Changes

The surface of an ordinary sphere, like the Earth, is called a two sphere, or <math>\mathbb{S}^2 \ </math>.  The defining quality of a sphere is that it is the set of all points in a Euclidean (i.e., normal) three dimensional space which are the same distance from a set point.  Note that the interior of a sphere is not <math>\mathbb{S}^2 \ </math>, and that like <math>\mathbb{S}^1 \ </math>, a deflated beach ball stretched into some strange shape is still <math>\mathbb{S}^2 \ </math>, so long as it does not self-intersect.

The surface of an ordinary sphere, like the Earth, is called a two sphere, or <math>\mathbb{S}^2 \ </math>.  The defining quality of a sphere is that it is the set of all points in a Euclidean (i.e., normal) three dimensional space which are the same distance from a set point.  Note that the interior of a sphere is not <math>\mathbb{S}^2 \ </math>, and that like <math>\mathbb{S}^1 \ </math>, a deflated beach ball stretched into some strange shape is still <math>\mathbb{S}^2 \ </math>, so long as it does not self-intersect.

The three sphere, or <math>\mathbb{S}^3 \ </math>, refers to the set of all points in a four dimensional Euclidean space which are the same distance from a single point.  Any deformation of that object is a <math>\mathbb{S}^3 \ </math>.

The three sphere, or <math>\mathbb{S}^3 \ </math>, refers to the set of all points in a four-dimensional Euclidean space which are the same distance from a single point.  Any deformation of that object is a <math>\mathbb{S}^3 \ </math>.

===Strings, surfaces, and looping around===

===Strings, surfaces, and looping around===