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
Triangle inequality is an exterior angle of a right triangle and therefore larger than , meaning the other base angle of the isosceles triangle also is greater than and their sum exceeds in violation of the triangle postulate). This theorem establishing inequalities is sharpened by Pythagoras' theorem to the equality that the square of the length of the hypotenuse equals the sum of the squares of the other two sides. Consider a triangle whose sides are in an arithmetic progression and let the sides be , , . Then the triangle inequality requires that The first of these quadratic inequalities requires to