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RETRACTED: New parallel theory
Applied Mathematics Letters 23 (2010) 1137–1139
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
New parallel theory M. Sivasubramanian
Article history: Received 26 March 2010 Accepted 1 May 2010 Keywords: Euclidean theory of parallel Spherical geometry New parallel theory
1. Introduction
abstract
info
It is well known that for a given line, there is only one parallel line through a point in Euclidean space, there are many parallel lines through a point in Lobachevskian space and there are no parallel through a point in spherical space. But in this work, the author has attempted and showed that there is a set of parallel segment spheres. © 2010 Elsevier Ltd. All rights reserved.
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article
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Department of Mathematics, Dr. Mahalingam College of Engineering and Technology, Pollachi, Tamil Nadu-642003, India
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The origin of geometry dates back to 32 000 BCE. Even before that, a number of geometrical figures were inscribed in African rock caves. It has been shown scientifically that these geometrical figures are more than 70 000 (seventy thousand) years old. The exact place of birth of geometry is not clearly known. Several geometrical theorems have been found by unknown mathematicians. For the first time in the history of mathematics, Euclid of Alexandria city in Egypt complied the then existing consistent propositions and gave a logical and mathematical form and proof. The first scientific treatise on geometry was Euclid’s Elements. In Elements, Euclid proposed five postulates on which the whole Elements were logically deduced. Before Euclid, nobody introduced those five postulates. The first four postulates are obvious. Euclid thought that his last postulate which also known as the parallel postulate needed a proof. Euclid trotted his brain to deduce his fifth postulate from the first four postulates. But unfortunately Euclid was unsuccessful in his rigorous and repeated attempts. Finally, Euclid abandoned his efforts. After Euclid almost all the great and famous mathematicians restarted to prove the parallel postulate as a special theorem. Those reattempts of famous mathematicians also ended in a fatal failure. But those investigations led to a number of equivalent propositions to the fifth postulate. The important outcome of the works of research scientist were the creation of two different types of non-Euclidean geometries namely the hyperbolic geometry by Gauss–Bolyai–Lobacheskvy and elliptic geometry by Riemann. Newtonian mechanics is based on the classical Euclidean geometry. This first branch of geometry has the wider applications in all the branches of science, engineering and technology. After 40 years of the birth of non-Euclidean geometries, Einstein applied non-Euclidean geometrical concepts to formulate his profound theory of gravitation. Beyond any questions, the research community physically concluded that Einstein’s new theory of gravitation is nothing but geometrical interpretation. Einstein imprisoned gravity in the trap of geometry. One cannot even imagine the field equations of general relativity without the beautiful application of non-Euclidean revolutionary results. Quantum mechanics was one of the most important scientific theories of the 20th century. To borrow from the Nobel physical laureate Richard Feynman that it is highly impossible to put quantum physics in classical ways. Newtonian mechanics does not hold in quantum world. This implies that the classical geometry faces severe set backs. That is why there are many chaos and confusion in quantum phenomena. It is generally believed by the physicists and other research workers that only a new field of geometry will describe the full interpretation of quantum science. Kalimuthu and Sivasubramanian have restudied postulate geometry and found several results [1–13].
E-mail address: [email protected]. 0893-9659/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2010.05.003
M. Sivasubramanian / Applied Mathematics Letters 23 (2010) 1137–1139
2. Construction and result
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Fig. 1. Spherical.
