Non-Euclidean Geometry [Topics in the History of Mathematics]

geometry the study of space might seem to be the best understood of all mathematical subject surely its foundations are the most secure but when mathematicians in the 18th and 19th centuries looked hard at these foundations they found things were not so comforting perhaps Euclidean geometry could be wrong what if alternative geometries could be found could the nature of space itself be in doubt the way I have taken seems not to lead to the goal but much rather to make the truth of geometry doubtful it's a strange phrase the truth of geometry what gauss was interested in when he wrote those words in 1799 was the deceptively simple question is Euclidean geometry true does it correctly describe the space we live in the answer might seem to be a simple yes it does seem rather plausible that Euclid's geometry is true doesn't it but Gauss wasn't satisfied with what appears to be true I come more and more to the opinion that the truth of our geometry cannot be proved perhaps in another life we will get another insight into the nature of space which is unattainable to us now but until then we must not rank geometry with arithmetic whose truth is a matter of logic but rather with mechanics to understand how Gauss could doubt the truth of geometry let's look a little more carefully at what it means to believe the Euclidean geometry is necessarily true how could such a claim be proved well there have to be some initial assumptions in order to get started this is just what Euclid's 5 postulates are statements we can't imagine being without and then we'd want to reason from then to the rest of our beliefs about physical space in that way we would have shown that those beliefs are necessarily true because they followed logically from our assumptions but the difficulty that Gauss and other mathematicians recognized was not the method of reasoning but one of these five postulates no one doubted that a line could be drawn between two points or that a circle could be drawn but the fifth postulate wasn't quite so obvious if in a plane a third line crosses two others and if the sum of the angles alpha and beta is less than two right angles then the lines will eventually meet it was the necessity of having to assume this the parallel postulate that seems to have annoyed mathematicians down the centuries it's not obviously true and when one is playing for such high stakes as obtaining certain knowledge of the world it is worrying Daffy to assume a not quite obvious truth so people tried instead to prove it that is to say they discarded the postulate and tried to deduce it using just the remaining assumptions of Euclid they make a sensible place from which to start for trouble came when people try to deduce theorems from this restricted list the parallel postulate is really very useful and it's hard to deduce theorems without it people found a variety of assumptions that are equivalent to it for instance let's see what we can deduce about the angle sum of a triangle using the parallel postulate if we assume the parallel postulate then when we draw two parallel lines these angles are up to 180 degrees and so these two angles are equal and likewise for this line these two angles are equal putting this information together since alpha beta and gamma lie on a line then alpha plus beta plus gamma is equal to 180 degrees or two right angles and conversely by assuming that the sum of the angles is 180 degrees we can deduce the parallel postulate but be careful that isn't what people were trying to do they wanted to deduce the parallel postulate without making any other assumptions in these four they didn't want to make another assumption even one as plausible as this one about the angles in a triangle in 1733 in italian Geronimo's Akari had pioneered the most fruitful way forward it was his idea to consider what a geometry could be like that was different from Euclid's he hoped to show that such a geometry could not exist evidently if Euclid is the only self-consistent geometry it must be the true one so Kari in fact found that there were just two other geometries to consider in both these geometries Sakaki accepted the first four postulates of Euclid but used alternatives for the fifth postulate his first geometry was one in which there are no parallel lines so any straight line through this point must meet this line somewhere along it now if you're having difficulty imagining a geometry with no parallel lines here's an example on the surface of this sphere we can imagine the straight lines as being great circles now let's consider a straight line parallel to this one but through this point on the equator the angles of these two points on the equator add up to 180 degrees but as I move up the globe the lines get closer and closer together and eventually meet at the pole and whatever line I draw through this point it must eventually meet this one somewhere so there are no parallel lines here there's an entire geometry associated with the surface of a sphere called spherical geometry and it was well known in the 18th century we can even have triangles on the sphere where each side is an arc of a great circle but spherical geometry requires a different set of postulates from those of you kids geometry so spherical geometry isn't a candidate for Sir Kerry's first alternative perhaps it's conceivable that something else could be devised well it wasn't to be the carry was able to show that the combination of Euclid's first four postulates and this property is untenable it does lead to a contradiction his second geometry was one in which there are many straight lines through P that never meet this line to show the results of junkies like this my straight lines will appear curved so the pictures will look rather odd but this second geometry proved to be much more obstinate it simply would not go away as the Swiss mathematician Johann Ludwig wrote in 1770 I have sought such consequences of this hypothesis to see if it did not contradict itself from them all I saw that this hypothesis would not destroy itself at all easily this is where Gauss came in he read Lambert's book when he first went to Gert Engine in 1795 and he shared with his predecessor the belief that if only this alternative hypothesis could be made to contradict itself then at last Euclid's geometry would have been shown to be true but as Lambert discovered this assumption may produce strange theorems but that does not seem to be a contradiction for example in this geometry the angle sum of a triangle is always less than 180 degrees or two right angles and figures cannot have the same shape without also having the same size unexpected but not logically impossible