Sphere
Elementary
Think of a ball, like a basketball. A sphere is the surface of the ball, like the skin of the basketball. A sphere is hollow: it does not include the inside.
High school
Consider a point $c$ in space (the center) and a distance $r$ (the radius). The sphere with center $c$ and radius $r$ is the set of all points in space at distance $r$ from $c$.
Undergraduate (metric spaces)
Consider a metric space with metric $d$, a center $c$, and a radius $r$. The sphere of radius $r$ around $c$ is
\[S(c, r) = \setdef{x \suchthat d(x, c) = r}.\]In $\reals^{n+1}$ with the Euclidean metric, the unit sphere is written $S^n$; it is $n$-dimensional.
Examples
Examples
- The surface of a basketball High school
Non-examples
Things that may be mistaken for examples.
-
A solid bowling ball
High school
A solid bowling ball is not a sphere, which would be only the surface.
-
The surface of a cube
High school
The surface of a cube is not a (geometric) sphere, because its points are not all at the same distance from the center.
Notes
The dimension convention is a common source of confusion: the ordinary sphere in $\reals^3$ is the $2$-sphere $S^2$, because its surface is two-dimensional.