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1138
In a sphere S, choose two points A and B not lying on the poles N and S as shown in the spherical Fig. 1. At A and B erect two perpendiculars contacting at C and D on the opposite side of the perimeter. Einstein geometrically proved that a light source which originates at a point O will reach back this point O. Einstein used spherical concepts for his proof. The same proof holds here also. So, the segments AC and BD are parallel. 3. Discussion
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There are many burning problems in physics such as missing mass problems, quantum gravity, why the universe expands, why two black holes attract and become one black hole, why a black hole does not split into two or more parts, why the physical existence of tiny dot singularity, how is it possible for the expansion of the universe without the outer space and tachyons. Einstein believed that a new algebraic idea would reply these challenging topics. He always pleaded his contemporaries to replace partial differential equations by algebraic equations. But the author firmly believes that a new geometrical proposals may solve one of the above mentioned serious phenomena. Gödel’s incompleteness theorems put an upper limit to scientists in knowing the ultimate reality of the nature. The theorems express and explain that mathematics cannot solve everything. The problems will remain and remain for ever. Taking this logical fact into account both possible and impossible dominate mathematics. The one side of the coin states that a particular statement is valid but the other side demands and deduces that the same statement is invalid. This is a peculiar truth. Both science and spirituality came from space. Science is based on equations and experiments whereas spirituality relies on beliefs. The spirituality promises that everything in this universe was created in pairs but in opposites. For example, origin and end, man and women, light and dark, day and night, sorrow and pleasure, loss and gain, God and devil, ugly and beauty, good and bad and so on. Similarly, possible and impossible are consistent in mathematics. Acknowledgement
The author wishes to thank S. Kalimuthu, originator of postulate geometry, SF. 211&212/4, Kanjampatti P.O., Pollachi via, Tamil Nadu-642003, India for his kind encouragement for the preparation of this paper. References [1] M. Sivasubramanian, S. Kalimuthu, On the new branch of mathematical science, Journal of Mathematics and Statistics (ISSN: 1549-3644) 4 (2) (2008) 122–123. http://www.scipub.org/fulltext/jms2/jms242122-123.pdf. [2] M. Sivasubramanian, L. Senthilkumar, K. Raghul Kumar, S. Kalimuthu, On the new branch of mathematical science—part 2, Journal of Mathematics and Statistics (ISSN: 1549-3644) 4 (3) (2008) 148–149. http://www.scipub.org/fulltext/jms2/jms243148-149.pdf. [3] M. Sivasubramanian, S. Kalimuthu, On algebra and tachyons, Journal of Mathematics and Statistics (ISSN: 1549-3644) 5 (2) (2009) 88–89. http://www.scipub.org/fulltext/jms2/jms25288-89.pdf. [4] M. Sivasubramanian, Application of Sivasubramanian Kalimuthu hypothesis to triangles, Journal of Mathematics and Statistics (ISSN: 1549-3644) 5 (2) (2009) 90–92. http://www.scipub.org/fulltext/jms2/jms243148-149.pdf. [5] M. Sivasubramanian, A phenomenon in geometric analysis, Indian Journal of Science and Technology (ISSN: 0974-5645) 2 (4) (2009) 23–24. http://indjst.org/archive/vol.2.issue.4/apr09sivasub.pdf.
M. Sivasubramanian / Applied Mathematics Letters 23 (2010) 1137–1139
1139
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R AC TE
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[6] M. Sivasubramanian, S. Kalimuthu, A computer application in mathematics, Computers and Mathematics with Applications, doi:10.1016/j. camwa2009.07.048, 2009, ISSN: 0898-1221. [7] M. Sivasubramanian, Application of algebra to geometry, Indian Journal of Science and Technology (ISSN: 0974-5645) 2 (10) (2009) 23–24. http:// indjst.org/current/oct09sivasub-18.pdf. [8] M. Sivasubramanian, S. Kalimuthu, A simple experimental verification of Einstein’s variance of mass with velocity equation, Researcher (ISSN: 15539865) 1 (5) (2009) 44–46. http://www.sciencepub.net/researcher/0105/07_0870_Kalimuthu_pub_research0105.pdf. [9] M. Sivasubramanian, On the parallel postulate, Researcher (ISSN: 1553-9865) 1 (5) (2009) 58–61. http://www.sciencepub.net/researcher/0105/10_ 0882_Siva7_pub_research0105.pdf. [10] M. Sivasubramanian, S. Kalimuthu, An easy experiment for dark matter, Nature and Science (ISSN: 1545-0740) 7 (12) (2009) 31–32. http://www. sciencepub.net/nature/ns0712/05_2012_easy_ns0712_31_32.pdf. [11] M. Sivasubramanian, Application of quadratic equation laws to geometry, European Journal of Scientific Research (ISSN: 1450-216X) 38 (2) (2009) 157–158. http://www.eurojournals.com/ejsr_38_2_01.pdf. [12] M. Sivasubramanian, S. Kalimuthu, Application of algebra to trisect and angle of 60 degree, Advances in Algebra (ISSN: 0973-6964) 2 (1) (2009) 43–47. http://www.ripublication.com/aa.htm. [13] M. Sivasubramanian, L. Senthil, P. Periasamy, K. Kalimuthu, Application of computer magnification to geometry, Scientific Research and Essay (ISSN: 1992-2248) 4 (11) (2009) 1408–1409. http://www.academicjournals.org/sre/PDF/pdf2009/Nov/Sivasubramanian et al.pdf.
