Hypersphere: Difference between revisions

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A '''hypersphere''' is a highly-symmetric object in each respective [[dimension]]. Hyperspheres are not polytopes, as they are round objects with no vertices or other flat elements. The surface of the hypersphere in n dimensions is the set of all points a fixed distance away from a given point (its center).
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.


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.
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]]).


==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 (n + 1)-ball is the closed (n + 1)-ball without its boundary.

Examples[edit | edit source]

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

Orthoplexes by dimension
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