- Volume 32 Issue 5
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피타고라스의 정리 II : 평행공리와의 관계
Pythagorean Theorem II : Relationship to the Parallel Axiom
- Jo, Kyeonghee (Division of Liberal Arts and Sciences, Mokpo National Maritime Univ.) ;
- Yang, Seong-Deog (Dept. of Math., Korea Univ.)
- 투고 : 2019.09.04
- 심사 : 2019.10.30
- 발행 : 2019.10.31
The proposition that the parallel axiom and the Pythagorean theorem are equivalent in the Hilbert geometry is true when the Archimedean axiom is assumed. In this article, we examine some specific plane geometries to see the existence of the non-archimidean Hilbert geometry in which the Pythagorean theorem holds but the parallel axiom does not. Furthermore we observe that the Pythagorean theorem is equivalent to the fact that the Hilbert geometry is actually a semi-Euclidean geometry.
피타고라스의 정리;평행공리;준-유클리드;아르키메데스의 공리
연구 과제 주관 기관 : 한국연구재단
- David E. DOBBS, A single instance of the Pythagorean theorem implies the parallel postulate, Internat. J. Math. Ed. Sci. Tech. 33(4) (2002), 596-600.
- EUCLID, The thirteen books of Euclid's Elements, Translated with introduction and commentary by Sir Thomas L. Heath, Vols. 1,2,3, Dover Publications, Inc., New York, 1956.
- H. EVES, An introduction to the History of Mathematics, Rinehart, New York, 1953.
- M. J. GREENBERG, (Translation in Korean by Lee, Woo Young), Euclidean and Non-Euclidean geometries, Kyung Moon Sa, 1997. M. J. Greenberg, 이우영 역, 유클리드 기하학과 비유클리드 기하학, 경문사, 2013.
- M. J. GREENBERG, Euclidean and Non-Euclidean Geometries : Development and History, W. H. Freeman and Company, San Francisco, 1980.
- Robin HARTSHORNE, Geometry: Euclid and beyond, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2000.
- Robin HARTSHORNE, Non-Euclidean III.36, The American Mathematical Monthly 110(6) (Jun/Jul 2003), 495-502.
- R. HARTSHORNE (難破誠번역), 幾何学I, II, 現代数学から見たユークリッド原論, 丸善出版 株式会社, 2012.
- T. L. HEATH, The thirteen books of Euclid's Elements, translated from the text of Heiberg, with introduction and commentary, 2nd ed., 3 vols, Cambridge University Press, 1926(Dover reprint 1956).
- D. HILBERT, The Foundations of Geometry, 2nd English Edition, Authorized translated by Leo Unger from the 10th German Edition, Revised and Enlarged by Dr. Paul Bernays, The open court publishing company, 1971.
- K. JO, A historical study of de Zolt's axiom, The Korean Journal for History of Mathematics 30(5) (2017), 261-287.
- K. JO, S.-D. YANG, Moulton Geometry, The Korean Journal for History of Mathematics 29(3) (2016), 191-216.
- K. JO, S.-D. YANG, Pythagorean Theorem I : In non-Hilbert Geometry, The Korean Journal for History of Mathematics 31(6) (2018), 315-337.
- R. KAYA, Area formula for Taxicab triangle, ME Journal 12(4) (2006), 219-220.
- R. KAYA, H. B. COLAKOGLU, Taxicab versions of some Euclidean theorems, Int. J. Pure Allo. Math. 26(1) (2006), 69-81.
- I. KOCAYUSUFOGLU, E. OZDAMAR, Isometries of Taxicab geometry, Commun. Fac. Sci. Univ. Ank. Series Al 47 (1998), 73-83.
- LEE Jong Woo, Historical Backgrounds and Developments of Geometries, Kyung Moon Sa, 1997. 이종우 편저, 기하학의 역사적 배경과 발달, 경문사, 1997.
- LEE Nany, Euclidean Geometry and Beyond, Kyo Woo Sa, 2018. 이난이, 유클리드 기하와 그너머, 교우사, 2018.
- E. S. LOOMIS, The Pythagorean Proposition, Classics in Mathematics Education Series., National Council of Teachers of Mathematics, 1968.
- Paolo MARANER, A Spherical Pythagorean Theorem, The Mathematical intelligencer 32(3) (2010), 46-50.
- W. PEJAS, Die Modelle des Hilbertschen Axiomensystems der absoluten Geometrie, Math. Annalen143 (1961), 212-235.
- K. P. THOMPSON, The nature of length, area, and volume in Taxicab geometry, Int. Electron. J. Geom. 4(2) (2011), 193-207.
- YUN Gabjin, Geometry, Kyo Woo Sa, 2009. 윤갑진, 기하학, 교우사, 2009.