The Second Postulate of Euclid and the Hyperbolic Geometry

Transcription

1 1 The Seond Postulate of Eulid and the Hyperboli Geometry Yuriy N. Zayko Department of Applied Informatis, Faulty of Publi Administration, Russian Presidential Aademy of National Eonomy and Publi Administration, Stolypin Volga Region Institute, Russia, Saratov. Abstrats. The artile deals with the onnetion between the seond postulate of Eulid and non-eulidean geometry. It is shown that the violation of the seond postulate of Eulid inevitably leads to hyperboli geometry. This eliminates misunderstandings about the sums of some divergent series. The onnetion between hyperboli geometry and relativisti omputations is noted. Keywords: Postulates of Eulid, non-eulidean geometry, metri, embedding, hyperboli geometry, relativisti omputation. 1. Introdution Historially, the appearane of non-eulidean geometry is assoiated with the realization of the possibility of not fulfilling the fifth priniple (postulate) of Eulid about parallel lines. Geometry introdued in the writings of Lobahevsky and Boyayi instead of the fifth postulate of Eulid takes the opposite one and is just as onsistent as the Eulidean geometry. It was named hyperboli [1]. The seletion of the fifth postulate laid the foundation for the aepted division of geometries into absolute one based on the first four postulates of Eulid, Eulidean geometry, in whih, in addition to the first four, the fifth postulate is added and the hyperboli geometry already mentioned. The distint feature of the fifth postulate from the others was stressed long before the appearane of non-eulidean geometry. The rest of the postulates did not ause suh inreased attention, and, espeially, doubts about their fairness, whih seems rather strange, sine there are numerous examples of violation of at least one of the rest postulates of Eulid - the seond postulate.

2 As shown in this paper, this indiates a possible deviation from the Eulidean geometry in the rejetion of the seond postulate, whih should also be taken into aount in mathematial and other studies. To bind it with the violation of the fifth postulate is not possible, therefore, it is neessary to arefully study the seond postulate, what is done below.. The seond postulate of Eulid Let us give below one of the formulations of the seond postulate [1] A finite straight line may be extended ontinuously in a straight line Like any statement expressed in verbal form, it differs in ambiguity and admits numerous variants. For example, in [] the word "ontinuously" is replaed by "unlimited". In order to eliminate this inauray, we resort to a tehnique ommon in the mathematis - the modeling of the statements of one region by the means of another [1]. In this ase, the seond postulate in the language of arithmeti is equivalent to the following A sum of an infinite divergent series of positive numbers is equal to infinity In aordane with it, for example, there should be K + K = K + n + K = (1) However, this ontradits known fats [3]. The sum of the first series is equal ζ ( 0) = 0.5, and of the seond, ζ ( 1) = 1/1 = , where ζ (s) -is the Riemann zeta-funtion [3] ζ ( s) = n s, s = u + iv () n= 1 n is an integer, u and v are real numbers. There are a lot of suh examples.

3 3 The very fat of the finiteness of the sum of a divergent series does not raise questions sine its meaning is different than for the sum of onvergent series [4]. Divergent series are used in various fields of siene, primarily in physis. In partiular, the sum of the seond series in (1) underlies many results of string theory [5]. However, until reently no one paid any attention to the fat that this and similar results violate the seond Eulidean postulate and the onsequenes of this fat. 3. Calulation of the sum of a divergent series Reall that the sums of divergent series are not omputed, sine diret omputation is usually impossible, but is determined either indiretly, for example, as in the ase of the series for ζ ( 1) - by analyti ontinuation of the zeta-funtion, or by using various summation methods [4]. The first attempt to alulate the sum of this series was undertaken in the author's work [6]. It turned out that for this it is neessary to introdue a metri on the numerial axis ds 1 x x = 1 + dt 1 dx x + x (3) where s is an interval; x, t are oordinate on the numerial axis and time; x some harateristi value for the given metri [6]. The alulation is realized as the motion of a material partile along the numerial axis aording to with the relativisti equations of motion written for the metri (3). Fatually, it is talking about the relativisti Turing mahine (a relativisti superomputer in the terminology of [7]) whose tape is represented by the numerial axis with the metri (3), and the role of moving head is played by the above-mentioned partile. Suh a superomputer is able to solve problems that are not omputable in the traditional sense, in this ase, to alulate the sum of a divergent series. The auray attained in [6] orresponds to an error in 3,5%. 4. Nature of Geometry

