Can a Triangle Have Angles Over 180 Degrees Without 3D Embedding?

[Pjpic]

Is it possible that a triangle with angles totaling over 180 degrees could exist without being embedded in a 3rd dimension?

 

[HallsofIvy]

If you are talking about Euclidean geometry then, no. Of course, you can have a hyperbolic two dimensional geometry without it being imbedded in a three dimensional Euclidean geometry.

 

[Pjpic]

a hyperbolic two dimensional geometry without it being imbedded

Does this have something to do with it being "intrinsic"?

 

[mathwonk]

to speak of the sum of angles in a triangle in a geometry you just need some way to compare angles, AND ADD THEM AND THEN TO SAY THAT THE SUM OF ANGLES IN A TRIANGLE IS MORE THAN (oops) a straight angle.

if you want to prove such examples exist, it depends what your standards of belief are. if you are someone who believes only in euclidean space, then for you it is necessary to find every other example embedded there.

it is quite consistent to imagine spherical or hyperbolic geometries, where triangles add to other than a straight angle, but to produce examples of them, we often look in euclidean space of higher dimension.

i think Halls meant that for a triangle with angle sum more than a straight angle, we probably look in a spherical geometry, and for less, in a hyperbolic geometry.

 

[HallsofIvy]

I always get "hyperbolic" and "elliptic" geometries confused! Of course, I should just associate "ellipsoid" with "spherical".