Combinatorial Intricacies of Labeled Fano Planes

Metod Saniga

doi:10.3390/e18090312

Abstract

Given a seven-element set X = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } , there are 30 ways to define a Fano plane on it. Let us call a line of such a Fano plane—that is to say an unordered triple from X—ordinary or defective, according to whether the sum of two smaller integers from the triple is or is not equal to the remaining one, respectively. A point of the labeled Fano plane is said to be of the order s, 0 ≤ s ≤ 3 , if there are s defective lines passing through it. With such structural refinement in mind, the 30 Fano planes are shown to fall into eight distinct types. Out of the total of 35 lines, nine ordinary lines are of five different kinds, whereas the remaining 26 defective lines yield as many as ten distinct types. It is shown that no labeled Fano plane can have all points of zero-th order, or feature just one point of order two. A connection with prominent configurations in Steiner triple systems is also pointed out.

Highlights

The Fano plane is the smallest projective plane
We find that nine ordinary lines are of five different kinds, whereas the remaining 26 defective lines yield as many as ten distinct types
The Fano plane occurs in algebraic geometry and geometric algebra in a number of disguises [9,10], providing a link between such important mathematical concepts as design theory, error-correcting codes, Latin squares, skew-Hadamard matrices, Klein’s quartic curve, and Leech’s eight-dimensional minimal sphere-packing lattice, being perhaps most recognized as a “gadget”

Summary

Introduction

The Fano plane is the smallest projective plane It consists of seven lines and seven points, with three points on a line and, dually, three lines per point, where every pair of points is connected by a line, every line intersects every other line, and there are four points such that no line contains more than two of them. It is well known [1,2,3] that there are thirty different Fano planes on a given seven-element set.

Refined Structure of Numbered Fano Planes

Conclusions