Fano Plane

The 2-D Projective Plane over (``of order two''), illustrated above. It is a Block Design with ,, , , and , and is also the Steiner Triple System .

The Fano plane also solves the Transylvania Lottery, which picks three numbers from the Integers1-14. Using two Fano planes we can guarantee matching two by playing just 14 times as follows. Label theVertices of one Fano plane by the Integers 1-7, the other plane by theIntegers 8-14. The 14 tickets to play are the 14 lines of the two planes. Then if is thewinning ticket, at least two of are either in the interval [1, 7] or [8, 14]. These two numbers are on exactly oneline 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 QuadraticForms. Here, the triples of forms used form the lines of the Projective Geometry on seven points, whoseplanes 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 isthe Simple Group of Order 168 (Klein 1870).

References

Guy, R. ``How to Factor a Number.'' Proc. Fifth Manitoba Conf. on Numerical Math., 49-89, 1975.

Lehmer, D. H. and Lehmer, E. ``A New Factorization Technique Using Quadratic Forms.'' Math. Comput. 28, 625-635, 1974.

Shanks, D. Solved and Unsolved Problems in Number Theory, 3rd ed. New York: Chelsea, pp. 202 and 238, 1985.

• Whitney-Graustein Theorem
• Whitney-Mikhlin Extension Constants
• Whitney Singularity
• Whitney Sum
• Whitney Umbrella
• Whittaker Differential Equation
• Whittaker Function
• Whole Number
• Width (Partial Order)
• Width (Size)
• Wiedersehen Manifold
• Wieferich Prime
• Wielandt's Theorem
• Wiener Filter
• Wiener Function
• Wiener-Khintchine Theorem
• Wiener Measure
• Wiener Space
• Wigner 3j-Symbol
• Wigner 6j-Symbol
• Wigner 9j-Symbol
• Wigner-Eckart Theorem
• Wilbraham-Gibbs Constant
• Wilcoxon Rank Sum Test
• Wilcoxon Signed Rank Test