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 as "Birkhoff's axioms". These postulates are all based on basic geometry that can be experimentally verified with a scale and protractor. In a radical departure from the synthetic approach of Hilbert, Birkhoff was the first to build the foundations of geometry on the real number system. It is this powerful assumption that permits the small number of axioms in this system. Birkhoff uses four undefined terms: "point", "line", "distance" and "angle". His postulates are: Postulate I: Postulate of Line Measure. The points "A", "B", ... of any line can be put into 1:1 correspondence with the real numbers "x" so