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The General Parametric Equation of Pythagoras Theorem and The General Connectedness Theorem

Year 2024, , 17 - 30, 30.03.2024
https://doi.org/10.46373/hafebid.1392222

Abstract

In this article, the Quarter Squares Rule is used to prove that it also satisfies the Pythagoras Theorem. Using this proof, it will be shown that there are parametric equations among sides and the radius of inner circle of a right triangle. Thus, it is proved that a quadratic equation which has some definite properties is connected with a right triangle. After that, firstly, utilizing this connection, it will be reached the general parametric equation of The Pythagoras Theorem. Secondly, it is going to be identified the general connectedness theorem. Further, it is going to be given relation between golden ratio and Earth’s axial tilt angle as an interesting example.

References

  • [1]. Barning, F. J. M., On Pythagorean and quasi-Pythagorean triangles and a generation process with the help of unimodular matrices (Dutch). Math. Centrum, Amsterdam, Afd. Zuivere Wisk. ZW, 011, (1963) 37.
  • [2]. Alperin, R. C., The Modular tree of Pythagoras. Preprint, (2000), http://www. arxiv.org/abs/math.HO/0010281.
  • [3]. Gollnick, J., Scheid, H. and Zöllner, J., Rekursive Erzeugung der primitiven pythagoreischen Tripel, Math. Semesterber, 39,(1992) 85–88.
  • [4]. Hall, A., Genealogy of Pythagorean triads, Math. Gazette, 54:390, (1970) 377– 379.
  • [5]. Jaeger, J., Pythagorean number sets, Nordisk Mat. Tidskr, 24,(1976) 56–60, 75.
  • [6]. Kanga, A. R., The family tree of Pythagorean triples, Bull. Inst. Math. Appl., 26, (1990) 15–17.
  • [7]. Préau, P., Un graphe ternaire associé à l’équation + = ,. C. R.Acad. Sci. Paris Ser. I Math., 319, (1994) 665–668.
  • [8]. Emelyanov, P. G., Path Reconstruction in the Barnig-Hall Tree, Journal of Mathematical Sciences, 202(1), (2014) 72-79.
  • [9]. Romik, D., The dynamics of Pythagorean triples, Trans. Am. Math. Soc.,360(11), (2008) 6045–6064.
  • [10].Price, H. L., The Pythagorean Tree: A New Species, (2008), arXiv:0809.4324.
  • [11].Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers, Oxford University Press, Oxford, (1985).
  • [12]. Roy, T., and Sonia, F. J., Direct Method To Generate Pythagorean Triples and Its Generalization To Pythagorean Quadruples and n-tuples, (2012), arxiv:1201.2145.
  • [13]. < https://www.researchgate.net/publication/308796761 >, (Accessed : Nov 2, 2023).
  • [14]. Beauregard, R. and Suryanarayan, E., Proof without words: Parametric representation of primitive pythagorean triples, Mathematics Magazine, 69(3),(1996) 189.
  • [15]. McFarland, D. D., Quarter-Square Revisited: Earlier Tables, Division of Labor in Table Construction, and Later Implementations in Analog Computers, California Center for Population Research, 16(2), (2007) 1-7.
  • [16]. <http://mathworld.wolfram.com/ RightTriangle.html.> (Accessed: Jan 15, 2023)

Pisagor Teoreminin Genel Parametrik Denklemi ve Genel Bağlantılık Teoremi

Year 2024, , 17 - 30, 30.03.2024
https://doi.org/10.46373/hafebid.1392222

Abstract

Bu makalede, Çeyrek Kareler Kuralının, Pisagor Teoremini de sağladığının kanıtlaması için kullanılmıştır. Bu ispat kullanılarak bir dik üçgenin kenarları ve iç teğet çemberinin yarıçapının aynı parametrelere bağlı olarak tanımlanabileceği gösterilecektir. Bu bağlantı ile belirli özelliklere sahip ikinci dereceden bazı denklemlerin, dik üçgenlerle bağlantılı olduğu kanıtlanmıştır. Daha sonra, bu ilişkilerden yararlanılarak Pisagor Teoreminin, genel parametrik denklemine ulaşılacaktır. Sonra da genel bağlantılık teoremi tanımlanacaktır. Ayrıca ilginç bir örnek olarak altın oran ile Dünya'nın eksen eğiklik açısı arasındaki ilişki gösterilecektir.

References

  • [1]. Barning, F. J. M., On Pythagorean and quasi-Pythagorean triangles and a generation process with the help of unimodular matrices (Dutch). Math. Centrum, Amsterdam, Afd. Zuivere Wisk. ZW, 011, (1963) 37.
  • [2]. Alperin, R. C., The Modular tree of Pythagoras. Preprint, (2000), http://www. arxiv.org/abs/math.HO/0010281.
  • [3]. Gollnick, J., Scheid, H. and Zöllner, J., Rekursive Erzeugung der primitiven pythagoreischen Tripel, Math. Semesterber, 39,(1992) 85–88.
  • [4]. Hall, A., Genealogy of Pythagorean triads, Math. Gazette, 54:390, (1970) 377– 379.
  • [5]. Jaeger, J., Pythagorean number sets, Nordisk Mat. Tidskr, 24,(1976) 56–60, 75.
  • [6]. Kanga, A. R., The family tree of Pythagorean triples, Bull. Inst. Math. Appl., 26, (1990) 15–17.
  • [7]. Préau, P., Un graphe ternaire associé à l’équation + = ,. C. R.Acad. Sci. Paris Ser. I Math., 319, (1994) 665–668.
  • [8]. Emelyanov, P. G., Path Reconstruction in the Barnig-Hall Tree, Journal of Mathematical Sciences, 202(1), (2014) 72-79.
  • [9]. Romik, D., The dynamics of Pythagorean triples, Trans. Am. Math. Soc.,360(11), (2008) 6045–6064.
  • [10].Price, H. L., The Pythagorean Tree: A New Species, (2008), arXiv:0809.4324.
  • [11].Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers, Oxford University Press, Oxford, (1985).
  • [12]. Roy, T., and Sonia, F. J., Direct Method To Generate Pythagorean Triples and Its Generalization To Pythagorean Quadruples and n-tuples, (2012), arxiv:1201.2145.
  • [13]. < https://www.researchgate.net/publication/308796761 >, (Accessed : Nov 2, 2023).
  • [14]. Beauregard, R. and Suryanarayan, E., Proof without words: Parametric representation of primitive pythagorean triples, Mathematics Magazine, 69(3),(1996) 189.
  • [15]. McFarland, D. D., Quarter-Square Revisited: Earlier Tables, Division of Labor in Table Construction, and Later Implementations in Analog Computers, California Center for Population Research, 16(2), (2007) 1-7.
  • [16]. <http://mathworld.wolfram.com/ RightTriangle.html.> (Accessed: Jan 15, 2023)
There are 16 citations in total.

Details

Primary Language English
Subjects Algebra and Number Theory, Algebraic and Differential Geometry
Journal Section Articles
Authors

Cengiz Şener

Publication Date March 30, 2024
Submission Date November 17, 2023
Acceptance Date February 9, 2024
Published in Issue Year 2024

Cite

APA Şener, C. (2024). The General Parametric Equation of Pythagoras Theorem and The General Connectedness Theorem. Haliç Üniversitesi Fen Bilimleri Dergisi, 7(1), 17-30. https://doi.org/10.46373/hafebid.1392222

T. C. Haliç Üniversitesi Fen Bilimleri Dergisi