On Number Theory, Algebra and Spherical Geometry

Transcription

1 On Number Theory, Algebra and Spherical Geometry [ S.Kalimuthu,SF 212/4, Kanjampatti P.O, Pollachi Via, Tamil Nadu , India ] unsolvedproblems@ymail.com Abstract The origin of number theory dates back to 3000 BC. The number system is the mother of all the branch of mathematics. Addition, subtraction, multiplication and division are the four fundamental operations of number theory. Algebra is the extension of number theory. In this work, by applying the addition operation of number theory, the interior angle sum of spherical triangles are transformed into linear algebraic equations. The fifth postulate is restudied and a new result is found. MSC: 08C99,51 M04,11A99 PACS: Dr,02.40.Ky Introduction The fifth Euclidean problem in geometry is 2300 years old. Almost all the mathematician tried their best to deduce Euclid s fifth postulate from the other four postulates, but unfortunately, their attempts yielded no results. Lobachevsky found hyperbolic geometry and Riemann formulated elliptic geometry. In Lobacheskian geometry, the sum of the interior angles of a triangle is less than 180 degrees. And in Riemannian geometry, the interior angle sum of a triangle is more than two right angles. The formulae of Lobachevskian geometry are widely used to study the properties of atomic objects in quantum mechanics. Alfred Einstein

2 used the principles of Riemannian geometry to formulate his general theory of relativity. A turning point in geometry will immediately reflect in physics. Further developments of physics mainly rely on new discoveries in geometry. By analyzing the properties of spherical geometry, a new result is proposed for the origin of a new field of mathematics. Construction Let S be the given sphere. Take a point N. Make NA = NB. Draw NS perpendicular to ANB. Join A and B contacting NS at O. On NB, choose a point C. Join C and O. Produce CO until it meets AS at D. Small letters denote the sum of the interior angles of triangles. Let a,b,c,d,e,f,g,h,i,j,k and l respectively refer to the sum of the interior angles in bi-angle ANCB, triangle AOC, triangle NOB, bi-angle ADSB, triangle AOS, triangle BOD, bi-angle DANC, triangle NOD, bi-angle CBSD, triangle COS, bi-angle NADS, bi-angle NCBS. The angles ANC, NCB, CBS, BSD, SDA, DAN, NOS and Cod are all straight angles and so their measures are all equal to 180 degrees. Let v be the value of this This 180 degrees (1) Results Applying the addition operation of number theory and by assuming (1) we have, x + y + r = 3v + a (2) v + b = x + y (3) y + r = v + c (4) 3v + d = m + n + p (5) m + n = v + e (6) v + f = p + n (7) m + x + y = 3v + g (8)

3 v + h = x + m (9) n + p + r = 3v + i (10) v + j = p + r (11) x + m + n = 3v + k (12) 3v + l = y + r + p (13) Adding (2) to (13),b+r + y +d + f + m + x + h + j + n + l = 4v + a + c + e + g + i +k+3p Putting (3) in (2), 2v + a = b + r v + e = m + n (6) Adding the above four eqns., 2v +d +f + j + l = m + c + i + k + 3p 3v + g = x + y + m (8) Applying (6) in (5), p + e = 2v + d m + n = v + e (6) Assuming (11) in (10), Adding the above four eqns., 2v + i = j + n v + f + l = k + 2p Applying (70 in LHS, n + l = p + k (14) (12) + (13) gives, x + m + n + l = y + r + p + k (15) Putting (14) in (15), x + m = y + r (16) Assuming (4) in RHS, x + m = v + c (17) By construction, NA = NB and NO is perpendicular to ANB. Therefore, by SAS correspondence, triangles AON and BON are congruent. So, x=c (18) Applying (18) in (17), m = v = 180 degrees [ from (1) ] (19) i.e the sum of the interior angles of spherical triangle AOD is equal to 180 degrees (20)

4 Discussion We have derived (20) without assuming the fifth Euclidean postulate which does NOT hold in space. [1] and [2]. But eqns (1) to (20) are consistent. There is NO doubt and No question about it. The application of modern algebra has shown that the famous classical problems such as trisection of a general angle with ruler and compass, squaring the circle duplicating the cube and drawing a regular septagon were impossible to solve. Here the application of addition operation of number theory and classical algebra produced a problematic problem. What went wrong? The author s polite reply is NOTHING. The author sincerely believes that there are only two possibilities: 1. The application of number theory does not hold in sphere 2. Further studies may give birth to a new field of mathematics. So, the author requests the research community to probe into this work. READERS COMMENTS ARE MOST WELCOME. Acknowledgments: The author wishes to thank the late professor of mathematics Palaniappan Kaliappan of Nallamuthu Gounder Mahalingam College,Pollachi, Tamil Nadu ,India for his perfectual encouragement. References: [ 1 ] Effimov,ND : Higher Geometry,Mir Publishers,Moscow,1972,pp 1 30 [ 2 ] Smilga KN: In the search for the beauty,mir publishers,moscow,1972,pp 1 50 Further Readings: [3] [4]

5 Figure 1 ( Spherical ) N C y x r A m n O p B D S