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→The three sphere: a "plane" is one of multiple conjectured "planes" that exist: needs indefinite article; "distance" is a noun that may convey a reference to a particular one (as by the adjective "some") as the adjective "distant" cannot: substituted
===The three sphere===
===The three sphere===
Mathematicians call the ordinary circle the "one sphere," which is written <math>\mathbb{S}^1 \ </math>. The defining quality of a circle is that it is the set of all points in plane which are the same distant from a specific point. Any deformation of this circle is a <math>\mathbb{S}^1 \ </math> - an object does not need to be a perfect circle to be called a one sphere by topologists, any closed loop which does not intersect itself is an example of <math>\mathbb{S}^1 \ </math>.
Mathematicians call the ordinary circle the "one sphere," which is written <math>\mathbb{S}^1 \ </math>. The defining quality of a circle is that it is the set of all points in a plane which are the same distance from a specific point. Any deformation of this circle is a <math>\mathbb{S}^1 \ </math> - an object does not need to be a perfect circle to be called a one sphere by topologists, any closed loop which does not intersect itself is an example of <math>\mathbb{S}^1 \ </math>.
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.