Hypersphere

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For spheres in hyperspace, see n-sphere. Hypersphere_sentence_0

In geometry of higher dimensions, a hypersphere is the set of points at a constant distance from a given point called its centre. Hypersphere_sentence_1

It is a manifold of codimension one—that is, with one dimension less than that of the ambient space. Hypersphere_sentence_2

As the hypersphere's radius increases, its curvature decreases. Hypersphere_sentence_3

In the limit, a hypersphere approaches the zero curvature of a hyperplane. Hypersphere_sentence_4

Hyperplanes and hyperspheres are examples of hypersurfaces. Hypersphere_sentence_5

The term hypersphere was introduced by Duncan Sommerville in his 1914 discussion of models for non-Euclidean geometry. Hypersphere_sentence_6

The first one mentioned is a 3-sphere in four dimensions. Hypersphere_sentence_7

Some spheres are not hyperspheres: If S is a sphere in E where m < n, and the space has n dimensions, then S is not a hypersphere. Hypersphere_sentence_8

Similarly, any n-sphere in a proper flat is not a hypersphere. Hypersphere_sentence_9

For example, a circle is not a hypersphere in three-dimensional space, but it is a hypersphere in the plane. Hypersphere_sentence_10


Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Hypersphere.