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
New parallel theory M. Sivasubramanian
Article history: Received 26 March 2010 Accepted 1 May 2010 Keywords: Euclidean theory of parallel Spherical geometry New parallel theory
1. Introduction
abstract
info
It is well known that for a given line, there is only one parallel line through a point in Euclidean space, there are many parallel lines through a point in Lobachevskian space and there are no parallel through a point in spherical space. But in this work, the author has attempted and showed that there is a set of parallel segment spheres. © 2010 Elsevier Ltd. All rights reserved.
R AC TE
article
D
Department of Mathematics, Dr. Mahalingam College of Engineering and Technology, Pollachi, Tamil Nadu-642003, India
R ET
The origin of geometry dates back to 32 000 BCE. Even before that, a number of geometrical figures were inscribed in African rock caves. It has been shown scientifically that these geometrical figures are more than 70 000 (seventy thousand) years old. The exact place of birth of geometry is not clearly known. Several geometrical theorems have been found by unknown mathematicians. For the first time in the history of mathematics, Euclid of Alexandria city in Egypt complied the then existing consistent propositions and gave a logical and mathematical form and proof. The first scientific treatise on geometry was Euclid’s Elements. In Elements, Euclid proposed five postulates on which the whole Elements were logically deduced. Before Euclid, nobody introduced those five postulates. The first four postulates are obvious. Euclid thought that his last postulate which also known as the parallel postulate needed a proof. Euclid trotted his brain to deduce his fifth postulate from the first four postulates. But unfortunately Euclid was unsuccessful in his rigorous and repeated attempts. Finally, Euclid abandoned his efforts. After Euclid almost all the great and famous mathematicians restarted to prove the parallel postulate as a special theorem. Those reattempts of famous mathematicians also ended in a fatal failure. But those investigations led to a number of equivalent propositions to the fifth postulate. The important outcome of the works of research scientist were the creation of two different types of non-Euclidean geometries namely the hyperbolic geometry by Gauss–Bolyai–Lobacheskvy and elliptic geometry by Riemann. Newtonian mechanics is based on the classical Euclidean geometry. This first branch of geometry has the wider applications in all the branches of science, engineering and technology. After 40 years of the birth of non-Euclidean geometries, Einstein applied non-Euclidean geometrical concepts to formulate his profound theory of gravitation. Beyond any questions, the research community physically concluded that Einstein’s new theory of gravitation is nothing but geometrical interpretation. Einstein imprisoned gravity in the trap of geometry. One cannot even imagine the field equations of general relativity without the beautiful application of non-Euclidean revolutionary results. Quantum mechanics was one of the most important scientific theories of the 20th century. To borrow from the Nobel physical laureate Richard Feynman that it is highly impossible to put quantum physics in classical ways. Newtonian mechanics does not hold in quantum world. This implies that the classical geometry faces severe set backs. That is why there are many chaos and confusion in quantum phenomena. It is generally believed by the physicists and other research workers that only a new field of geometry will describe the full interpretation of quantum science. Kalimuthu and Sivasubramanian have restudied postulate geometry and found several results [1–13].
E-mail address: [email protected]. 0893-9659/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2010.05.003
M. Sivasubramanian / Applied Mathematics Letters 23 (2010) 1137–1139
2. Construction and result
R AC TE
Fig. 1. Spherical.
