planes then all the spheres of the pencil are planes, otherwise there is only one plane (the radical ...
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Spherical geometry or spherics (from Ancient Greek σφαιρικά) is the geometry of the two-dimensional surface of a sphere [a] or the n-dimensional surface of higher dimensional spheres.
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 ...
The upper bound for the density of a strictly jammed sphere packing with any set of radii is 1 – an example of such a packing of spheres is the Apollonian sphere packing. The lower bound for such a sphere packing is 0 – an example is the Dionysian sphere packing. [27]
Some candidates proposed for exotic 4-spheres are the Cappell–Shaneson spheres (Sylvain Cappell and Julius Shaneson ) and those derived by Gluck twists . Gluck twist spheres are constructed by cutting out a tubular neighborhood of a 2-sphere S in S 4 and gluing it back in using a diffeomorphism of its boundary S 2 ×S 1.
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The -spheres admit several other topological descriptions: for example, they can be constructed by gluing two -dimensional spaces together, by identifying the boundary of an -cube with a point, or (inductively) by forming the suspension of an -sphere.