Editing Finite geometry (section)

===The smallest projective three-space===
[[File:fano3space.png|thumb|PG(3,2) but not all the lines are drawn]]

The smallest 3-dimensional projective space is over the field [[GF(2)]] and is denoted by [[PG(3,2)]].  It has 15 points, 35 lines, and 15 planes.  Each plane contains 7 points and 7 lines.  Each line contains 3 points.  As geometries, these planes are [[Isomorphism|isomorphic]] to the [[Fano plane]].

[[File:150614-PG-3-2-schoolgirls-arrangement.png|thumb|Square model of Fano 3-space]]

Every point is contained in 7 lines.  Every pair of distinct points are contained in exactly one line and every pair of distinct planes intersects in exactly one line.

In 1892, [[Gino Fano]] was the first to consider such a finite geometry.

====Kirkman's schoolgirl problem====

PG(3,2) arises as the background for a solution of [[Kirkman's schoolgirl problem]], which states: "Fifteen schoolgirls walk each day in five groups of three.  Arrange the girls’ walk for a week so that in that time, each pair of girls walks together in a group just once." There are 35 different combinations for the girls to walk together.  There are also 7 days of the week, and 3 girls in each group. Two of the seven non-isomorphic solutions to this problem can be stated in terms of structures in the Fano 3-space, PG(3,2), known as ''packings''. A [[spread (projective geometry)|spread]] of a projective space is a [[partition (set theory)|partition]] of its points into disjoint lines, and a packing is a partition of the lines into disjoint spreads. In PG(3,2), a spread would be a partition of the 15 points into 5 disjoint lines (with 3 points on each line), thus corresponding to the arrangement of schoolgirls on a particular day. A packing of PG(3,2) consists of seven disjoint spreads and so corresponds to a full week of arrangements.