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Fano Plane


FanoPlane

The Fano plane is the configuration consisting of the two-dimensional finite projective plane the Galois field of order 2 GF(2). It is not realizable over the real or rational numbers (Gropp 1997). The incidence structure of the Fano is plane illustrated above.

It is a block design with nu=7, k=3, lambda=1, r=3, and b=7, the Steiner triple system S(7), and the unique 7_3 configuration. The Levi graph of the Fano plane is the Heawood graph.

The connectivity of the Fano plane corresponds to the order-2 two-dimensional Apollonian network.

The Fano plane also solves the Transylvania lottery, which picks three numbers from the integers 1-14. Using two Fano planes we can guarantee matching two by playing just 14 times as follows. Label the graph vertices of one Fano plane by the integers 1-7, the other plane by the integers 8-14. The 14 tickets to play are the 14 lines of the two planes. Then if (a,b,c) is the winning ticket, at least two of a,b,c are either in the interval [1, 7] or [8, 14]. These two numbers are on exactly one line of the corresponding plane, so one of our tickets matches them.

The Lehmers (1974) found an application of the Fano plane for factoring integers via quadratic forms. Here, the triples of forms used form the lines of the projective geometry on seven points, whose planes are Fano configurations corresponding to pairs of residue classes mod 24 (Lehmer and Lehmer 1974, Guy 1975, Shanks 1985). The group of automorphisms (incidence-preserving bijections) of the Fano plane is the simple group of group order 168 (Klein 1870).


See also

Configuration, Design, Fano's Geometry, Heawood Graph, Projective Plane, Steiner Triple System, Transylvania Lottery

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References

Coxeter, H. S. M. "Self-Dual Configurations and Regular Graphs." Bull. Amer. Math. Soc. 56, 413-455, 1950.Gropp, H. "Configurations and Their Realization." Discr. Math. 174, 137-151, 1997.Grünbaum, B. Configurations of Points and Lines. Providence, RI: Amer. Math. Soc., pp. 67-68, 2009.Guy, R. "How to Factor a Number." Proc. Fifth Manitoba Conf. on Numerical Math., 49-89, 1975.Klein, F. "Zur Theorie der Liniencomplexe des ersten und zweiten Grades." Math. Ann. 2, 198-226, 1870.Lehmer, D. H. and Lehmer, E. "A New Factorization Technique Using Quadratic Forms." Math. Comput. 28, 625-635, 1974.Pisanski, T. and Randić, M. "Bridges between Geometry and Graph Theory." In Geometry at Work: A Collection of Papers Showing Applications of Geometry (Ed. C. A. Gorini). Washington, DC: Math. Assoc. Amer., pp. 174-194, 2000.Shanks, D. Solved and Unsolved Problems in Number Theory, 3rd ed. New York: Chelsea, pp. 202 and 238, 1985.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 72, 1991.

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Fano Plane

Cite this as:

Weisstein, Eric W. "Fano Plane." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FanoPlane.html

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