# Hypersphere: Difference between revisions

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A '''hypersphere''' is a |
A '''hypersphere''' is a generalization of the [[circle]] and [[sphere]] to any number of dimensions. The '''<nowiki/>''n''-sphere''' or <math>S^n</math> comprises the set of points in (''n'' + 1)-dimensional Euclidean space that are all a fixed nonzero distance from a given point, which is designated as its center. The fixed distance is the '''radius''', and the resulting shape is an ''n''-dimensional manifold. |
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The ''n''-sphere is the boundary of the '''closed (''n'' + 1)-ball''', which is the set of points in whose distance from the center is less than or equal to a given radius. The '''open (''n'' + 1)-ball''' is the closed (''n'' + 1)-ball without its boundary. |
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It is the most common practice to refer to a hypersphere by the dimensionality of its surface, which is one less than the space it occupies. For example, a 3-sphere refers to the hypersphere in 4D space (a [[glome]]). |
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==Examples== |
==Examples== |
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## Latest revision as of 19:59, 11 February 2024

A **hypersphere** is a generalization of the circle and sphere to any number of dimensions. The ** n-sphere** or comprises the set of points in (

*n*+ 1)-dimensional Euclidean space that are all a fixed nonzero distance from a given point, which is designated as its center. The fixed distance is the

**radius**, and the resulting shape is an

*n*-dimensional manifold.

The *n*-sphere is the boundary of the **closed ( n + 1)-ball**, which is the set of points in whose distance from the center is less than or equal to a given radius. The

**open (**is the closed (

*n*+ 1)-ball*n*+ 1)-ball without its boundary.

## Examples[edit | edit source]

Excluding the point, the hyperspheres up to 10D are the following:

Rank | Name | Picture | Rank | Name | Picture | |
---|---|---|---|---|---|---|

1 | Dyad | 6 | Hexasphere | |||

2 | Circle | 7 | Heptasphere | |||

3 | Sphere | 8 | Octasphere | |||

4 | 3-sphere (glome) | 9 | Enneasphere | |||

5 | Pentasphere | 10 | Decasphere |

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