gradually Gauss came to believe that perhaps this alternative geometry might be logically consistent but although Gauss correspondent with those who are working on the matter he did little to develop their ideas to follow those who really did investigate in new geometry we must turn to the distant corners of Europe to Russia and to Hungary Gauss's old student friend Wolfgang boy I had gone back to Hungary and become a professor there his son janosh turned out to be a mathematical prodigy and learned of the problem of the truth of geometry from his father I entreat you to leave the science of parallels alone I thought I would sacrifice myself for the sake of truth I was ready to become a martyr who would remove the floor from geometry and return it purified to mankind I turned back when I saw that no man can reach the bottom of this night the Sun didn't listen and his father tried again I have traveled past all reefs of this infernal dead sea and have always come back with broken mast and torn sail imagine his consternation when his son replied in 1823 I have not yet made the discovery but the path which I have followed is almost certain to lead me to the goal provided the goal is possible all that I can say now is that I have created a new and different world out of nothing all that I have sent you thus far is like a house of cards compared to a tower in the two phrases I have not yet made a discovery and I have created a new world out of nothing lies the whole story of what we can begin to call non Euclidean geometry for once a new geometry can be conceived it is possible that this geometry is the true one to determine the truth of geometry would henceforth be a task for the experimenter and geometry would be like mechanics but the discovery is not yet made for sure that could still be a contradiction just around the corner in 1831 suppressing their remaining doubts they went ahead and published ganaches work it came out as an appendix to the father's book on geometry a copy was sent to gauss who replied if i commence by saying that I'm unable to praise this work he would certainly be surprised but I cannot say otherwise to praise it would be to praise myself indeed the whole contents of the work the path taken by your son the results to which he is led coincide almost entirely with my meditations understandably janosh never forgave gauss for what he saw as an attempt by the most famous mathematician of the day to claim priority falsely over a completely unknown mathematician in russia quite independently a similar story was unfolding a professor a mathematician at the University of Kazan Nikolai Ivanovich Lobachevsky was also trying to prove that there was a second logically possible geometry in 1829 he published the first description of a non Euclidean geometry two years before Janna Fry's publication but there was no way that Janos boy I could have known of Lobachevsky x' work so what is it that these mathematicians had achieved since their descriptions of non-euclidean geometry are so similar it's convenient to describe their work together and because their geometry turns out to be somewhat different from that of you kids we have to suspend many of our conceptions of the world and prepare to enter a new universe perhaps the first surprise is at Lobachevsky and Yayoi both began by describing a three-dimensional geometry in this three-dimensional world they assumed that a line and a point defined a plane and that in such a plane there are always infinitely many lines through the point that do not meet the given line they found that on this assumption there are two really interesting lines through the point there is one in each direction that is asymptotic to the initial line that is gets closer and closer to it but never meets it these lines we call the asymptotic parallels to the initial line so that all these lines eventually meet the baseline and all these lines eventually diverge from it in both directions but it's these the asymptotic parallels that will be particularly interested in they wanted to work out the trigonometry associated with this new geometry for example they asked given this distance what is this angle known as the angle of parallelism and they wanted the answer as a function of the distance a and here's something relatively new the use of functions to do geometry but first to orient ourselves will turn this picture around the initial line now becomes a perpendicular standing on this plane and here's an asymptotic parallel to the line now in this novel geometry they picked another point so they got a right-angled non-euclidean triangle with it's right angle here and through this point they drew the asymptotic parallel to the original perpendicular over this vertex they added a tiny little hemisphere and between these three lines on the surface of the sphere they drew a tiny spherical triangle now for the wonderful idea remember the spherical geometry I showed you earlier it doesn't depend on Euclid's postulate or indeed on any assumption we make about parallel lines so any trigonometric formula describing the spherical triangle are true even in this peculiar non Euclidean world now it turns out the shape of one of these triangles determines the shape of the other so if we know all about the shape of the spherical triangle it should be possible to determine the shape of the non-nuclear Diem triangle so boy I and Lobachevsky could assume the spherical trigonometric form you were true and from them did use what formula would describe a triangle in the non Euclidean plane they found they were able to deduce all they wanted about the new geometry for example by cheating the original figure as being one of a triangle with a vertex at infinity there formally tell you what this angle is as a function of this distance and that's just what was required for that gave them the angle of parallelism there's a wonderful extra to be got out of this improving their results they made use of this bowl shaped surface which meets every possible asymptotic parallel to the original perpendicular at right angles here and here it turns out oddly enough that this triangle has an angle sum of 180 degrees exactly and indeed any triangle we draw on this surface has an angle sum of exactly 180 degrees so within this strange three dimensional non Euclidean space a boy on a Bukowski there is also an accurate picture of two-dimensional Euclidean geometry represented on this bowl it would seem then that by iana birch s ki had finally resolved the problem of the status of geometry by exhibiting Lobachevsky called an imaginary geometry different from Euclid's surely they had shown that Euclidean geology was not after all necessarily true but by and large the