PG(3,2)

In finite geometry, PG(3,2) is the smallest three-dimensional projective space. It can be thought of as an extension of the Fano plane. It has 15 points, 35 lines, and 15 planes.[1] It also has the following properties:[2]

  • Each point is contained in 7 lines and 7 planes
  • Each line is contained in 3 planes and contains 3 points
  • Each plane contains 7 points and 7 lines
  • Each plane is isomorphic to the Fano plane
  • Every pair of distinct planes intersect in a line
  • A line and a plane not containing the line intersect in exactly one point
PG(3,2) depicted as a tetrahedron (see text)

Construction from K6

Take a complete graph K6. It has 15 edges, 15 perfect matchings and 20 triangles. Create a point for each of the 15 edges, and a line for each of the 20 triangles and 15 matchings. The incidence structure between each triangle or matching (line) to its three constituent edges (points), induces a PG(3,2). (Sylvester's work on duads and synthemes, 1859)

Construction from Fano planes

Take a Fano plane and apply all 5040 permutations of its 7 points. Discard duplicate planes to obtain a set of 30 distinct Fano planes. Pick any of the 30, and pick the 14 others that have exactly one line in common with the first, not 0 or 3. The incidence structure between the 1+14 = 15 Fano planes and the 35 triplets they mutually cover induces a PG(3,2).[3]

Tetrahedral depiction

PG(3,2) can be represented as a tetrahedron. The 15 points correspond to the 4 vertices + 6 edge-midpoints + 4 face-centers + 1 body-center. The 35 lines correspond to the 6 edges + 12 face-medians + 4 face-incircles + 4 altitudes from a face to the opposite vertex + 3 lines connecting the midpoints of opposite edges + 6 ellipses connecting each edge midpoint with its two non-neighboring face centers. The 15 planes consist of the 4 faces + the 6 "medial" planes connecting each edge to the midpoint of the opposite edge + 4 "cones" connecting each vertex to the incircle of the opposite face + one "sphere" with the 6 edge centers and the body center.[4]

Square representation
Square model of Fano 3-space

A 3-(16,4,1) block design has 140 blocks of size 4 on 16 points, such that each triplet of points is covered exactly once. Pick any single point, take only the 35 blocks containing that point, and delete that point. The 35 blocks of size 3 that remain comprise a PG(3,2) on the 15 remaining points. If these 16 points are arranged in a 4x4 grid and assigned 4-bit binary coordinates like in a Karnaugh map for example, one gets the square representation. Geometrically, the 35 lines are represented as a bijection with the 35 ways to partition a 4x4 grid into 4 regions of 4 cells each, if the grid represents an affine space and the regions are 4 parallel planes.

Kirkman's schoolgirl problem

PG(3,2) arises as a background in some solutions of Kirkman's schoolgirl problem. Two of the seven non-isomorphic solutions to this problem can be embedded as structures in the Fano 3-space. In particular, a spread of PG(3,2) is a partition of points into disjoint lines, and corresponds to the arrangement of girls (points) into disjoint rows (lines of a spread) for a single day of Kirkman's schoolgirl problem. There are 56 different spreads of 5 lines each. A packing of PG(3,2) is a partition of the 35 lines into 7 disjoint spreads of 5 lines each, and corresponds to a solution for all seven days. There are 240 packings of PG(3,2), that fall into two conjugacy classes of 120 under the action of PGL(4,2) (the collineation group of the space); a correlation interchanges these two classes.[5]

Doily depiction
The Doily. This is also a representation of the strongly regular graph srg(15,6,1,3) drawn with overlapping edges.

The Doily diagram often used to represent the generalized quadrangle GQ(2,2) is also used to represent PG(3,2).[6]

Automorphisms

The automorphism group of PG(3,2) maps lines to lines. The number of automorphisms is given by finding the number of ways of selecting 4 points that are not coplanar; this works out to 15⋅14⋅12⋅8 = 20160 = 8!/2. It turns out that the automorphism group of PG(3,2) is isomorphic to the alternating group on 8 elements A8.

Coordinates

It is known that a PG(n,2) can be coordinatized with (GF(2))n + 1, i.e. a bit string of length n + 1. PG(3,2) can therefore be coordinatized with 4-bit strings. A common mapping between these coordinates and the tetrahedral representation above is for tetrehedron vertices to have Hamming weight 1, such as 0001, 0010, etc., and for other points to be the XOR thereof. Thus edge midpoints get Hamming weight 2, face centers get Hamming weight 3, and the body center gets Hamming weight 4.

In addition, the line joining points (a1, a2, a3, a4) and (b1, b2, b3, b4) can be naturally assigned Plücker coordinates (p12, p13, p14, p23, p24, p34) where pij = aibj − ajbi, and the line coordinates satisfy p12 p34 + p13 p24 + p14 p23 = 0. Each line in projective 3-space thus has six coordinates, and can be represented as a point in projective 5-space; the points lie on the surface p12 p34 + p13 p24 + p14 p23 = 0.

Notes
  1. Meserve, Bruce E. (1983) [1955], Fundamental Concepts of Geometry, Dover, p. 29, ISBN 0-486-63415-9
  2. Polster 1998, p. 69
  3. Polster 1998, p. 77
  4. Polster 1998, pp. 82-83
  5. Hirschfeld 1985, p. 73
  6. Polster 1998, p. 69
gollark: This is as it lacks lambdas.
gollark: I would only rewrite 22% of things at most in it.
gollark: I rate it π/7+2i.
gollark: Like you, you mean, yes.
gollark: I'm sure you'd like to think so. However, GTech™ intrusion detection systems detected your futile attempt and provided a decoy.

References
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.