Projective geometry has simple origins from Renaissance artists who portrayed the rim of a cup as an ellipse in their paintings to show perspective. Furthermore, projective geometry is independent from the theory of parallels if not, the consistency of projective geometry would be questioned. According to John Stillwell’s translation of On so-called Noneuclidean Geometry, Klein started by showing that each non-euclidean geometry is a special case of projective geometry he then ended by concluding that non-euclidean geometry can be derived from projective geometry. Klein’s contribution to geometry is not only the famous Klein Bottle, but also his proof of the extension of the Cayley Measure on projective geometry to non-euclidean geometry. Now that we’ve gone over the founding of hyperbolic geometry and have some sense of what it is, we’ll talk about two more influential mathematicians in the field: Felix Klein and Henri Poincare. įigure 5: Hyperbolic geodesic pictured as a euclidean semi-circle centered at a point on x. Note, that spherical geometry has constant positive curvature. It has constant negative Gaussian curvature, which resembles a hyperboloid (See Figure 2).
Now that a brief history of the sources of hyperbolic geometry has been provided, we will define hyperbolic geometry. It was definitive that without the parallel postulate, the remaining four postulates created a geometry that is equally consistent. With the work of these three mathematicians, the controversy of the parallel postulate was put to rest. Bolyai and Lobachevsky published their works on noneuclidean geometry independently but around the same time however, their findings were not popularized until after 1862, when Gauss’s private letter to Schumacher about his “meditations” on hyperbolic geometry was published. Yet, Gauss extended geometry to include what he coined “non-euclidean,” or the contradiction of the parallel postulate. Gauss published little of his work on it because he, allegedly, feared disrespecting Euclid by disproving the parallel postulate. Lobachevsky are considered the fathers of hyperbolic geometry. Over 2,000 years after Euclid, three mathematicians finally answered the question of the parallel postulate.
However, it turns out this postulate determines whether we are in euclidean or noneuclidean geometry. For centuries, mathematicians and amateurs alike attempted to prove that the fifth postulate is a consequence of the first four postulates and other established theorems. Unlike the other postulates, the fifth is not obvious (to satisfy your curiosity, the rest of Euclid’s postulates are attached at the end of this essay). See Figure 1(B) below.įigure 1: Visualization of parallel postulate. This is also called the parallel postulate because it is equivalent to the following statement: “if one draws a straight line and a point not on that line, there is only one unique line that is parallel to given line through the point”. 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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