The Conformal Hyperbolic Square and Its Ilk: Unterschied zwischen den Versionen
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== Used References == | == Used References == | ||
Aktuelle Version vom 27. Dezember 2016, 13:20 Uhr
Inhaltsverzeichnis
Reference
Chamberlain Fong: The Conformal Hyperbolic Square and Its Ilk. In: Bridges 2016, Pages 179–186.
DOI
Abstract
We present explicit equations for three different mappings between the disk and the square. We then use these smooth and invertible mappings to convert the Poincaré disk into a square. In doing so, we come up with three square models of the hyperbolic plane. Although these hyperbolic square models probably have limited use in mathematics, we argue that they have artistic merit. In particular, we discuss their use for aesthetic visualization of infinite patterns within the confines of a square region.
Extended Abstract
Bibtex
@inproceedings{bridges2016:179,
author = {Chamberlain Fong},
title = {The Conformal Hyperbolic Square and Its Ilk},
pages = {179--186},
booktitle = {Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture},
year = {2016},
editor = {Eve Torrence, Bruce Torrence, Carlo S\'equin, Douglas McKenna, Krist\'of Fenyvesi and Reza Sarhangi},
isbn = {978-1-938664-19-9},
issn = {1099-6702},
publisher = {Tessellations Publishing},
address = {Phoenix, Arizona},
url = {http://de.evo-art.org/index.php?title=The_Conformal_Hyperbolic_Square_and_Its_Ilk },
note = {Available online at \url{http://archive.bridgesmathart.org/2016/bridges2016-179.html}}
}
Used References
[1] V. Bulatov, Conformal Models of the Hyperbolic Geometry. MAA-AMS Joint Math. Meeting. 2010
[2] D. Dunham, Hamiltonian Paths and Hyperbolic Patterns, Contemporary Mathematics, Vol. 479, 2009
[3] M. Fernandez-Guasti, Analytic Geometry of Some Rectilinear Figures, International Journal of Mathematical Education in Science and Technology. 23, pp. 895-901. 1992.
[4] C. Fong, Analytical Methods for Squaring the Disc. (poster) Seoul International Congress of Mathematicians, 2014. http://arxiv.org/abs/1509.06344
[5] J. Langer, The Conformal Vega Disk. Bridges 2011, Coimbra Portugal, pp.483-484
[6] H. Müller, The Conformal Mapping of a Circle Onto a Regular Polygon, with an Application to Image Distortion. http://herbert-mueller.info
[7] P. Nowell, Mapping a Square to a Circle (blog) http://mathproofs.blogspot.com/2005/07/mapping-square-to-circle.html
[8] P. Ouyang, F. Ding, X. Wang, Beautiful Math, Part 4: Polygonal Aesthetic Patterns Based on the Schwarz-Christoffel Mapping. IEEE Computer Graphics & Applications, pp. 22-25, July-Aug. 2015
[9] D. Schattschneider, Escher’s Metaphors. Scientific American, pp.66-71, November 1994
[10] D. Swart, Warping Pictures Nicely. Bridges 2011, Coimbra Portugal, pp.303-310
Links
Full Text
http://archive.bridgesmathart.org/2016/bridges2016-179.pdf
