An equivalent way to express this is that the angle sum of a triangle is two right angles. See Figure 1(A) below for an illustration of this. These “other” geometries come from Euclid’s fifth postulate: “If a straight line falling on two straight lines makes 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 the angles less than two right angles”. Nonetheless, there are a few other lesser-known, but equally important, geometries that also have many applications in the world and the universe. It is the most intuitive geometry in that it is the way humans naturally think about the world. Lastly, Poincare makes some notable contributions to solidifying hyperbolic geometry as an area of academic study.Įuclidean geometry came from Euclid’s five postulates. Later, Klein settled any doubt of noneuclidean consistency.
Then we will look at the effect of Gauss’s thoughts on Euclid’s parallel postulate through noneuclidean geometry. We will discuss some of their influences in the following sections, starting with Euclid and his postulates that defined geometry. Euclid, Gauss, Felix Klein and Henri Poincare all made major contribution to the field. This essay is an introduction to the history of hyperbolic geometry.
The essay has been lightly edited before being published here. Sami was a student in the Fall 2016 course “Geometry of Surfaces” taught by Scott Taylor at Colby College.
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