D
1138
In a sphere S, choose two points A and B not lying on the poles N and S as shown in the spherical Fig. 1. At A and B erect two perpendiculars contacting at C and D on the opposite side of the perimeter. Einstein geometrically proved that a light source which originates at a point O will reach back this point O. Einstein used spherical concepts for his proof. The same proof holds here also. So, the segments AC and BD are parallel. 3. Discussion
R ET
There are many burning problems in physics such as missing mass problems, quantum gravity, why the universe expands, why two black holes attract and become one black hole, why a black hole does not split into two or more parts, why the physical existence of tiny dot singularity, how is it possible for the expansion of the universe without the outer space and tachyons. Einstein believed that a new algebraic idea would reply these challenging topics. He always pleaded his contemporaries to replace partial differential equations by algebraic equations. But the author firmly believes that a new geometrical proposals may solve one of the above mentioned serious phenomena. Gödel’s incompleteness theorems put an upper limit to scientists in knowing the ultimate reality of the nature. The theorems express and explain that mathematics cannot solve everything. The problems will remain and remain for ever. Taking this logical fact into account both possible and impossible dominate mathematics. The one side of the coin states that a particular statement is valid but the other side demands and deduces that the same statement is invalid. This is a peculiar truth. Both science and spirituality came from space. Science is based on equations and experiments whereas spirituality relies on beliefs. The spirituality promises that everything in this universe was created in pairs but in opposites. For example, origin and end, man and women, light and dark, day and night, sorrow and pleasure, loss and gain, God and devil, ugly and beauty, good and bad and so on. Similarly, possible and impossible are consistent in mathematics. Acknowledgement
The author wishes to thank S. Kalimuthu, originator of postulate geometry, SF. 211&212/4, Kanjampatti P.O., Pollachi via, Tamil Nadu-642003, India for his kind encouragement for the preparation of this paper. References [1] M. Sivasubramanian, S. Kalimuthu, On the new branch of mathematical science, Journal of Mathematics and Statistics (ISSN: 1549-3644) 4 (2) (2008) 122–123. http://www.scipub.org/fulltext/jms2/jms242122-123.pdf. [2] M. Sivasubramanian, L. Senthilkumar, K. Raghul Kumar, S. Kalimuthu, On the new branch of mathematical science—part 2, Journal of Mathematics and Statistics (ISSN: 1549-3644) 4 (3) (2008) 148–149. http://www.scipub.org/fulltext/jms2/jms243148-149.pdf. [3] M. Sivasubramanian, S. Kalimuthu, On algebra and tachyons, Journal of Mathematics and Statistics (ISSN: 1549-3644) 5 (2) (2009) 88–89. http://www.scipub.org/fulltext/jms2/jms25288-89.pdf. [4] M. Sivasubramanian, Application of Sivasubramanian Kalimuthu hypothesis to triangles, Journal of Mathematics and Statistics (ISSN: 1549-3644) 5 (2) (2009) 90–92. http://www.scipub.org/fulltext/jms2/jms243148-149.pdf. [5] M. Sivasubramanian, A phenomenon in geometric analysis, Indian Journal of Science and Technology (ISSN: 0974-5645) 2 (4) (2009) 23–24. http://indjst.org/archive/vol.2.issue.4/apr09sivasub.pdf.
M. Sivasubramanian / Applied Mathematics Letters 23 (2010) 1137–1139
1139
R ET
R AC TE
D
[6] M. Sivasubramanian, S. Kalimuthu, A computer application in mathematics, Computers and Mathematics with Applications, doi:10.1016/j. camwa2009.07.048, 2009, ISSN: 0898-1221. [7] M. Sivasubramanian, Application of algebra to geometry, Indian Journal of Science and Technology (ISSN: 0974-5645) 2 (10) (2009) 23–24. http:// indjst.org/current/oct09sivasub-18.pdf. [8] M. Sivasubramanian, S. Kalimuthu, A simple experimental verification of Einstein’s variance of mass with velocity equation, Researcher (ISSN: 15539865) 1 (5) (2009) 44–46. http://www.sciencepub.net/researcher/0105/07_0870_Kalimuthu_pub_research0105.pdf. [9] M. Sivasubramanian, On the parallel postulate, Researcher (ISSN: 1553-9865) 1 (5) (2009) 58–61. http://www.sciencepub.net/researcher/0105/10_ 0882_Siva7_pub_research0105.pdf. [10] M. Sivasubramanian, S. Kalimuthu, An easy experiment for dark matter, Nature and Science (ISSN: 1545-0740) 7 (12) (2009) 31–32. http://www. sciencepub.net/nature/ns0712/05_2012_easy_ns0712_31_32.pdf. [11] M. Sivasubramanian, Application of quadratic equation laws to geometry, European Journal of Scientific Research (ISSN: 1450-216X) 38 (2) (2009) 157–158. http://www.eurojournals.com/ejsr_38_2_01.pdf. [12] M. Sivasubramanian, S. Kalimuthu, Application of algebra to trisect and angle of 60 degree, Advances in Algebra (ISSN: 0973-6964) 2 (1) (2009) 43–47. http://www.ripublication.com/aa.htm. [13] M. Sivasubramanian, L. Senthil, P. Periasamy, K. Kalimuthu, Application of computer magnification to geometry, Scientific Research and Essay (ISSN: 1992-2248) 4 (11) (2009) 1408–1409. http://www.academicjournals.org/sre/PDF/pdf2009/Nov/Sivasubramanian et al.pdf.