Patterns on the Genus-3 Klein Quartic

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Carlo H. Séquin: Patterns on the Genus-3 Klein Quartic. In: Bridges 2006. Pages 245–254



Projections of Klein's quartic surface of genus 3 into 3D space are used as canvases on which we present regular tessellations, Escher tilings, knot- and graph-embedding problems, Hamiltonian cycles, Petrie polygons and equatorial weaves derived from them. Many of the solutions found have also been realized as small physical models made on rapid-prototyping machines.

Extended Abstract


Used References

[1] C. Adams, The Knot Book. W. H. Freeman and Co., New York, 1994.

[2] D. Dunham, 168 Butterflies on a Polyhedron of Genus 3. Bridges 2002, Baltimore, Conf. Proc., pp 197–204.

[3] A. Holden, Orderly Tangles. Columbia University Press, New York, 1983.

[4] A. Hurwitz, Ueber algebraische Gebilde mit eindeutingen Transformationen unter sich. Math. Annalen 41 (1893) pp 403-442.

[5] F. Klein, Ueber die Transformationen siebenter Ordnung der elliptischen Funktionen. Math Ann. Vol 14, 1879. (Translation into English by S. Levy [4]).

[6] S. Levy, The Eightfold Way: The Beauty of Klein’s Quartic Curve. Cambridge University Press, 1999.

[7] C. H. Séquin, Viae Globi - Pathways on a Sphere. Proc. Mathematics and Design Conference, pp 366–374, Geelong, Australia, July 3-5, 2001.

[8] C. H. Séquin, and L. Xiao, K12 and the Genus-6 Tiffany Lamp. Proc. ISAMA CTI 2004, pp 49–52, Chicago, June. 17-19, 2004.

[9] C. H. Séquin, Tilings on Klein’s quartic and on the Poincaré disk. -- http://www/~sequin/GEOM/TILES/

[10] J. Yen and C. H. Séquin, Escher Sphere Construction Kits. Proc. Interactive 3D Graphics Symposium, pp 95- 98, Research Triangle Park, NC, March 19-21, 2001.


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