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Dynamic Surfaces

  • Simon Salamon
Conference paper
Part of the Springer Proceedings in Mathematics book series (PROM, volume 18)

Abstract

After discussing some well-known examples in geometry and function theory, we study surfaces in space that are defined by the vanishing of the torsion of integral curves of a given vector field.

Keywords

Vector Field Space Curve Ricci Flow Space Curf Lorenz Attractor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

I first started using iterated exponentiation to illustrate bifurcation in a project for a Maple course I taught in Oxford in the 1990s. Remnants of this course are available on my homepage.

The study of space curves with assigned torsion and curvature is described in [3], and I used it as a student competition to find an interesting space curve by guessing simple candidate functions in a differential geometry course in Turin.

The vector field Q a is the basis of the author’s animation

Vector field Kaleidoscope on YouTube. This video contains frames of null-torsion surfaces for about 500 values of a and Figs. 19, 20 and 23 are instances of that.

Figure 25 arises from the vector field (4) with cubic components. As personal computing power increases, it will be easier to study more and more complicated vector fields and their associated surfaces.

The ESMA conference taught me not just to better appreciate artistic aspects inherent in modern mathematics, and but also the importance of explaining to a wider public the underlying geometrical ideas I deal with on a daily basis. I found myself having to work on a number of deep mathematical problems to answer some of the questions that arose. In this respect, I wish to express gratitude to the participants, and in particular François Apery, Claude Bruter, Eugenia Emets, George Hart, and Jos Leys. Thanks are also due to Claude for his afternoon visits from Gometz to Bures encouraging me to finish the text.

References

  1. 1.
    Ghys, E.: Lorenz and modular flows, a visual introduction, www.ams.org/samplings/feature-column/fcarc-lorenz
  2. 2.
    Gilmore, C.R., Letellier, C.: The Symmetry of Chaos. Oxford University Press, London (2007)Google Scholar
  3. 3.
    Gray, A., Abbena, E., Salamon, S.: Modern Differential Geometry of Curves and Surfaces with Mathematica. CRC Press, FL (2006)Google Scholar
  4. 4.
    Pressley, A.: Elementary Differential Geometry, Springer Undergraduate Mathematics Series. Springer, Berlin (2001)Google Scholar
  5. 5.
    Struik, D.: Lectures on Classical Differential Geometry. Dover, NY (1988)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Politecnico di TorinoTorinoItalia

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