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 a more abstract treatment of one of Pieri's systems (from 1909) based on the primitives "point" and "congruence". Starting with Peano, there has been a parallel thread of interest amongst logicians concerning the axiomatic foundations of Euclidean geometry. This can be seen, in part, in the notation used to describe the axioms. Pieri claimed that even though he wrote in the traditional language of geometry, he was always thinking in terms of the logical notation introduced by Peano, and used that formalism to see how to prove things. A typical example of this type of notation can be found in