Ultimate fate of the universe eventually stops the expansion of the universe, after which it starts to contract until all matter in the universe collapses to a point, a final singularity termed the "Big Crunch", the opposite of the Big Bang. Some new modern theories assume the universe may have a significant amount of dark energy, whose repulsive force may be sufficient to cause the expansion of the universe to continue forever—even if formula_5. If formula_7, the geometry of space is open, i.e., negatively curved like the surface of a saddle. The angles of a triangle sum to less than 180 degrees, and lines that
Skinny triangle based on the small-angle approximations; and, when formula_5 is in radians. The proof of the skinny triangle solution follows from the small-angle approximation by applying the law of sines. Again referring to figure 1; The term formula_7 represents the base angle of the triangle and is this value because the sum of the internal angles of any triangle (in this case the two base angles plus "θ") are equal to "π". Applying the small angle approximations to the law of sines above results in; the desired result. This result is equivalent to assuming that the length of the base of