Do we know everything about 2d geometry?

[kramer733]

I guess this is classified as euclidean geomtry right? So do we? If not, what else is there to know?

 

[pwsnafu]

I guess this is classified as euclidean geomtry right? So do we? If not, what else is there to know?

Well there http://en.wikipedia.org/wiki/Rational_trigonometry" for a start.

 

[Eynstone]

Obviously, we don't - people still hit upon new theorems in plane geometry.

 

[TylerH]

Eynstone, can you give an example of a recent theorem in Euclidean geometry? Not that I don't believe you, just that it makes me wonder what is it that we didn't know about it until recently.

Kramer, If you like Euclidean geometry you might like finite(discrete) geometry. IMHO, it's a lot more fun. Have you studied it any?

 

[micromass]

Well, one thing that still deludes us is the classification of polynomial equations.
For polynomials of degree one, that is: aX+bY+c, we know what it looks like graphically: lines.
Polynomials of degree two have the general form

$$aX^2+bXY+cY^2+dX+eY+f=0$$

These things are conic sections and can be classified as ellipses, parabolas and hyperbolas.
Third degree polynomials are far less understood, but can still be classified.
But in general, I don't think there's a general classification for general n-degree polynomials...

 

[TylerH]

Well, one thing that still deludes us is the classification of polynomial equations.
For polynomials of degree one, that is: aX+bY+c, we know what it looks like graphically: lines.
Polynomials of degree two have the general form

$$aX^2+bXY+cY^2+dX+eY+f=0$$

These things are conic sections and can be classified as ellipses, parabolas and hyperbolas.
Third degree polynomials are far less understood, but can still be classified.
But in general, I don't think there's a general classification for general n-degree polynomials...

Wouldn't that be algebraic geometry? That's a couple thousand years past Euclidean geometry. (I guess discrete geometry is too... but I'm hypocritical.)

 

[micromass]

Wouldn't that be algebraic geometry? That's a couple thousand years past Euclidean geometry. (I guess discrete geometry is too... but I'm hypocritical.)

Well, yes, these questions are studied in algebraic geometry. However, the OP asked for things in 2D-geometry, and these questions are 2D-geometry.

If you only want Euclidean geometry, that is geometry that Euclid performed, then I guess that some research might still be possible in topics like triangle centers ( http://faculty.evansville.edu/ck6/encyclopedia/ETC.html )

Also, algebraic geometry isn't as young as one might think. The study of conic sections is as old as the elements of Euclid. The study of cubics however, was already performed by Newton, thus it's already about 500 years old.
Modern algebraic geometry is only about 50 years old, but it's roots lie very deep!

 

[wisvuze]

how can you ever justify a statement like is that ____ field is done? You cannot be done with a field of study , if you have yet to know all the questions!