where homogeneous coordinates have been used here so that can be considered a parameter (the plot above shows the curve
for a number of values of
between
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 is isomorphic to but the natural map is purely inseparable.
The surface in complex projective coordinates (Levy 1999, p. ix; left figure), and with the ideal surface determined by the equation
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). McCooey
(2009) gave a non-self-intersecting integer-coordinate polyhedral surface
realizing the regular Klein map. It has 56 polyhedron
vertices, 84 polyhedron edges, and 24 planar
nonconvex heptagonal faces. Its dual
map is the Klein map ,
and its skeleton is the 56-vertex member of the Klein
graphs. The subscript 8 is the length of a Petrie
polygon.
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 th "ring" is, amazingly, equal to , where is a Fibonacci number
(Thurston 1999, p. 5).
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).
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, Germany:
Springer-Verlag, pp. 90-136, 1973.Klein, F. "On the Order-Seven
Transformation of Elliptic Functions" (Trans. S. Levy). 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. I. "A Non-Self-Intersecting Polyhedral
Realization of the All-Heptagon Klein Map." Symmetry: Culture and Science20,
247-268, 2009. https://journal-scs.symmetry.hu/content-pages/volume-20-numbers-1-4-pages-001-448-2009/.McCooey,
D. I. "Klein Map (Basic Shape)." https://dmccooey.com/polyhedra/KleinBasic.html.Thurston,
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.
defined by
can be considered a parameter (the plot above shows the curve
for a number of values of 
between 
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 
is isomorphic to 
but the natural map 
is purely inseparable.
. It has 56 polyhedron
vertices, 84 polyhedron edges, and 24 planar
nonconvex heptagonal faces. Its dual
map is the Klein map 
,
and its skeleton is the 56-vertex member of the Klein
graphs. The subscript 8 is the length of a Petrie
polygon.
th "ring" is, amazingly, equal to 
, where 
is a Fibonacci number
(Thurston 1999, p. 5).
(Basic Shape)."