Changes

I am going to bypass Euclid's other assumptions until we need them.

I am going to bypass Euclid's other assumptions until we need them.

* Suppose, since this is plane geometry, that we have a plane with no points in it "given". Then we can perform none of these three constructions. We can, if we like, consider a geometry with 0 points and 0 lines as a trivial Euclidean geometry.

* Suppose, since this is plane geometry, that we have a plane with no points in it "given". Then we can perform none of these three constructions. We can, if we like, consider a geometry with 0 points and 0 lines as a trivial Euclidean geometry.

* One point? Nope. We need two points or a line segment to do anything. Still trivial.

* One point? Nope. We need two points or a line segment to do anything. Still trivial.

* Two points? Now we are getting somewhere. We can connect the two points with a straight line segment, and then extend the line segment. We can't go anywhere in particular by doing so. But we can draw two circles, one with each of the given points as center, and the line segment joining them as radius. Those two circles intersect in more points that are not on the original line, and also intersect the extensions of the original line (which now does go somewhere of interest), so that we can draw more lines and more circles, and so on and on forever, with segments as long as we want and as short as we want, and as close as we want to any direction. This is the smallest set of points that make up a non-trivial Euclidean geometry. Pythagoras thought that it included every point in the plane, but it was proved later on in Greek times that it omits infinitely many points, such as the cube root of 2. In the 19th century, Georg Cantor proved that point sets of this kind omit "almost all" points in the plane, since the number of points omitted is vastly greater than the number of points included. Infinitely greater, in fact. However, that is a topic for another time, after we lay a proper foundation for such ideas. That should not prevent you from asking yourself what it could mean for one infinite number to be infinitely larger than another, a question that Greek mathematicians could not have dreamed of, much less ask seriously.

 

* Two points? Now we are getting somewhere. We can connect the two points with a straight line segment, and then extend the line segment.  

 

 

We can't go anywhere in particular by doing so. But we can draw two circles, one with each of the given points as center, and the line segment joining them as radius. Those two circles intersect in more points that are not on the original line, and also intersect the extensions of the original line (which now does go somewhere of interest).

 

 

Then we can draw more lines and more circles, and so on and on forever, with segments as long as we want and as short as we want, by bisecting lines,

 

 

 

and as close as we want to any direction, by bisecting angles.

 

[[File:TAEuclidBisectAngle.png]]

 

This is the smallest set of points that make up a non-trivial Euclidean geometry. Pythagoras thought that it included every point in the plane, but it was proved later on in Greek times that it omits infinitely many points, such as the cube root of 2. In the 19th century, Georg Cantor proved that point sets of this kind omit "almost all" points in the plane, since the number of points omitted is vastly greater than the number of points included. Infinitely greater, in fact. However, that is a topic for another time, after we lay a proper foundation for such ideas. That should not prevent you from asking yourself what it could mean for one infinite number to be infinitely larger than another, a question that Greek mathematicians could not have dreamed of, much less ask seriously.

* If we started with just one given line segment, that would give us its two endpoints, with the same result as just above.

* If we started with just one given line segment, that would give us its two endpoints, with the same result as just above.