Fano Plane - Configurations

Configurations

The Fano plane contains the following numbers of configurations of points and lines of different types. For each type of configuration, the number of copies of configuration multiplied by the number of symmetries of the plane that keep the configuration unchanged is equal to 168, the size of the entire symmetry group.

• There are 7 points, and 24 symmetries fixing any point.
• There are 7 lines, and 24 symmetries fixing any line.
• There are 21 unordered pairs of points, each of which may be mapped by a symmetry onto any other unordered pair. For any unordered pair there are 8 symmetries fixing it.
• There are 21 flags consisting of a line and a point on that line. Each flag corresponds to the unordered pair of the other two points on the same line. For each flag, 8 different symmetries keep it fixed.
• There are 21 ways of selecting a quadrangle of four cyclically ordered points no three of which are collinear, and eight symmetries that fix any such quadrangle. For each flag consisting of a point p and a line l, there is a unique quadrangle in which the four points of the quadrangle are disjoint from l and the four edges of the quadrangle do not pass through p, and every quadrangle corresponds to a flag in this way.
• There are 28 triangles, which correspond one-for-one with the 28 bitangents of a quartic (Manivel 2006). For each triangle there are six symmetries fixing it, one for each permutation of the points within the triangle.
• There are 28 ways of selecting a point and a line that are not incident to each other, and six ways of permuting the Fano plane while keeping a configuration of this type fixed. For every non-incident point-line pair (p,l), the three points that are unequal to p and that do not belong to l form a triangle, and for every triangle there is a unique way of grouping the remaining four points into a non-incident point-line pair.
• There are 28 ways of specifying a hexagon in which no three consecutive vertices lie on a line, and six symmetries fixing any such hexagon.
• There are 42 ordered pairs of points, and again each may be mapped by a symmetry onto any other ordered pair. For any ordered pair there are 4 symmetries fixing it.
• There are 84 ways of specifying a triangle together with one point on that triangle, each of which has two symmetries fixing it.
• There are 84 ways of specifying a pentagon in which no three consecutive vertices lie on a line, and two symmetries fixing any pentagon.
• There are 168 different ways of specifying a triangle together with an ordering for its three points, and only the identity symmetry fixes this configuration.