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Aktuelle Version vom 30. Oktober 2015, 22:24 Uhr

Reference

Carlo H. Séquin: 2-Manifold Sculptures. In: Bridges 2015. Pages 17–26

DOI

Abstract

Many abstract geometrical sculptures have the shape of a (thickened) 2D surface embedded in 3D space. A fundamental theorem about such surfaces states that their topology is captured with just three parameters: orientability, genus, and number of borders. When trying to apply this classification to interesting sculptures of famous artists depicted on the Web, a first non-trivial task is to establish an unambiguous 3D model based on which the three topological parameters can be determined. This paper describes some successful, practical approaches and gives the results for sculptures by M. Bill, C. Perry, E. Hild, and others. It also discusses the surprising topological equivalences that arise from such an analysis.

Extended Abstract

Bibtex

@inproceedings{bridges2015:17,
 author      = {Carlo H. S\'equin},
 title       = {2-Manifold Sculptures},
 pages       = {17--26},
 booktitle   = {Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture},
 year        = {2015},
 editor      = {Kelly Delp, Craig S. Kaplan, Douglas McKenna and Reza Sarhangi},
 isbn        = {978-1-938664-15-1},
 issn        = {1099-6702},
 publisher   = {Tessellations Publishing},
 address     = {Phoenix, Arizona},
 note        = {Available online at \url{http://archive.bridgesmathart.org/2015/bridges2015-17.html }},
 url         = {http://de.evo-art.org/index.php?title=2-Manifold_Sculptures },
}

Used References

[1] M. Bill, Endless Ribbon (1953-56). – Middelheim Open Air Museum for Sculpture, Antwerp.

[2] M. Bill, Tripartite Unity (1948-49). – http://www.lacma.org/beyondgeometry/artworks1.html

[3] B. Collins, Heptoroid (1998). – http://bridgesmathart.org/bcollins/gallery6.html

[4] E. Catmull and J. Clark. Recursively generated B-spline surfaces on arbitrary topological meshes. Computer-Aided Design 10 (1978), pp 350-355.

[5] W. Cavendish and J. H. Conway, Symmetrically Bordered Surfaces. Math Monthly 117 (2010).

[6] W. Dyck, Beiträge zur Analysis situs. I, Math. Ann. 37 (1888) no.2, pp 457–512.

[7] G. K. Francis, A Topological Picturebook. Springer, New York, 1987, p 101, fig. 33.

[8] G. K. Francis and J. R. Weeks, Conway’s zip proof, Amer. Math. Monthly 106 (1999) pp 393–399.

[9] J. Hass and J. Hughes, Immersions of Surfaces in 3-Manifolds. Topology 24 (1985) no.1, pp 97-112.

[10] E. Hild, Homepage. – http://evahild.com/#

[11] D. Lerner and D. Asimov, The Sudanese Mobius Band. SIGGRAPH Electronic Theatre, 1984.

[12] T. Marar, Projective planes and Tripartite Unity. – http://www.mi.sanu.ac.rs/vismath/marar/ton_marar.html

[13] C. Perry, Topological sculpture – http://www.charlesperry.com/sculpture/style/topological/

[14] C. Perry, Sculpture List. – http://www.charlesperry.com/sculpture/list/

[15] U. Pinkall, Regular homotopy classes of immersed surfaces, Topology 24 (1985) pp 421–434.

[16] R. Roelofs, Mobius Torus. – http://www.rinusroelofs.nl/rhinoceros/rhinoceros-m13.html

[17] C. H. Séquin, Sculpture Generator I. – http://www.cs.berkeley.edu/~sequin/GEN/Sculpture%20Generator/bin/

[18] C. H. Séquin, Tori Story. Bridges Conf. Proc., pp 121-130, Coimbra, Portugal, July 27-31, 2011.

[19] C. H. Séquin, From Moebius Bands to Klein-Knottles. Bridges Conf. Proc., pp 93-102, Towson, July 2012.

[20] C. H. Séquin, Cross-Caps – Boy Caps – Boy Cups. Bridges Conf. Proc., pp 207-216, Enschede, the Netherlands, July 26-31, 2013.

[21] H. Shoryu, Galaxy; Bamboo Basket (2001). Museum of Asian Art, San Francisco (Exhibit 2014).

[22] J. Smith, SLIDE design environment. (2003). – http://www.cs.berkeley.edu/~ug/slide/


Links

Full Text

http://archive.bridgesmathart.org/2015/bridges2015-17.pdf

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