4 4 Now it is understood the perplexity that aused by the reeived result - the magnitude of the sum of ζ ( 1) for a number of representatives of the aademi ommunity [8]. It was aused by an attempt to interpret it in aordane with the so-alled ommon sense, i.e. in fat, an attempt to omprehend it within the framework of Eulidean geometry. In any ase, no deviations from it, not related to the violation of the fifth postulate, no one expeted. Nevertheless, this is so. The harater of the resulting geometry an be easily determined if one tries to solve the problem of embedding a one-dimensional manifold with the metri (3) into a manifold of higher dimension - the plane. As will be shown below, in whole, for < x < this is possible, only for a plane with hyperboli geometry. Before takling diretly the solution of this problem, we reall that the awareness of the impossibility of a smooth embedding of a subspae into a Eulidean spae of higher dimension in its time ontributed to the disovery of non-eulidean geometry [9]. Thus, onsider a two-dimensional plane with a spatial metri defined on it dl dy = ± dx ± dy = ± 1 ± dx (4) dx Comparing with (3), find that the embedding leads to the equation for y(x) dy ± 1± dx 1 = x x (5) whih is easily integrated. Below are the results, aording to whih there are three areas, eah of whih is haraterized by its own funtion y(x) I. II. < x < x ; dl y( x) = ± x x y( x) = ± x = dx dy ( arh z + ( z) z ) < x < + ; dl = dy dx ( arsh z ) + ( z)( + z) ; ; (6)

5 5 III. x < x < 0; dl y( x) = ± x = dx + dy ( arsin z z( z) ) ; where z = x / x. This is illustrated in Figure 1, where some branhes of the embedding (6) are shown Fig. 1. Areas of embedding of a one-dimensional manifold with metri (3) into a twodimensional plane. The Roman numerals next to eah branh of y (x) refer to eah orresponding region: I-solid line, II-dash-dotted, III- dashed. From these results, it follows that one an ompletely embed a onedimensional manifold with the metri (3) only in a two-dimensional plane with hyperboli geometry (branhes I and II). Into a plane domain with Eulidean geometry, only part of it orresponding to the domain III an be embedded. 5. Disussions. The fat that the violation of the seond postulate of Eulid leads to non- Eulidean geometry, restores the "symmetry" between postulates, abolishing the "monopoly" of the fifth postulate, whih lasted about two hundred years. This is a omparatively short period of time if one ompares it with the time during whih Eulidean geometry dominated. It should be noted that the rejetion of both postulates leads to the same onsequenes.

6 6 It is now diffiult to foresee the other onsequenes to whih this result will lead. Right now it helps to eliminate misunderstandings about the sums of some divergent series mentioned above. As for the possibilities in the field of omputational mathematis, and in partiular the reation of relativisti superomputers, it is worth noting the undoubted advantages of this approah in omparison with the traditional one where it is proposed to use Kerr-Newman blak holes for the realization of relativisti alulations [7]. Calulation of the ζ ( 1), made in [6], as already noted above, has insuffiient auray. If one onsiders it as another onfirmation of the general theory of relativity, then it is inferior in auray in omparison with all the other (experimental). However, in view of their fewness, it should not be disounted. It must be said that the alulation method of ζ ( 1) used in [6] is physial, and based on the fat that the formula for the partial sums of the series (1) oinides with the expression for the distane traversed by a partile moving with a onstant (nonrelativisti) aeleration. Stritly speaking, it was this analogy that made it possible to obtain expressions for the metri (3). It is not yet lear how this result an be obtained from purely geometri onsiderations. One an only hope that the establishment of a loser relationship between the two approahes will allow for greater progress in relativisti alulations and, in partiular, to improve their auray. 6. Conlusion The paper shows that geometry in whih the seond postulate of Eulid is not satisfied is hyperboli. This allows eliminating the misunderstandings assoiated with the alulation of the sums of some divergent series. A metri is given on a numerial axis in whih the speified alulations an be performed. The embedding of a numerial axis with a given metri into a two-dimensional plane was performed and it is shown that the ondition for smooth embedding of

7 7 the entire numerial axis is the hyperboli geometry on the plane. Various appliations of the obtained results are onsidered. Referenes. 1. H.S.M. Coxeter, F.R.S. Introdution to Geometry, New York, London, John Wiley&Sons, In.. S.N. Byhkov, E.A. Zaitsev, Mathematis in world ulture. Tutorial, Mosow: Russian State Humanitarian Univ. Publishers, 006 (Russian). 3. E. Janke, F. Emde, F. Lösh, Tafeln Höherer Funktionen, B.G. Teubner Verlagsgeselshaft, Stuttgart, G.H. Hardy, Divergent series,oxford, B. Zwiebah, A First Course in String Theory, -nd Ed., MIT, Y.N. Zayko, The Geometri Interpretation of Some Mathematial Expressions Containing the Riemann ζ-funtion, Mathematis Letters, 016; (6): H. Andr eka, I. N emeti, P. N emeti, General relativisti hyperomputing and foundation of mathematis, Natural Computing, 009, V. 8, 3, pp D. Berman, M. Freiberger, Infinity or -1/1?, + plus magazine, Feb. 18, 014, 9. S. Weinberg, Gravitation and Cosmology. Priniples and Appliations of the General Theory of Relativity, MTI, John Wiley&Sons, 197.