「 克萊恩瓶」主題變奏曲

20150115第三期

作者:弗蘭佐尼 Gregorio Franzoni

譯者:王夏聲

全文閱覽

          • 作者簡介
            弗蘭佐尼在義大利的中學及卡格利亞里大學(University of Cagliari)教數學。
          • 譯者簡介
            王夏聲為新竹交通大學應用數學系副教授
          • 致謝
            本文的圖形是由Wolfram Research Inc. 的Mathematica 所繪製,一些3D 數據是以McNeel 的Rhinoceros, v.4 做處理和修飾。
          • 本文出處
            Notices 59 (2012) No.8 AMS.
          • 延伸閱讀
            ◊ 一段簡單的短片,顯示如何由長方形黏貼出克萊恩瓶,再切出莫比烏斯帶。https://www.youtube.com/watch?v=sRTKSzAOBr4
            ◊ 用MAYA 動畫軟體製作的克萊恩瓶短片,內容較多,還可以在瓶內外「開車」。https://www.youtube.com/watch?v=sRTKSzAOBr4&spfreload=10
          • 參考資料
            [1] F. Apéry, Models of the Real Projective Plane, Vieweg, 1987.
            [2] F. Apéry and G. Franzoni, Il rovesciamento della sfera: un modello materiale della fase centrale, Rendiconti del Seminario della Facoltà di Scienze dell’Università di Cagliari (1999), 1–18.
            [3] T. Banchoff, Minimal submanifolds of the bicylinder boundary, Boletim da Sociedade Brasileira de Matematica 7 (1976), 37–57.
            [4] Steven Feiner, David Salesin, and Thomas Banchoff, Dial: A Diagrammatic Animation Language, IEEE Computer Graphics and Applications, September (1982), 43–54.
            [5] T. Banchoff, Beyond the Third Dimension: Geometry, Computer Graphics and Higher Dimensions, Second Edition, Freeman, 1990.
            [6] , private communication.
            [7] C. P. Bruter, Mathematics and Art, Springer, Paris, 2002.
            [8] R. Caddeo and A. Gray, Curve e Superfici, Volume I, CUEC, 2000.
            [9] Concise Encyclopedia of Mathematics, Second Edition, CRC Press LLC, 2003.
            [10] P. Chang, Klein bottle in four parts, 1993, http:// www.ifp.illinois.edu/
            [11] S. Dickson, Klein bottle graphic, 1991, http:// library.wolfram.com
            [12] Ivars Peterson, Plastic Math, Science News, vol. 140, no. 5, Aug. 3, 1991, 72–73.
            [13] G. Fischer, Mathematische Modelle: Mathematical Models, Vieweg, Inc., 1986.
            [14] G. K. Francis, A topological picturebook, SpringerVerlag, New York, 1987.
            [15] D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, Chelsea Publishing Co., New York, 1952.
            [16] A. Jackson, Dirk Struik celebrates his 100th, Notices of the AMS 42 (1995), 43–45.
            [17] I. James and E. Thomas, Note on the classification of cross-sections, Topology 4 (1966), 351–359.
            [18] F. Klein, Über Riemann’s theorie der algebraischen Funktionen und ihrer Integrale, Teubner Verlag, Leipzig, 1882.
            [19] H. B. Lawson, Complete minimal surfaces in S 3, Ann. of Math. 92 (1970), 335–374.
            [20] R. S. Palais, The visualization of mathematics: Towards a mathematical exploratorium, Notices of the AMS 46 (1999), 647–658.
            [21] U. Pinkall, Regular homotopy classes of immersed surfaces, Topology 24 (1985), 421–432.
            [22] C. H. Séquin, Art, math, and computers: New ways of creating pleasing shapes, Bridges 1998, Mathematical Connections in Art, Music, and Science , Reza Sarhangi, Editor, 1998, 1–10.
            [23] , From Möbius bands to Klein-Knottles, to appear in the Proceedings of the Bridges Conference, 2012.
            [24] M. Trott, Constructing an algebraic Klein bottle, Mathematica in Education and Research 8 (1999), 24–27.
            [25] http://www.wolfram.com.
            [26] http://www.zcorp.com.

            作者希望感謝艾沛瑞,班喬夫,狄克遜,法蘭西斯,及瑟昆有益的討論以及這篇論文的審稿群。

(Visited 20 times, 1 visits today)