Hyperbolic geometry or hypercycle, then the triangle has no circumscribed circle. As in spherical and elliptical geometry, in hyperbolic geometry if two triangles are similar, they must be congruent. A special polygon in hyperbolic geometry is the regular apeirogon, a uniform polygon with an infinite number of sides. In Euclidean geometry, the only way to construct such a polygon is to make the side lengths tend to zero and the apeirogon is indistinguishable from a circle, or make the interior angles tend to 180 degrees and the apeirogon approaches a straight line. However, in hyperbolic geometry, a regular apeirogon has sides of
Foundations of geometry or "parallel" lines (when this last term is used, these are the only parallel lines). All other lines through "P" which do not meet formula_8 are called "non-intersecting" or "ultraparallel" lines. Since hyperbolic geometry and Euclidean geometry are both built on the axioms of absolute geometry, they share many properties and propositions. However, the consequences of replacing the parallel postulate of Euclidean geometry with the characteristic postulate of hyperbolic geometry can be dramatic. To mention a few of these: Advocates of the position that Euclidean geometry is the one and only "true" geometry received a setback when, in a memoir