Negative Curvature: Can We Draw an Infinite Triangle?

[randa177]

Can we draw an infinite equlateral triangle on a negatively curved surface?

 

[tiny-tim]

Hi randa177!

Can we draw an infinite equlateral triangle on a negatively curved surface?

Yes.

Start here.

Draw three "straight" lines at angles of 120º, all with length L.

Join the three endpoints.

That's an equilateral triangle, and its sides are longer than L√3.

Make L as large as you like.

 

[Ben Niehoff]

Using the Poincare disc model, this may be clearer. You can make any regular n-gon out of limit rays, which makes the n-gon infinite in area (I think), with vertex angles of 0 degrees.

Not only can you make an infinite equilateral triangle, you can actually tile the hyperbolic plane with them! See an animation here:

http://mathworld.wolfram.com/PoincareHyperbolicDisk.html

 

[randa177]

This is very interesting. I've never heard about the poincare disc, is it negatively curved? In the 3-d picture it looks more like a positively curved surface...

 

[randa177]

Actually what confuses me is that the negativle curved surfaces have always a finite toatal area, so how would it be posible to draw an infinite triangle on it? ( we run out of space! )

 

[Doodle Bob]

Actually what confuses me is that the negativle curved surfaces have always a finite toatal area, so how would it be posible to draw an infinite triangle on it? ( we run out of space! )

Actually, this is not true. The Poincare disk has infinite area. That's why the triangles in the tesselation are getting smaller and smaller as they approach the boundary: The boundary circle is the boundary of the hyperbolic plane at infinity.

All triangles -- including those with vertices on the line at infinity -- have finite area -- but possibly infinitely long sides. But the area of the entire plane is infinite.

Of course, if you are looking at a compact surface with negative curvature, then all of the measurements of any given triangle are finite.

 

[tiny-tim]

the hyperbolic plane

Actually what confuses me is that the negativle curved surfaces have always a finite toatal area, so how would it be posible to draw an infinite triangle on it? ( we run out of space! )

Hi randa177!

The Poincaré disc is one of the representations of the hyperbolic plane.

See wikipedia: http://en.wikipedia.org/wiki/Hyperbolic_geometry#Models_of_the_hyperbolic_plane"

It has negative curvature.

It has infinite area, since each of the black or white triangles in the diagram are indentical in both shape and size, and there are infinitely many of them.

The "infinite" triangles, of course, also have the same area, which I believe is finite.

 

[HallsofIvy]

Actually what confuses me is that the negativle curved surfaces have always a finite toatal area, so how would it be posible to draw an infinite triangle on it? ( we run out of space! )

No. surfaces of positive constant curvature (spheres) have finite area, not sufaces of negative curvature.

 

[randa177]

yes,,, that's right,,, I just got confused,,,
so there is no limit for the area of a equileteral triangle on a negatively curved space. Is that correct?

 

[randa177]

but what are the 3 angles of the equilateral triangle drawn on a negatively curved surface?

 

[tiny-tim]

The 3 angles of an infinite equilateral triangle drawn on a homogenous negatively curved surface are zero.

(The angle between any two straight lines on that surface which meet "at infinity" is zero.)

 

[HallsofIvy]

In either hyperbolic or ellipitic geometry (positive or negative curvature) the angle sum in a triangle depends upon the size of the triangle. Thus, while it can be proven that the three angles in an equilateral triangle are the same, what they are depends upon the size of the triangle. As tiny tim said, the measure of the angles of an "infinitely large" equilateral triangle are 0. The measure of the angles of an "infinitesmally small" triangle are "infinitesmally" close to 60 degrees or $\pi/3$ radians.

 

[DeaconJohn]

In either hyperbolic or ellipitic geometry (positive or negative curvature) the angle sum in a triangle depends upon the size of the triangle. Thus, while it can be proven that the three angles in an equilateral triangle are the same, what they are depends upon the size of the triangle. As tiny tim said, the measure of the angles of an "infinitely large" equilateral triangle are 0. The measure of the angles of an "infinitesmally small" triangle are "infinitesmally" close to 60 degrees or $\pi/3$ radians.

Now, it that cool, or, what?

I'm really impressed. Hyperbolic geometry, elliptic geometry, Wow!

"What won't they think of next?"

DJ

 

[tiny-tim]

"What won't they think of next?"

anabolic geometry! ​

 

[DeaconJohn]

anabolic geometry! ​

ROTFL loud and hard! Tiny Tim, you made my day!