Have You Come up With a Conclusion?

I hope you are now able to determine what non-Euclidean geometry is and how it came into existence throughout the minds of brilliant mathematicians. Of course, we are not saying that Euclidean geometry is useless; after all, its practices were numerous and its services were 'uncountable'. But as our math professor, Dr. Abi Khuzam in AUB once told us, "it all depends on the way you wish to view the world". We started off with linear algebra and extended it to non-linear algebra and we do the same with geometry and practically all other fields. The beauty of our universe is held in the fact that it is special, also non-linear and math is simply trying to simplify its complexity. No matter how hard we try to approach the meaning of life and grab to it, we must acknowledge that other people will have a different approach than ours. Of course, our approaches are correct as much as we want them to be correct, however, they must be flexible and adaptive, so as not to hold us back in our short simple life.

Hyperbolic Geometry

In hyperbolic geometry the two rays extending out in either direction from a point P and not meeting a line L are considered distinct parallels to L, hence came the definition of infinity as the point of intersection of two parallel lines. This geometry results in the theorem that the sum of the angles of a triangle is less than 180°. Shockingly, this geometry gave rise to a special triangle which sides are all parallels to one another and which angles are all zeros. To better imagine the setting of this geometry, think of the lines and figures drawn on a saddle shaped surface. Looking closely at a certain triangle on that saddle, it would seem clear that the middle parts of the lines are bent in towards the center making the angles at the tips less than what they are for a Euclidean triangle. This result in that the sum of these angles will be less than 180.

References: (2009). Infinity. Columbia Electronic Encyclopedia, 6th Edition, 1. http://search.ebscohost.com
(2009). Non-Euclidean geometry. Columbia Electronic Encyclopedia, 6th Edition, 1. http://search.ebscohost.com
MARGENSTERN, M. (2008). Fig. 1. International Journal of Foundations of Computer Science, 19(5), 1237. http://search.ebscohost.com

Elliptical Geometry

In elliptic geometry there are no parallels to a given line L through an external point P, and the sum of the angles of a triangle is greater than 180°. Consider the geometry on the surface of a sphere, the shortest distance between two points on a sphere is an arc of a great circle dividing the sphere in exactly two halves. Since any two great circles always meet, no parallel lines are possible. The angles of a triangle formed by arcs of three great circles always add up to more than 180°. To better understand this, imagine a triangle on the earth's surface bounded by a portion of the equator and two meridians of longitude connecting its end points to one of the poles. Since the two angles at the equator are each 90°, the sum of the angles is determined by the angle at which the meridians meet at the pole and is most certainly greater than 180.
Reference: (2009). Non-Euclidean geometry. Columbia Electronic Encyclopedia, 6th Edition, 1. http://search.ebscohost.com/
n.d., n.t., google image, retrieved on 4/11/2009
http://www.gap-system.org/~john/geometry/Diagrams/x25-1.gif

Non-Euclidean geometry

Non-Euclidean geometry replaces the fifth postulate of the Euclidean geometry by one of two alternative postulates. The first postulate allows two parallels through any external point and is known as the hyperbolic geometry which is attributed to Lobachevski and Bolyai independently. The second postulate allows no parallels through any external point and is known as the elliptic geometry attributed to Riemann. The results of these two types of non-Euclidean geometry contradict those of Euclidean geometry only when dealing with parallel lines, either explicitly or implicitly, which alters the theorem of the sum of the angles in a triangle.

Reference: (2009). Non-Euclidean geometry. Columbia Electronic Encyclopedia, 6th Edition, 1. http://search.ebscohost.com/
n.d, n.t., google image, retrieved on 4/11/2009 from
http://images.google.com.lb/imgres?imgurl=http://images.absoluteastronomy.com/images/encyclopediaimages/b/bu/butterfly_theorem.svg.png&imgrefurl=http://www.absoluteastronomy.com/topics/Butterfly_theorem&usg=__daegwZp_GwSH9qVLtmJ82ezMpwc=&h=224&w=245&sz=10&hl=en&start=8&tbnid=vn_P-DjmKIBg4M:&tbnh=101&tbnw=110&prev=/images%3Fq%3Dbutterfly%2Bgeometry%26gbv%3D2%26ndsp%3D20%26hl%3Den

