Klein Quartic
Klein Quartic is a topological surface, given by a parametrisation, genus 3, immersed.
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aperiodic closed closed-orientable def-parametric immersed implemented topological tradition-classical tradition-sculptural
Properties
- Family
- topological
- Given by
- a parametrisation
- Curvature
- no condition imposed
- Periodicity
- not periodic
- Genus
- 3
- Ends
- 0
- Embedding
- immersed
- Orientable
- yes
- Fidelity
- exact
Definition
x^3 y + y^3 z + z^3 x = 0 in the complex projective plane -- not a real surface in R^3, so the realisation is a genus-3 handlebody carrying its 336-fold symmetry.
Sources
Derived from the Math Art generator registries and cross-referenced to the literature; not transcribed from any gallery's compilation.
- S. Levy (ed.), "The Eightfold Way: The Beauty of Klein's Quartic Curve", MSRI Publications 35, Cambridge University Press (1999) -- the volume around Helaman Ferguson's sculpture of the 24-heptagon tiling, which the dual view shows.
- J. C. Baez, "Klein's Quartic Curve", math.ucr.edu/home/baez/klein.html, and G. Egan, "Klein's Quartic Curve", gregegan.net/SCIENCE/KleinQuartic/KleinQuartic.html -- the symmetry story and tetrahedral realizations of the tilings.
- Klein bottle: F. Klein (1882). The default classical bottle shape is built by the tube scheme of G. Franzoni, "The Klein bottle in its classical shape: a further step towards a good parametrization", arXiv:0909.5354 (2009): a tube of varying radius swept along a plane directrix, with the dumbbell-curve directrix of the paper's section 4 (which closes) as the default and its section-3 piriform directrix and the older polynomial immersion as alternatives. A converted copy is in research/papers/ surfaces-and-immersions/franzoni-2009-klein-bottle-classical-shape/.
- Mobius band (the plain ruled one-sided strip): A. F. Mobius (1858) and J. B. Listing (1858), as the standard half-twist ruled parametrization.
- Boy's surface: W. Boy, Math. Ann. 57 (1903), here via the R. Bryant - R. Kusner parametrization.
- Cross-cap and Roman surface: two immersions of RP^2 due to J. Steiner (Rome, 1844).
- F. Klein, 'Ueber die Transformation siebenter Ordnung der elliptischen Functionen', Math. Ann. 14 (1878).