Mathematical methods for CAD: the method of proportional division of thewhole into two unequal parts

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dc.citation.epage89
dc.citation.issue908
dc.citation.journalTitleВісник Національного університету “Львівська політехніка”. Серія: Комп’ютерні системи проектування теорія і практика
dc.citation.spage75
dc.contributor.affiliationThe City University of New York
dc.contributor.authorKosobutskyy, P.
dc.contributor.authorKarkulovska, M.
dc.contributor.authorMorgulis, A.
dc.coverage.placenameЛьвів
dc.coverage.placenameLviv
dc.date.accessioned2020-03-06T09:20:09Z
dc.date.available2020-03-06T09:20:09Z
dc.date.created2018-02-26
dc.date.issued2018-02-26
dc.description.abstractПодано аналіз законів квадратичної ірраціональності коренів квадратного рівняння з коефіцієнтами, що описують пропорційний розподіл цілого числа на дві нерівні частини та характеристичне рівняння рекурентного співвідношення другого порядку. Показано, що на фазовій діаграмі існує безліч ірраціональних значень коренів з властивостями, подібними до властивостей класичних “золотих” чисел φ + = 2 (-1 + V5 )= +0.618... and φ+ = 2 (1 + V5 )= +1.618...
dc.description.abstractIn this paper an analysis of the laws of quadratic irrationality of the roots of the quadratic equation with modulus coefficients is described , which describes the proportional division of the whole into two unequal parts and the characteristic equation of the second order recurrence relation. It is shown that in the phase diagram there exists a set of irrational values of the roots with properties similar to those of the classical “golden” numbers φ + = 2 (-1 + V5 )= +0.618... and φ+ = 2 (1 + V5 )= +1.618...
dc.format.extent75-89
dc.format.pages15
dc.identifier.citationKosobutskyy P. Mathematical methods for CAD: the method of proportional division of thewhole into two unequal parts / P. Kosobutskyy, M. Karkulovska, A. Morgulis // Вісник Національного університету “Львівська політехніка”. Серія: Комп’ютерні системи проектування теорія і практика. — Львів : Видавництво Львівської політехніки, 2018. — № 908. — С. 75–89.
dc.identifier.citationenKosobutskyy P. Mathematical methods for CAD: the method of proportional division of thewhole into two unequal parts / P. Kosobutskyy, M. Karkulovska, A. Morgulis // Visnyk Natsionalnoho universytetu "Lvivska politekhnika". Serie: Kompiuterni systemy proektuvannia teoriia i praktyka. — Lviv : Vydavnytstvo Lvivskoi politekhniky, 2018. — No 908. — P. 75–89.
dc.identifier.urihttps://ena.lpnu.ua/handle/ntb/46919
dc.language.isoen
dc.publisherВидавництво Львівської політехніки
dc.relation.ispartofВісник Національного університету “Львівська політехніка”. Серія: Комп’ютерні системи проектування теорія і практика, 908, 2018
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dc.relation.references34. Vajda S. Fibonacci & Lucas Numbers, and the Golden Section. Theory and Applications. Ellis Horwood limited, 1989.
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dc.relation.references40. Bodnar О. The Golden Section and Non-Euclidean Geometry in Science and Art. Lviv: Publishing House “Ukrainian Technologies”, 2005 (Ukrainian).
dc.relation.references41. P.Kosobutskyy. On the Possibility of Constructinga Set of Numbers with Golden Section Properties. International Conference: Algebra and Analysis with Application. July 1-4 2018, Ohrid, Republic of Macedonia.
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dc.relation.referencesen1. Shene P., Kosnar M., Gardian I. et. al. Mathematics and CAD. In two books. Book 1. Basic methods of the theory of poles. Moscow: Myr, 1988.
dc.relation.referencesen2. Zharmen-Lakur P., Georges P., Pistre F., Bezier P. Mathematics and CAD. In two books. Book 2. Basic methods of the theory of poles. Moscow: Myr, 1989.
dc.relation.referencesen3. Pacioli di Borgo L. De Divina Proportione. On Divine Proportion. 1509.
dc.relation.referencesen4. Zeising A. Neue Lehre von den Proportionen des menschlischen Körpers. New Theory of the Proportions of Human Bodies. Leipzig: Weigel, 1854.
