To construct a non-Euclidean Geometry, we must deny the existence of a unique parallel. The hyperbolic axiom assumes two or more parallels, but there is one other logical possibility – no parallels, which lead us to the following axiom:
Thus, there are actually two non-Euclidean Geometries: Spherical and Hyperbolic.
As the name suggests, Spherical Geometry can be realized on the surface of a sphere – denoted in the case of the unit sphere – which we may picture as the surface of the Earth. On this sphere, what should be the analogue of a ‘straight line’ connecting two points on the surface?
The answer is – it is the shortest route between them! But if you wish to sail or fly from one place to another (Say from India to London), what is the shortest route?
The answer, already known to the ancient mariners, is that the shortest route is an arc of a great circle (such as the equator), obtained by cutting the sphere with the (unique) plane passing through its center and the given two points on the spherical surface. It is clear that the surface of a sphere satisfies the spherical axiom for: every line through a given point meets the line
at a pair of antipodals, i.e. diametrically opposite points.
In an Euclidean plane, the shortest route is also the straightest route, and in fact the same is true on a sphere: in a precise sense to be discussed later, the great circle trajectory bends neither to the right nor to the left as it traverses the spherical surface.
There are other ways of constructing the great circles on the Earth that do not require thinking about planes passing through the completely inaccessible center of the Earth, For example, on a globe you may map out your great circle journey by holding down one end of a piece of string on India and pulling the string tightly over the surface so that the other end is on London. The taut string has automatically found the shortest, straightest route – the shorter of the two arcs into which great circle through the two places is divided, by those two places.
With the analogue of straight lines now found, we can ‘do geometry’ within the spherical surface. For example, given three points on the surface of the Earth, we can connect them together with arcs of great circles (which are analogues of straight line segments on a sphere), to obtain a triangle. The following figure illustrates such a triangle drawn on a spherical surface (Notice that the angle sum of this spherical triangle is greater than But if this non-Euclidean Spherical Geometry was already used by ancient mariners to navigate the oceans, and by astronomers to map the spherical night sky, what then was so shocking and new about the non-Euclidean Geometry discovered by Lobachevsky and Bolyai? The answer is that this Spherical Geometry was merely considered to be inherited from the Euclidean Geometry of the ambient 3-dimensional space in which the sphere resides. No thought was given in those times to the sphere’s internal 2-dimensional geometry as representing an alternative to Euclid’s plane. Not only did it violate Euclid’s fifth axiom (the parallel axiom), it also violated a much more basic one – Euclid’s first axiom which says: we can always draw a unique straight line connecting two points, for this fails when the two points on the sphere are antipodal. On the other hand, the Hyperbolic Geometry of Lobachevsky and Bolyai was a much more serious affront to Euclidean Geometry, containing familiar lines of infinite length, yet flaunting multiple parallels, ludicrous angle sums, and many other seemingly nonsensical results. Yet the 21-year-old Bolyai was confident and exuberant in his discoveries, writing to his father, “From nothing I have created another entirely new world”. We end with a tale of tragedy. Bolyai’s father was a friend of Gauss, and sent him what his son had achieved. By this time Gauss had himself made some important discoveries in the area, but had kept them secret. In any case, young Bolyai had seen further than Gauss. A kind word in public from Gauss, the most famous mathematician in the world, would have assured young Bolyai a bright future. But Nature and nurture sometimes conspire to pour extraordinary mathematical gifts into a vessel marred by very ordinary human flaws, and Gauss’s reaction to Bolyai’s marvelous discoveries was mean-spirited and self-serving in the extreme. First, Gauss kept Bolyai in suspense for six months, then he replied to his father as follows: “Now something about the work of your son. You will probably be shocked for a moment when I begin by saying that ‘I cannot praise it’, but I cannot do anything else, since to praise it would be to praise myself. The whole content of the paper, the path that your son has taken, and the results to which he had been led, agree almost everywhere with my own meditations, which have occupied me in part for 30-35 years.” Gauss did however ‘thank’ young Bolyai for having ‘saved him the trouble’ of having to write down theorems he had known for decades. Bolyai never recovered from the surgical blow delivered by Gauss, and he abandoned mathematics for the rest of his life! Great minds like Gauss are not always free from the shadow of arrogance. Gauss had previously denigrated Abel‘s discovery of elliptic functions in precisely the same manner. In the few subsequent posts, we will dive deeper into the properties of spherical triangles. In particular, in the upcoming post, we will try to see how the angle sum property in Spherical Geometry deviates from its Euclidean counterpart. We will also deduce a beautiful relation between the area of a spherical triangle and the angular deviation. So, stay tuned!.
(German mathematician and astronomer)
(Norwegian mathematician)
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