As part of the flattening, many of the lines in the hyperbolic plane appear curved in the model. (In the standard "Mercator projection" maps, distances near the Poles are greatly distorted.) In the Poincaré half-plane model, the hyperbolic plane is flattened into a Euclidean half-plane. For example, if you fly a plane in a straight line from London to San Francisco, and then draw your route on a map, the route won't look straight any more, because maps distort straight The relationship between this model and "real" hyperbolic space is similar to that between flat maps and our spherical world. One way of visualising hyperbolic geometry is called the Poincaré half-plane model. We assume that there exists a line and a point not on the line with at least two parallels to the given line passing through the In hyperbolic geometry, as in spherical geometry, Euclid's first four postulates hold, but the fifth does not. Hyperbolic geometry isn't so easy to visualise as spherical geometry is, because it can't be modelled in three-dimensional Euclidean space without distortion. Now you might ask, is there a geometry in which Euclid's fifth postulate fails, but in the opposite way? That is, is there a geometry in which the angles of a triangle sum to less than 180 degrees? Escher works © 2002 Cordon Art - Baarn - Holland (All rights reserved. Escher's "Sphere with Angels and Devils".Īll M.C. The sum of the angles of a triangle is equal to two right angles. In the nineteenth century, this postulate was shown by Legendre to be equivalent to the statement that In it, he sets out five geometric "postulates", the fifth of which is this: If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. Euclid's work is discussed in detail in The OriginsĮuclid's famous treatise, the Elements, was most probably a summary of what was known about geometry in his time, rather than being his original work. Based on these axioms, he proved theorems - some of the earliest uses of proof in the history of mathematics. The famous mathematician Euclid is credited with being the first person to axiomatise the geometry of the world we live in - that is, to describe the geometric rules which govern it.
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