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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 ...
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.
Very little is known about irregular hypersphere packings; it is possible that in some dimensions the densest packing may be irregular. Some support for this conjecture comes from the fact that in certain dimensions (e.g. 10) the densest known irregular packing is denser than the densest known regular packing. [12]
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The volume can be computed without use of the Gamma function. As is proved below using a vector-calculus double integral in polar coordinates, the volume V of an n-ball of radius R can be expressed recursively in terms of the volume of an (n − 2)-ball, via the interleaved recurrence relation:
A horosphere within the Poincaré disk model tangent to the edges of a hexagonal tiling cell of a hexagonal tiling honeycomb Apollonian sphere packing can be seen as showing horospheres that are tangent to an outer sphere of a Poincaré disk model. In hyperbolic geometry, a horosphere (or parasphere) is a specific hypersurface in hyperbolic n ...
Created Date: 8/30/2012 4:52:52 PM
The deNeve/Hills sphere eversion: video and interactive model; Patrick Massot's project to formalise the proof in the Lean Theorem Prover; An interactive exploration of Adam Bednorz and Witold Bednorz method of sphere eversion; Outside In: A video exploration of sphere eversion, created by The Geometry Center of The University of Minnesota