Definition:Sphere

Geometry
A sphere is a surface in solid geometry such that every straight line falling upon it from one particular point inside it are equal.



Center
That point is called the center of the sphere.

Radius
A radius (plural radii, pronounced ray-dee-eye) of a sphere is a straight line segment whose endpoints are the center and the surface of the sphere.

The radius of a sphere is the length of one such radius.


 * Sphere.png

Thus it is the three-dimensional version of the circle.

Every point on the sphere is at the same distance from its center.

Topology
In the field of topology the concept is generalized.

The $n$-dimensional sphere, or $n$-sphere, is the set:


 * $\Bbb S^n = \left\{{x \in \R^{n+1} : \left|{x - y}\right| = r}\right\}$

where $r \in \R_+$ is called the radius of the sphere and $y \in \R^{n+1}$ is called the center of the sphere.

Frequently, the radius is taken as $1$ and the center as the origin.

Note
As the sphere is defined here, it is specified as being the surface only, that is, not the inside.