利用數學計算研究紐結圖像

Visualization of knots with Mathematical computation

張鳳蘭

指導教授 ： 陳永富

國立交通大學/理學院/理學院科技與數位學習學程/碩士(2012年)

https://doi.org/10.6842/NCTU.2012.00170

摘要

本文主要是利用數學軟體來探討「紐結(knots)」。首先探討二維的利薩如(Lissajous)圖形、擺線、次擺線等二維平面曲線圖形。其次，延伸到三維的利薩如紐結、擺線紐結、環面紐結、裝飾紐結等紐結。針對利薩如紐結，進行一系列不同相位及頻率的分析；接著觀察擺線紐結、環面紐結的三角函數參數方程中各項係數改變時，所得的規律及變化；以及研究在裝飾紐結上微調數學式時所產生的特殊紐結型式。此外，還會探討經由SU(2)群所進行從利薩如紐結轉換到擺線紐結的SU(2)紐結。最後將進一步試著將紐結圖形的變化與樂理的音階理論做對照，結合數學軟體及數學原理，設計出一個簡易的紐結樂譜或紐結音符。

關鍵字

紐結

並列摘要

The paper is the research of knots with the use of Mathematical software. Firstly, we introduce two-dimensional curves such as Lissajous figures, cycloids, and trochoids. Then, we extend to the three- dimensional knots such as Lissajous knots, trochoid knots, torus knots, and the decorative knots. In the study of Lissajous knots, we have a series of analysis between the different phases and frequencies. For the trochoid knots and the torus knots, we focus on the affection of the various coefficients in the formulas to the knot patterns. We also try to have the formulas varied in order to result in the special shape of knots. Besides, there is a heavy discussion about SU(2). Furthermore, we can generate the SU(2) knots using the idea of SU(2) transformations combined with the 3D knot generating formula. Finally ,we introduce the musical theory about scales for the goal of designing knot notes.

並列關鍵字

knot

參考文獻

[7] 周鴻案,“從利沙球圖形到擺線間的幾何轉換,” 交大理學院碩士在職專班應用科技組碩士論文,民98 (2009).

國際股份有限公司台灣分公司,台灣 (2002).

[8] G. B. Arfken and H. J. Weber, “Mathcmatical Methods for Physicists,” Elsevier, sixth edition, New York (2005).

[9] Y. F. Chen, “Geometry of classical periodic orbits and quantum coherent states in coupled oscillators with SU(2) transformations,” Phys. Rev. A 83, 032124 (2011).

[12] A. stoimenow, “Tait’s Conjectures and Odd Crossing Number Amphicheiral Knots,” Amer. Math. Soc. 45, 285 (2008).