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A Mathematician and an Artist. The Story of a Collaboration Read chapter Read first chapter

2012 | OriginalPaper | Chapter

Pleasing Shapes for Topological Objects

Author : John M. Sullivan

Published in: Mathematics and Modern Art

Publisher: Springer Berlin Heidelberg

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Abstract

Topology is the study of deformable shapes; to draw a picture of a topological object one must choose a particular geometric shape. One strategy is to minimize a geometric energy, of the type that also arises in many physical situations. The energy minimizers or optimal shapes are also often aesthetically pleasing. This article first appeared in an Italian translation [Sullivan, Affascinanti forme per oggetti topologici, 145–156 (2011)].

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previous chapter Dynamic Surfaces
next chapter Rhombopolyclonic Polygonal Rosettes Theory
Literature
1.
Almgren, F.J. Jr., Sullivan, M.: Visualization of soap bubble geometries. Leonardo 24(3/4), 267–271 (1992)CrossRef
2.
3.
Cantarella, J., Kusner, R.B., Sullivan, J.M.: On the minimum ropelength of knots and links. Inventiones Math. 150(2), 257–286 (2002), arXiv:math.GT/0103224MathSciNetMATHCrossRef
4.
Cantarella, J., Fu, J., Kusner, R., Sullivan, J.M., Wrinkle, N.: Criticality for the Gehring link problem. Geom. Topology 10, 2055–2115 (2006), arXiv.org/math.DG/0402212MathSciNetMATHCrossRef
5.
Emmer, M. (ed.): The Visual Mind: Art and Mathematics. MIT, Cambridge (1993)
6.
7.
Francis, G., Sullivan, J.M., Kusner, R.B., Brakke, K.A., Hartman, C., Chappell, G.: The Minimax Sphere Eversion. In: Hege, H.-C., Polthier, K.(eds.) Visualization and Mathematics, pp. 3–20. Springer, Heidelberg (1997)CrossRef
8.
Gunn, C., Sullivan, J.M.: The Borromean rings: a new logo for the IMU. In: Polthier, K., Aigner, M., Apostol, T.M., Emmer, M., Hege, C.-H., Weinberg, U. (eds.) MathFilm Festival. Springer, Berlin (2008); 5-minute video
9.
Gunn, C., Sullivan, J.M.: The Borromean rings: a video about the new IMU logo. Bridges Proceedings (Leeuwarden), pp. 63–70 (2008)
10.
Karcher, H., Pinkall, U.: Die Boysche Fläche in Oberwolfach. Mitteilungen der DMV 97(1), 45–47 (1997)
11.
Kusner, R., Sullivan, J.M.: Comparing the Weaire-Phelan equal-volume foam to Kelvin’s foam. Forma 11(3), 233–242 (1996)MathSciNetMATH
12.
Morgan, F.: Proof of the double bubble conjecture. Am. Math. Monthly 108(3), 193–205 (2001)MATHCrossRef
13.
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Sullivan, J.M.: Generating and rendering four-dimensional polytopes. Math. J. 1(3), 76–85 (1991)
15.
Sullivan, J.M.: The geometry of bubbles and foams. In: Rivier, N., Sadoc, J.-F. (eds.) Foams and Emulsions. NATO Advanced Science Institute Series E: Applied Sciences, vol. 354, pp. 379–402. Kluwer, Dordrecht (1998)
16.
Sullivan, J.M.: The Optiverse and other sphere eversions. Bridges Proceedings (Winfield), pp. 265–274 (1999), arXiv:math.GT/9905020
17.
Sullivan, J.M.: Minimal flowers. Bridges Proceedings (Pécs), pp. 395–398 (2010)
18.
Sullivan, J.M.: Affascinanti forme per oggetti topologici. In: Emmer, M. (ed.) Matematica e cultura 2011, pp. 145–156. Springer, Italia (2011)CrossRef
19.
20.
Sullivan, J.M., Francis, G., Levy, S.: The Optiverse. In: Hege, H.-C., Polthier, K. (eds.) VideoMath Festival at ICM’98, p. 16. Springer, Berlin (1998); plus 7-minute video, torus.math.uiuc.edu/optiverse/
21.
Thompson, W. (Lord Kelvin), On the division of space with minimum partitional area. Philos. Mag. 24, 503–514 (1887), also published in Acta Math. 11, 121–134
22.
Weaire, D. (ed.): The Kelvin Problem. Taylor & Francis, London (1997)
23.
Weaire, D., Phelan, R.: A counter-example to Kelvin’s conjecture on minimal surfaces. Phil. Mag. Lett. 69(2), 107–110 (1994)CrossRef
24.
Willmore, T.J.: A survey on Willmore immersions. In: Geometry and Topology of Submanifolds, IV (Leuven, 1991), pp. 11–16. World Scientific, Singapore (1992)
Metadata
Title
Pleasing Shapes for Topological Objects
Author
John M. Sullivan
Copyright Year
2012
Publisher
Springer Berlin Heidelberg
Book
Mathematics and Modern Art

Print ISBN: 978-3-642-24496-4

Electronic ISBN: 978-3-642-24497-1

Copyright Year: 2012

DOI
https://doi.org/10.1007/978-3-642-24497-1_13

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