Hyperbolic Geometry and Omega Triangles

HYPERBOLIC GEOMETRY AND OMEGA TRIANGLES Hyperbolic geometry was first discovered and explored by Omar Khayyam in the 9th century and Giovanni Gerolamo Saccheri in the 15th century. Both were attempting to prove Euclid’s parallel postulate by proving the concept of hyperbolic geometry to be inconsistent, and ironically they discovered it to be a new type of geometry. It wasn’t until the 19th century that it became fully developed with help from Karl Friedrich Gauss, Janos Bolyai, and Nikolai Ivanovich Labachevsky. Later on, Eugenio Beltrami developed models of it and used these to prove that hyperbolic geometry is consistent if Euclidean geometry is. Hyperbolic geometry is a form of non-Euclidean geometry. It upholds all of Euclid’s principles except the parallel postulate that says that if given a line [pic]and a point[pic] not on [pic], there is exactly one line through[pic] that doesn’t intersect [pic]. Hyperbolic geometry instead has the following modified postulate: “given any line [pic], and point[pic]not on [pic], there are exactly two lines through[pic]which are hyperparallel to [pic], and an infinite number of lines through[pic]ultraparallel to [pic]” (Wikipedia). Hyperbolic geometry has become well-understood in two dimensions; however, not much is known about it in three dimensions except that it can make dodecahedra. Hyperparallel lines, sometimes referred to as sensed parallel lines, are defined as “the first lines in either direction through a point that do not intersect a given line.” Ultraparallel lines or nonintersecting lines are defined as “lines through a point not intersecting the given line and not parallel to it” (Smart 372). The two hyperparallel lines are referred to as the left-hand parallel and right-hand parallel. They form the angles called the angles of parallelism. It can be proven that these angles cannot be right angles because of the characteristic postulate of hyperbolic geometry or obtuse because it would

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