dc.relation.referencesen5. Zeising A. Äesthetische Forschungen. Aesthetic Research. Frankfurt: Medinger, 1855.
dc.relation.referencesen6. Fechner G. Über die frage des golden Schnitts. On the question of the golden section. Arch Zeich Künste, 1865.
dc.relation.referencesen7. Vorobyov N.N., Fibonacci Numbers. Moscow: Nauka, 1961.
dc.relation.referencesen8. V.Hoggat V. Fibonacci and Lucas Numbers. MA: Houghton Mifflin, Boston, 1969.
dc.relation.referencesen9. Shechtman D. Crystals of golden proportions. Ann Fernholm, The Nobel Prize in Chemistry 2011, The Swedish Academy of Sciences.
dc.relation.referencesen10. Gratia D. Quasicrystals, Successes of physical sciences 1988, 156 (2): 347-364.
dc.relation.referencesen11. Grushina N. V., Korolenko P. V., Perestoronin P. A. Fractal structures and "Golden" ratio in optics, Preprint of the Physics Faculty of M. V. Lomonosov Moscow State University 2007, 6: 58.
dc.relation.referencesen12. Heyrovska, S. Narayan, et. al. Doi:arXiv:physics/0509207; http://arxiv.org/abs/physics/0509207
dc.relation.referencesen13. Daqiu Yu, Dongfeng Xue, and Henryk Ratajczak. Golden ratio and bond-valence parameters of hydrogen bonds of hydrated borates, Journal of Molecular Structure 2006, 783: 210–214.
dc.relation.referencesen14. Sherbon M. Fine-Structure Constant from Golden Ratio Geometry, International Journal of Physical Research 2017, 2 (1): 89.
dc.relation.referencesen15. Affleck I. Solid-state physics: golden ratio seen in a magnet, Nature 2010, 464 (7287): 362–363.
dc.relation.referencesen16. Semenyuta N.F. On the relationship of parameters of chain circuits with recurrent numerical sequences, Theoretical electrical engineering, Lviv: High school 1974, 17: 23–25.
dc.relation.referencesen17. Semenuta N.F. Analysis of electrical circuits by the recurrence number method, Electric communication on railway transport. Proceedings of the Belarusian Institute of Railway Transport Engineers. Gomel: Bel.IRTE, 1974, 134: 3–19.
dc.relation.referencesen18. Gazale Midhat J. Gnomon. From Pharaohs to Fractals. Princeton, New Jersey: Princeton University Press, 1999.
dc.relation.referencesen19. Omotehinwa T. O., Ramon S. O., International Journal of Computer and Information Technology 2013, 02(04): 630.
dc.relation.referencesen20. Pletser V., https://arxiv.org/ftp/arxiv/papers/1801/1801.01369.pdf.
dc.relation.referencesen21. Nicholas J. Rose. The Golden Mean and Fibonacci Numbers. From Web Resource: http://www4.ncsu.edu/~njrose/pdfFiles/GoldenMean.pdf.
dc.relation.referencesen22. Philip J. Davis. Spirals from Theodorus to Chaos. A. K. Peterss, Wellesley, MA., 1993.
dc.relation.referencesen23. Matthew Oster. A spiral of triangles related to the great pyramid, The Mathematical Spectrum 2005-06, 38: 108–112.
dc.relation.referencesen24. Prodinger H. The asymptotic behavior of the golden numbers, Fibonacci Quarterly 1996, 34.3: 224–225; fq.math.ca/Scanned/34-3/prodinger.pdf.
dc.relation.referencesen25. V. de Spinadel. On characterization of the onset to chaos, Chaos, Solitons and Fractals 1997, 8 (10): 1631–1643.
dc.relation.referencesen26. V. de Spinadel. The metallic means family and multifractal spectra, Nonlinear Analysis 1999, 36: 721–745.
dc.relation.referencesen27. Bradley S. A geometric connection between generalized Fibonacci sequences and nearly golden sections, Fibonacci Quarterly 2000, 38.2: 174–179. –fq.math.ca/Scanned/38-2/bradley.pdf.
dc.relation.referencesen28. Stakhov A. Generalized Golden Sections and a new approach to the geometric definition of a number, Ukrainian Mathematical Journal 2004, 56 (8): 1143–1150.
