rigons with different areas can exist in Euclidean geometry. It is not possible in hyperbolic or elliptic geometry. In two-dimensional geometries, lines that are perpendicular to the same given line are parallel in abstract geometry, are neither parallel nor intersecting in hyperbolic geometry, and intersect at the pole of the given line in elliptic geometry. The appearance of the lines as straight or curved depends on the postulates for the space.At the present time mathematics is trying to figure out which of the three is the best representation of the universe. They know that Euclidean geometry provides an excellent representation for our part of the world. Many people feel that abstract geometry is the only geometry and object to measurement geometry. The only problem is that all measurements are approximates, which means the margin for error is great. Maybe one-day mathematicians will find out which is the best....
