Is this a new projection of a 4D cube ??

[Marek14]

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.

[Keiji]

Yes, but since it grows towards zero as the line goes to infinity, that means there is an infinite length of side with distance to the adjacent side, doesn't it? Therefore the area would be infinite. In order for the area not to be infinite you would have to have an infinite length of side with infinately-small distance to the adjacent side, which would make the area undefined (but not infinite).

[wendy]

In Hyperbolic space, lines that converge on infinity do so in only one direction. The distance between the pair of lines grows exponentially until it hits infinity.

So even though the lines run to infinity, the area between them go to zero.

See for instance, these pages

http://www.superliminal.com/geometry/ty ... cycle.html

The first picture shows several horocycles embedded in each other. You see that the number of features halve for each inward-layer, and 1 + 1/2 + 1/4 + 1/8 &c = 2.

So it is indeed quite possible for infinite lines to have finite area.

Wendy