iNVERTED wrote:As the angle sum of triangle goes to zero, its sides grow to infinity, but the area stays within the limit.
If the sides are infinitely long, and the area stays finite, that means that there must be an infinate length of each side where the distance to the adjacent side is 1/inf (I use 1/inf instead of 0 to mean an infinately small number greater than zero), which makes the area undefined...
Not really. The thing is that while there are two different ways to arrange two straight lines in E-space (intersecting or parallel), there are three ways in H-space: intersecting, parallel, or ultraparallel.
How does it look? For parallel lines in H-space, there is no constant distance between them. Instead, the distance grows indefinitely as you go in one direction, but shrinks, with limit at 0, as you go in the other. (ultraparallel lines have a "shortest chord" somewhere, and when you go to either side, their distance increases). Basically, parallel lines cross at a point at infinity.
Infinite triangle is made of three mutually parallel lines. As you near any of its "tips", the distance between the relevant pair of lines goes to zero.