dc.relation.referencesen29. Stakhov A. The Generalized Principle of the Golden Section and its Applications in Mathematics, Science and Engineering, Chaos, Solitons & Fractals 2005, 26(2): 263–289.
dc.relation.referencesen30. Stakhov A. Fundamentals of a new kind of Mathematics based on the Golden Section, Chaos, Solitons & Fractals 2005, 27(5): 1124–1146.
dc.relation.referencesen31. V. de Spinadel. The Metallic means Family and Art Aplimat, Journal of Appl. Mathematics 2010, 3(1): 53–64.
dc.relation.referencesen32. Vasilenko S. Generalized recursions with attractor of the golden section. From Web Resource: https://artmatlab.ru/templates/text/r_display/editor/ac49/or01.pdf.
dc.relation.referencesen33. Huntley H. The Divine Proportion: A Study in Mathematical Beauty. Dover Publications, Inc., New York, 1970.
dc.relation.referencesen34. Vajda S. Fibonacci & Lucas Numbers, and the Golden Section. Theory and Applications. Ellis Horwood limited, 1989.
dc.relation.referencesen35. Dunlap R. The golden ratio and Fibonacci numbers. World Scientific Publishing Co. Pte. Ltd. 1997.
dc.relation.referencesen36. Stakhov A. The Golden Section and Modern Harmony Mathmatics, Applications of Fibonacci numbers, Vol. 7 (Graz, 1996), Kluwer Academic Publishers, 1998, pp. 393–399.
dc.relation.referencesen37. Kappraff J. Connections. The geometric bridge between Art and Science. Second Edition. Singapore, New Jersey, London. World Scientific, 2001: 490.
dc.relation.referencesen38. Kappraff J. Beyond Measure A Guided Tour Through Nature, Myth and Number. Singapore, New Jersey, London, Hong Kong. World Scientific, 2002: 584.
dc.relation.referencesen39. Koshy T. Fibonacci and Lucas numbers with application. A Wiley-Interscience Publication: New York, 2001.
dc.relation.referencesen40. Bodnar O. The Golden Section and Non-Euclidean Geometry in Science and Art. Lviv: Publishing House "Ukrainian Technologies", 2005 (Ukrainian).
dc.relation.referencesen41. P.Kosobutskyy. On the Possibility of Constructinga Set of Numbers with Golden Section Properties. International Conference: Algebra and Analysis with Application. July 1-4 2018, Ohrid, Republic of Macedonia.
dc.relation.referencesen42. Schneider R. A Golden Product Identity for e, Mathematics Magazine 2014, 87(2): 132–134.
dc.relation.referencesen43. Shneider R. Fibonacci numbers and the golden ratio. VarXiv: 1611.07384v1 [math.HO] 22 Nov 2016.
dc.relation.urihttp://arxiv.org/abs/physics/0509207
dc.relation.urihttps://arxiv.org/ftp/arxiv/papers/1801/1801.01369.pdf
dc.relation.urihttp://www4.ncsu.edu/~njrose/pdfFiles/GoldenMean.pdf
dc.relation.urihttps://artmatlab.ru/templates/text/r_display/editor/ac49/or01.pdf
dc.rights.holder© Національний університет “Львівська політехніка”, 2018
dc.rights.holder© Kosobutskyy P., Karkulovska M., Morgulis A., 2018
dc.subjectзолота пропорція (ЗП)
dc.subjectпропорційний розподіл
dc.subjectквадратична ірраціональність
dc.subjectgolden ratio (GR)
dc.subjectproportional division
dc.subjectthe quadratic irrational
dc.subject.udc621
dc.subject.udc511.176
dc.subject.udc511.41
dc.subject.udc511.13
dc.subject.udc004.89
dc.subject.udc612.82
dc.subject.udc510.6
dc.subject.udc378.016
dc.subject.udc510
dc.subject.udc004
dc.subject.udc519
dc.subject.udc615.84
dc.titleMathematical methods for CAD: the method of proportional division of thewhole into two unequal parts
dc.title.alternativeМатематичні методи САПР: метод пропорційного поділу цілого на дві нерівні частини
dc.typeArticle

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Комп'ютерні системи проектування теорія і практика. – 2018. – №908