George Friedrich Bernhard Riemann (1826-1866)

It suffices to say that Riemann, a German mathematician, was Gauss's brilliant student. He soon became a surprising phenomenon for that great mathematician throughout his long teaching career. One of Riemann's revolutionary ideas was that space and time need not be finite, though they are unbounded, suggesting indirectly the setting for a geometry in which "no two lines are parallel and the sum of the angles of a triangle is greater than two right angles" (Wolfe, 1945, p.61). Riemann set the Riemann Sphere and made it the foundation of the elliptical geometry, a crucial branch of the non-Euclidean geometry.

Reference: Wolf, H.E. (1945). Introduction to non-Euclidean geometry
Figure s087750a, n.d., n.t., google image, retrieved on 4/11/2009
http://eom.springer.de/common_img/s087750a.gif

Nikolai Ivanovich Lobachevski (1792-1856)

Lobachevski, as many other mathematicians at that time, tried to address the dilemma in the Euclid's fifth postulate. He recognized that no actual proofs have been written about it and that most of what mathematicians came up with were nothing more than mere explanations. Although he became blind a year before his death, he wrote about his research on the new theory of parallels. His book was published in French and is entitled: "Pangeometrie ou précis de gometrie fondee sur une theorie general et rigoureuse des paralleles" (Wolfe, 1945, p.55). Lobachevski did not live to witness his success, however, Gauss has described his work as "[…] contain[ing] the elements which must hold, and can with strict consistency hold, if the Euclidean is not true […]" (Wolfe, 1945, p.55)

Reference: Wolf, H.E. (1945). Introduction to non-Euclidean geometry
Nikolai Ivanovich, n.d., n.t., google image, retrieved on 4/11/2009
http://images.google.com.lb/imgres?imgurl=http://www.math.uconn.edu/General/Pictures/Mathematicians160x214/Nikolai_Ivanovich_Lobachevsky.jpg&imgrefurl=http://www.math.uconn.edu/MathLinks/mathematicians_gallery.php&usg=__cj-AbYeXrEGJnAWrTBhx3UAEywg=&h=214&w=160&sz=7&hl=en&start=60&tbnid=lOypJdEqO8PWIM:&tbnh=106&tbnw=79&prev=/images%3Fq%3Dlobachevsky%26gbv%3D2%26ndsp%3D20%26hl%3Den%26sa%3DN%26start%3D40

Johann Bolyai (1802-1860)

The Hungarian Wolfgang Bolyai, Gauss's colleague at Gottingen, was trying with Gauss to prove the fifth postulate. Although Wolfgang Bolyai never succeeded, his true success was carried through his child that he described as having "deep blue eyes, which at time sparkle like two jewels" (Wolfe, 1945, p.49) in a letter that he sent to Gauss in 1803. And sparkle he did, Johann Bolyai was capable of proving the fifth postulate by denying it when he was only twenty-one years old. He wrote a letter to his father telling him that “out of nothing, [he] has created a strange new universe” (Wolfe, 1945, p.51). Gauss was very proud of the young Bolyai and exclaimed that he was "[…] overjoyed that it happens to be the son of [his] old friend who outstrip[ed] [him] in such a remarkable way" (Wolfe, 1945, p.52).

Reference: Wolf, H.E. (1945). Introduction to non-Euclidean geometry

Janos Bolyai, n.d., n.t., google image, retrieved on 4/11/2009
http://images.google.com.lb/imgres?imgurl=http://library.thinkquest.org/06aug/00905/pic/bolyaij.jpg&imgrefurl=http://library.thinkquest.org/06aug/00905/eng/bolyai.html&usg=__y0fScCyERABgwVhcRL74JP9mabE=&h=340&w=280&sz=8&hl=en&start=2&tbnid=zRn5BdwBPPLV7M:&tbnh=119&tbnw=98&prev=/images%3Fq%3Dbolyai%26gbv%3D2%26hl%3Den

