Klein Quartic


Consider the plane quartic curve X defined by


where homogeneous coordinates have been used here so that z can be considered a parameter (the plot above shows the curve for a number of values of z between -2 and 2), over a field of characteristic 3. Hartshorne (1977, p. 305) terms this "a funny curve" since it is nonsingular, every point is an inflection point, and the dual curve X^* is isomorphic to X but the natural map X->X^* is purely inseparable.

Klein quarticKleinQuarticCurve

The surface in complex projective coordinates (Levy 1999, p. ix; left figure), and with the ideal surface determined by the equation


(Thurston 1999, p. 3; right figure) is more properly known as the Klein quartic or Klein curve. It has constant zero Gaussian curvature.

Klein (1879; translation reprinted in 1999) discovered that this surface has a number of remarkable properties, including an incredible 336-fold symmetry when mirror reflections are allowed (Levy 1999, p. ix; Thurston 1999, p. 2), a number that later was found to be the maximum possible for a curve of its type (Hurwitz 1893; Karcher and Weber 1999, p. 9). Klein arrived at this equation as a quotient of the upper half-plane by the modular group of fractional linear transformations whose coefficients are integers and that reduce to the identity modulo 7 (Levy 1999, p. ix).


The abstract surface cannot be rendered exactly in three-dimensional space, but topologically, the Klein quartic is a three-holed torus (Thurston 1999, pp. 1 and 4). In 2008, the surface was rendered as a toroid with 24 planar heptagons on rational coordinates (McCooey 2009, Szilassi Lajos, pers. comm., Jan. 22, 2009).


The Klein quartic can be viewed as an extension of the concept of the Platonic solids to a hyperbolic heptagonal tiling, as illustrated above (Coxeter 1956; Thurston 1999, p. 7; Wolfram 2002, p. 1050). In the tiling, the number of heptagons in the nth "ring" is, amazingly, equal to 7F_n, where F_n is a Fibonacci number (Thurston 1999, p. 5).

The Eightfold Way, by Helaman Ferguson

The surface has been sculpted by Helaman Ferguson in marble and serpentine, and was unveiled at the Mathematical Sciences Research Institute in Berkeley on November 14, 1993 (Levy 1999, Plate 1 following p. 142; Borwein and Bailey 2003, p. 55, color plate IV, and back cover).

See also

Hyperbolic Tiling, Quartic Curve, Riemann Surface

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Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, pp. 87-88, 2003.Coxeter, H. S. M. "Regular Honeycombs in Hyperbolic Space." In Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, Vol. 3. Groningen, Netherlands: Noordhoff, pp. 155-169, 1956. Reprinted as Ch. 10 in The Beauty of Geometry: Twelve Essays. New York: Dover, pp. 200-214, 1999.Hartshorne, R. Problem 2.4 in Algebraic Geometry. New York: Springer-Verlag, pp. 305 and 385, 1977.Hurwitz, A. "Über algebraische Gebilde mit eindeutigen Transformationen in sich." Math. Ann. 41, 403-442, 1893.Karcher, H. and Weber, M. "The Geometry of Klein's Riemann Surface." In The Eightfold Way: The Beauty of the Klein Quartic (Ed. S. Levy). New York: Cambridge University Press, pp. 9-49, 1999.King, R. B. "Chemical Applications of Topology and Group Theory, 29, Low Density Polymeric Carbon Allotropes Based on Negative Curvature Structures." J. Phys. Chem. 100, 15096, 1996.King, R. B. "Novel Highly Symmetrical Trivalent Graphs Which Lead to Negative Curvature Carbon and Boron Nitride Chemical Structures." Disc. Math. 244, 203-210, 2002.Klein, F. "Über die Transformationen siebenter Ordnung der elliptischen Funktionen." Math. Ann. 14, 428-471, 1879. Reprinted in Gesammelte Mathematische Abhandlungen, 3: Elliptische Funktionen etc. (Ed. R. Fricke et al. ). Berlin: Springer-Verlag, pp. 90-136, 1973.Klein, F. Translated by S. Levy. "On the Order-Seven Transformation of Elliptic Functions." In The Eightfold Way: The Beauty of the Klein Quartic. New York: Cambridge University Press, pp. 287-331, 1999.Levy, S. (Ed.). The Eightfold Way: The Beauty of the Klein Quartic. New York: Cambridge University Press, 1999.McCooey, D. "Toroidal Solids.", W. P. "The Eightfold Way: A Mathematical Sculpture by Helaman Ferguson." In The Eightfold Way: The Beauty of the Klein Quartic (Ed. S. Levy). New York: Cambridge University Press, pp. 1-7, 1999.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1050, 2002.

Cite this as:

Weisstein, Eric W. "Klein Quartic." From MathWorld--A Wolfram Web Resource.

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