MathFoundations25: The basic framework for geometry (III)

[njwildberger]

The definition of parallel is not that the two lines don't meet. It is rather an algebraic condition---see the previous videos. if two lines are the same, then they are parallel, even though they pass through many common points.

[AdolfOliverBush4]

Hey buddy... Your last theorem says that A1,A3 is parallel to A2,A3. Earlier you said that A3 is a point ... parallel lines do not intersect at any point whatever... so how can the the line described by point, A1, be collinear with the point, A3, and also be parallel with the line described by the point, A2, and the same point, A3?

[njwildberger]

Not quite. Lines which are the same are also parallel. So A1A2 being parallel to A1A3 is just the same as saying that A1,A2 and A3 are collinear.

I stated things that way to emphasize that Pythagoras theorem and the Triple quad formula are really analogous theorems.

[njwildberger]

Yes they do. However the more interesting question about the existence of a regular polygon with n sides is more subtle.

Later on I will show how to easily extend the geometry over the rational numbers to `extension fields'. This doesn't change the underlying theorems too much, but it does enlarge the kinds of configurations we can discuss.

In any case, rational approximations are always available to us, these suffice in engineering and scientific practice.

[Pebbe496]

Do any equilateral triangles exist in rationol geometry?

Because if not, I don't think it is very useful...