Carl Friedrich Gauss (1777-1855)

During his lifetime, Gauss published only few results of his research on the problems of the fifth postulate. However, further investigations concerning his work made it clear that he was the first one to discover the "non-Euclidean" geometry and called it as such. He wrote a letter that was addressed to F.A. Taurinus (a fellow mathematician) on November 8, 1824 telling him the following: "[…] it is true that your demonstration of the proof that the sum of the three angles of a plane triangle cannot be greater than 180 is somewhat lacking in geometrical rigor […] I imagine that this problem has not engaged you very long. I have pondered it for over thirty years, and I do not believe that anyone can have given more thought to this second part but I, though I have never published anything on it. The assumption that the sum of the three angles is less than 180 leads to a curious geometry, quite different from ours (the Euclidean), but thoroughly consistent, which I have developed to my entire satisfaction […]" (Wolfe, 1945, p.46-47).

Reference: Wolf, H.E. (1945). Introduction to non-Euclidean geometry

n.d, n.t, google image, retrieved on 4/11/2009

http://images.google.com.lb/imgres?imgurl=http://www.ce.memphis.edu/1112/FAQs/gauss.jpg&imgrefurl=http://www.ce.memphis.edu/1112/FAQs/gauss.htm&usg=__18AmrEmlAxwVuHq-WwtWQeDSX8Y=&h=350&w=296&sz=24&hl=en&start=2&tbnid=f8OpiEXGWd_LUM:&tbnh=120&tbnw=101&prev=/images%3Fq%3DGauss%26gbv%3D2%26hl%3Den

The Fifth Postulate

"One of Euclid's postulates – his postulate 5 - had the fortune to be an epoch-making statement – perhaps the most famous single utterance in the history of science" (Wolfe, 1945, p.17). The fifth postulate exclaims that "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 angles less than the two right angles" (Wolf, 1945, p.4), (see fig 2). In other words, the fifth postulate exclaims that one and only one line can be drawn parallel to a given line through a given external point. With the many attempts mathematicians made to prove this thesis over the years, it seems ironic that what stole the lights with a flawless well-rounded proof were actually its antitheses. Although this new geometry attracted little attention for over thirty-five years, the honored mathematicians, Gauss and Riemann in Germany, Bolyai in Hungary and Lobachevski in Russia were busy trying to take a strategic place in the history as the founders of non-Euclidean geometry.

Reference: Wolf, H.E. (1945). Introduction to non-Euclidean geometry
Euclid's fifth, n.d, n.t, google image, retrieved on 4/11/2009
http://www-math.cudenver.edu/~wcherowi/courses/m3210/hghw23p1.gif

Euclid’s Postulates

Euclid, a Greek mathematician of Alexandria was the founder of the Euclidean geometry about 300 B.C. He acknowledged -as most philosophers and mathematicians- that not everything can be proved and hence the need to make assumptions is not an option. Euclid set 10 assumptions, 5 of which were classified as common notions and the others as postulates. These postulates were usually driven by instinct and intuition; their validity would only appear in that of their results. Even though the postulates may seem empirically flawless, the mere change of one of them directly alters the Euclidean geometry. The 5th postulate of Euclid held that chance; in fact, it held so many chances that mathematicians never stopped investing in it bringing to life various geometrical settings.

Reference: Wolf, H.E. (1945). Introduction to non-Euclidean geometry.
Euclid's postulates, n.d., n.t., google image, retrieved on 4/11/2009

http://images.google.com.lb/imgres?imgurl=http://upload.wikimedia.org/wikipedia/commons/a/a0/Euclid%27s_postulates.png&imgrefurl=http://commons.wikimedia.org/wiki/File:Euclid%27s_postulates.png&usg=__o526OWmpWzhr3dXU5c0D1RJOirY=&h=450&w=300&sz=10&hl=en&start=1&tbnid=Y8tggUuFzgiHxM:&tbnh=127&tbnw=85&prev=/images%3Fq%3DEuclid%2527s%2Bpostulates%26gbv%3D2%26hl%3Den