克萊因四次曲線可以被視為柏拉圖立體概念到雙曲七邊形鑲嵌的擴充套件，如上圖所示（Coxeter 1956; Thurston 1999, p. 7; Wolfram 2002, p. 1050）。在鑲嵌中，第 個“環”中的七邊形數量驚人地等於 ，其中 是一個斐波那契數（Thurston 1999, p. 5）。
該曲面已被 Helaman Ferguson 用大理石和蛇紋石雕刻出來，並於 1993 年 11 月 14 日在伯克利的數學科學研究所揭幕（Levy 1999, Plate 1 following p. 142; Borwein and Bailey 2003, p. 55, 彩色圖版 IV, 以及封底）。
Borwein, J. 和 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 和 385, 1977.Hurwitz, A. "Über algebraische Gebilde mit eindeutigen Transformationen in sich." Math. Ann.41, 403-442, 1893.Karcher, H. 和 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." http://homepage.mac.com/dmccooey/polyhedra/Toroidal.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.
由下式定義
可以被視為一個引數（上面的圖顯示了曲線在 
在 
和 2 之間的一些值時的情形），在一個特徵為 3 的域上。Hartshorne (1977, p. 305) 稱之為“有趣的曲線”，因為它是非奇異的，每個點都是拐點，並且對偶曲線 
與 
同構，但是自然對映 
是純不可分的。
個“環”中的七邊形數量驚人地等於 
，其中 
是一個斐波那契數（Thurston 1999, p. 5）。