response to Lobachevsky employers work was poor and in a sense we can see why after all they had both proceeded from an initial assumption about parallels strictly speaking their work doesn't show that that assumption is logically possible what was required was a complete to rethink something that could embrace both the old way of doing geometry and the new ideas the manner was to provide this was a student of Gauss the shy but gifted than a dreamin remands arguments were to revolutionize the way in which geometry was perceived he didn't start with the belief that we all know what Euclidean geometry is or even with the view that we know what straight lines and angles are riemann argued forcefully that you could do geometry on any surface such as this pear-shaped surface he started instead with the idea that we know how to measure length that's something he said you can do in any geometry if we have a curve on the surface then we use the calculus to measure its length riemann was also able to define what a straight line is in terms of length the straight line between two points is just the curve of shortest length on the surface between the points or probably most excited Gauss who is very impressed by Riemann's ideas was that this new idea of what geometry is put all geometries on a par Euclidean geometry is just now one of many geometries the geometry of a flat surface and what precisely is a flat surface to see their surfaces flat we can just step off it and look but if we want to do geometry in riemann sense we have to do it using properties that lies solely within the surface well it was Gauss himself would shown how to define what's called the curvature of a surface in such a way that it can be determined from properties within the surface alone when we find as we do that for any triangle we can draw on this surface the angles add up to 180 degrees that amounts to showing in Gauss's language the surface has zero curvature in other words it's flat in the same way if we find ourselves on a surface in which the angles of every triangle add up to more than a hundred eighty degrees then we're on a surface which has positive curvature such as this sphere and here's a triangle on a surface we wish to understand it's easy enough to construct a bit of such a surface and we say it has negative curvature because it curves towards us of one direction but away from us in the other but what's worrying about the surface is that when you try to extend it it might not be able to grow beyond a certain point that would be dreadful for any surface that was trying to be a model of even two-dimensional space because nobody believed that space came to an end the man who was able to overcome this dilemma was an Italian aoj near Beltrami our homage solution was ingenious he observed that an atlas is as good a description of a sphere as any if you know how the scale varies from point to point you can work out from the Atlas how far apart these points are on a sphere so an Atlas is a perfectly good description of a surface of constant positive curvature it has some disagreeable features for example on this one equal distances on the sphere appear to stretch more and more as you move outwards belch Army's idea was to construct an atlas for a surface of constant negative curvature if he could construct such an act list then he could hope to show that there was such a surface and that there was no problem about it stopping at a boundary well Charlie did indeed succeed in constructing such a map but it's the version constructed by only Frank RA that I'd like to show you like the maps in the usual atlas of a globe distances are distorted in this case the whole of two-dimensional non Euclidean space is depicted inside this disc as I move outwards distances appear to shrink but that's a distortion of the map making process we non Euclidean z-- don't actually shrink nor do we ever run up against an edge now looking from above we can see how on this map non Euclidean straight lines appear as arcs of circles perpendicular to the boundary circle or as diameters but angles appear actual size so you can see that in this triangle the angle sum really is less than 180 degrees or two right angles and here you can see that in non Euclidean geometry there really are two asymptotic parallels to this line through this point of course the lines meet at the boundary but that's a non Euclidean analog of the way we talk of Euclidean parallel lines meeting at infinity so maps like this disk indeed show that a surface of constant negative curvature exists the first time mathematicians could be certain that an alternative geometry was possible so now not only who won out the truth of Euclidean geometry one happened to doubt it but what a three-dimensional space could a three-dimensional non Euclidean universe be depicted well here is such a model the hole of a non Euclidean universe is inside this ball here in this slice is our two-dimensional representation with the triangle that we so earlier in the lobby jetski model you can see the asymptotic parallels which now meet the original line at the top the asymptotic parallels appear curved because of the way the map has been made again this apparent point of contact is actually infinitely far away in a non Euclidean space as are all the points of this enclosing sphere and this sphere inside the large one is the surface which is perpendicular to all the possible asymptotic parallels to that first line and on this surface the angle sum of any triangle is 180 degrees so this three-dimensional atlas confirms that a three dimensional non Euclidean space is possible it now does become a matter for the experimenter to determine which geometry is the true one finally going back to this model there are two equivalent ways of thinking of this configuration either you can see it from outside as a map of the three-dimensional non Euclidean space discovered by by iana Bukowski or you can see it from inside as a non Euclidean three-dimensional space in which we have an accurate picture of another geometry perhaps unfamiliar to earth but logically possible the remarkable world of two-dimensional Euclidean geometry you

Channel: marshare

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Keywords: open university, ou, maths, math, euclid, gauss, euclidean geometry, non-euclidean geometry, geometry, carl friedrich gauss, saccheri, giovanni saccheri, lambert, johann lambert, bolyai, janos bolyai, János Bolyai, parallel postulate, lobachevsky, nikolai lobachevsky, bernhard riemann, riemann, curvature, beltrami, eugenio beltrami, poincare, henri poincare, Henri Poincaré, Poincaré, mathematics

Id: an0dXEImGHM

Length: 24min 26sec (1466 seconds)

Published: Wed Aug 21 2013