As I aforementioned, there is a fine line between axiom and Definition. In my Holt geometry book there is an axiom that states,  if two points are on a plane then the line containing those points is also on the plane. One implication of this axiom is that planes are smooth. If a plane is curved or has a bump, then it would be possible to make a line that only has two of its points on the surface:In my opinion, this axiom about lines and planes is what Euclid says in his definition of a plane, "a surface that lies evenly with all the lines on itself." Last August, I rewrote all of my definitions, etc, in anticipation of our class debates. As, I contemplated the definition of a plane, I wrote: "A plane is a two dimensional surface that extends indefinitely in both dimensions, such that for any two points on the plane the line containing those two points is also on the plane." Of course, that is not what you saw on the table I made in my last blog. What I decided to write on the table was the definition I choose to give to my students, which was what my coworkers and I decided to agree upon for our classes.

However, other descriptions of flatness exist. Here is another axiom used by Holt and Mr. Hilbert, "through any three non-colinear points there exists exactly one plane."  To discuss this with my students I describe a plane as a triangle that is either tessellated or expanded(i.e. dilated) indefinitely. I believe that this description of a flat surface is what Mr. Playfair's meant in his definition of a plane, "a surface that can be derived by moving a straight line along two straight lines that make an angle." This definition describes dilating a triangle. Note that we may similarly describe the straightness of a line; my Holt text reads, "through any two points there is exactly one line." Similar statements are made about lines in all of my geometry source texts.


Other descriptions of both flatness for a plane and the straightness of a line can be found in these common axioms:
a) The intersection of two lines is a single point; i.e. lines can't bend or weave back across each other.
b) The intersection of two planes is a line; i.e. planes can't bend or weave back and forth across each other.

I have also thought of this latter axiom as a description of the flatness or straightness of space itself; space doesn't curl around to on itself.

There are also some other less obvious descriptions of flatness that have a direct bearing on modern geometry, but I will save those for my next blog.