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
Tropical geometry Tropical geometry Tropical geometry is a relatively new area in mathematics, which might loosely be described as a piece-wise linear or skeletonized version of algebraic geometry, using the tropical semiring instead of a field. Algebraic varieties can be mapped to a tropical counterpart and since this process still retain some geometric information about the original variety, it can be used to help prove classic results from algebraic geometry, such as Brill–Noether theorem using the tools of tropical geometry. The basic ideas of tropical analysis have been developed independently in the same notations by mathematicians working in various fields (see and