Hyperbolic geometry also known as Lobachevskian Geometry is a non-Euclidean geometry that rejects the validity of Euclid’s fifth, the “parallel,” postulate. The first published works of hyperbolic geometries can be traced to Russian Mathematician, Nikolay Ivanovich Lobachevsky, who initially wrote on the subject in 1829. Euclidean geometry and Hyperbolic Geometry are similar but have few differences which include:

 

•        Two parallel lines are taken to be everywhere equidistant in Euclidean geometry. In hyperbolic geometry, two parallel lines are taken to converge in one direction and diverge in the other.

•        In Euclidean, the sum of the angles in a triangle is equal to two right angles; in hyperbolic, the sum is less than two right angles.

•        In Euclidean, polygons of differing areas can be similar; and in hyperbolic, similar polygons of differing areas do not exist.

 

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