Introduction

The story of mathematics is as much the story of the human tendency to stay firmly grounded in “common sense” as it is the story of simple development of ideas. This is especially true with elliptical and hyperbolic geometries. Both these types of geometries simply don’t seem reasonable from a common sense perspective.

Euclidean geometry started with practical, direct observations of simple figures in a plane. A point, a straight line, and a circle are easy to imagine and draw. In addition, Euclidean geometry describes the ordinary physical world very well. For most purposes, the world looks flat like a Euclidean plane. Parallel lines do not intersect. Triangles, to the best measurements possible, have angles that add up to precisely 180 degrees. These are consistent with common sense.  They are also consistent with Euclidean geometry.

For 2000 years, Euclidean geometry matched the physical world perfectly well. Mathematicians often described new mathematical theories in terms of how well they described the physical world. In the 17^th century, when Newton developed the laws of motion and gravitation, mathematics became even more tied to descriptions of the physical world. Newton actually developed one form of calculus in order to find a way to justify his new gravitational theory.  When physical reality didn’t match new mathematical ideas, those ideas were often scorned. As noted in  Greenberg (2008, p. 244), even Gauss was reluctant to publish his work in novel geometries so he would not have to face “the howl from the Boeotians.”

As with earlier mathematics, Non-Euclidean geometries are good tools for understanding many problems in physics and computer science ( Sahin, 2010). This paper discusses how non-Euclidean geometries apply to modern physics problems.

Historical Background

Riemannian geometries (such as elliptical geometry and hyperbolic geometry) began in the fifth century with Proclus.  He criticized Euclid’s Parallel Postulate. The Parallel Postulate says that for any point not on a line, exactly one line parallel to the original line goes through that point. The postulate implies that if two lines get closer, at some point they must cross each other if they extend to infinity. Proclus noted that a simple hyperbola contradicts that because the hyperbola never actually meets its asymptote, even if they extend to infinity. Though not a formal start to non-Euclidean geometry, Proclus’s made it clear that not everything necessarily has to conform to Euclid (Greenberg, 2008, 210).

In the early eighteenth century, Saccheri developed the notion of a “Saccheri quadrilateral” in which the base has two right angles, and congruent sides. He considered the cases with right angles at the summit (making a rectangle) or with obtuse angles (contradictory and thus not possible). When he considered the case in which the summit angles were acute, he could not show that this led to contradictions. Instead, he got interesting results about parallel lines that eventually led to elliptical geometry (Ibid., 218).

In the mid-1700s, Lambert noted that the sum of the angles in triangles on the surface of a sphere is greater than 180 degrees. He also considered a Saccheri quadrilateral on the surface of sphere of an imaginary radius (i.e., ir). In such a case, he showed that the problems Saccheri had with acute angles disappeared.  This implied that acute-angle Saccheri quadrilaterals correspond to a geometry in an imaginary realm. (Ibid., 224).

By early in the nineteenth century, two mathematicians made independent breakthroughs about the Parallel Postulate. The young Hungarian Janos Bolyai assumed that it was not absurd (i.e., meaningless) to suppose that the postulate was wrong. When he worked out the results of that assumption, he created “a strange new universe” (Ibid., 226). Independently, Gauss concluded the same thing. Both men realized that non-Euclidean geometries were internally consistent and logical.  That was not “politically correct” at the time. The dominant philosopher of the age, Immanuel Kant, had said that Euclidean geometry was in the structure of the human mind. If Kant was correct, non-Euclidean geometry was impossible (Ibid., 245). Kant was wrong.

The first published version of non-Euclidean geometry was by Lobachevsky. Like Gauss, he labeled it “imaginary.” As Gauss had predicted, Lobachevsky’s work was ignored (because it was published in Russian) and scorned (because it was non-Euclidean) (Ibid., 245-246). Lobachevsky’s work, like Bolyai’s, included a mysterious constant he could not associate with anything physical. He described the properties of a non-Euclidean geometry—if one believed in such “imaginary” worlds (Ibid., 247).

In the mid-nineteenth century Riemann imagined a space that was not Euclidean. Riemann defined the curvature of this space by a parameter.  When analyzed, that curvature parameter was the mysterious constant Bolyai and Lobachevsky had puzzled over. Riemann realized that Bolyai and Lobachevsky had described a surface of constant negative curvature. This turned out to be a hyperbolic space similar to a “saddle” (Ibid., 248).  A positive constant gave an elliptical geometry. Riemann’s work meant that the universe itself could be finite. It also implied that no requirement limited space to only three spatial dimensions (Ibid., 248).

The development of non-Euclidean geometries initially met with scorn and disbelief. Yet, more than a century later, Riemannian geometries are important to physics.