Infinite geometry,PG(3, 2) is the smallest three-dimensionalprojective space. It can be thought of as an extension of theFano plane,PG(2, 2).

It has 15 points, 35 lines, and 15 planes. 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.^[1]

These can be summarized in a rank 3configuration matrix counting points, lines, and planes on the diagonal. The incidences are expressed off diagonal. The structure is self dual, swapping points and planes, expressed by rotating the configuration matrix 180 degrees.

It has the following properties:^[2]

• Each plane isisomorphic to the Fano plane.
• Every pair of distinct planes intersects in a line.
• A line and a plane not containing the line intersect in exactly one point.

PG(3, 2) has 20160automorphisms. The number of automorphisms is given by finding the number of ways of selecting 4 points that are not coplanar; this works out to (2⁴-1)(2⁴-2)(2⁴-2²)(2⁴-2³)/(2-1) = 15⋅14⋅12⋅8.

The 15 planes can be generated by a block designdifference set (0,1,2,4,5,8,10). The 35 lines can be generated by pairwise plane point intersections, each containing 3 points.^[3]

If one plane is removed (and its 7 points and 7 lines), we create theaffine space AG(3,2), composed of 7 sets of 2 parallel planes (each K₄ graphs). The 8 points and 28 lines alone make acomplete graph K₈ graph. It has 20160/15 = 1344 automorphisms.

Removing one point (and its 7 lines and 7 planes) further creates a smaller self dual rank 3 configuration of 7 points, 21 lines and 7 K₄ graph planes. Automorphisms reduce to 168 (1344/8).

This design is 7₄, and also set complement tofano plane, 7₃. Since 7₃ has difference set (1,2,4), 7₄ planes are the complement set (0,3,5,6). The 21 lines contain 2 points as pairwise intersections of the planes.

Take acomplete graphK₆. It has 15 edges, 15perfect 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. Theincidence structure between each triangle or matching (line) and its three constituent edges (points) induces aPG(3, 2).

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. Theincidence structure between the1 + 14 = 15 Fano planes and the 35 triplets they mutually cover induces aPG(3, 2).^[4]

PG(3, 2) can be represented as atetrahedron. 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. This was described byBurkard Polster.^[5] The tetrahedral depiction has the same structure as the visual representation of the multiplication table for thesedenions.^[6]

Numbering the points 0...14 (4 (v)ertices, 6 mid-(e)dges, 4 mid-(f)aces, and 1 (c)entral), the 15 planes and 35 lines of theconfiguration can be grouped by symmetry positions in the tetrahedron:

Another depiction of PG(3, 2) as a tetrahedron

PG(3, 2) can be represented as a square. The 15 points are assigned 4-bit binary coordinates from 0001 to 1111, augmented with a point labeled 0000, and arranged in a 4×4 grid. Lines correspond to the equivalence classes of sets of four vertices thatXOR together to 0000. With certain arrangements of the vertices in the 4×4 grid, such as the "natural"row-major ordering or theKarnaugh map ordering, the lines form symmetric sub-structures like rows, columns, transversals, or rectangles, as seen in the figure. (There are 20160 such orderings, as seen below inthe section on Automorphisms.) This representation is possible because geometrically the 35 lines are represented as abijection with the 35 ways to partition a 4×4 affine space into 4 parallel planes of 4 cells each. This was described by Steven H. Cullinane.

The Doily diagram often used to represent thegeneralized quadrangleGQ(2, 2) is also used to representPG(3, 2). This was described by Richard Doily.^[2]

PG(3, 2) arises as a background in some solutions ofKirkman'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, aspread ofPG(3, 2) is a partition of points into disjoint lines, and corresponds to the arrangement of girls (points) into disjoint rows (lines of aspread) for a single day of Kirkman's schoolgirl problem. There are 56 different spreads of 5 lines each. Apacking ofPG(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 ofPG(3, 2), that fall into two conjugacy classes of 120 under the action ofPGL(4, 2) (the collineation group of the space); a correlation interchanges these two classes.^[7]

It is known that aPG(n, 2) can be coordinatized with (GF(2))ⁿ⁺¹, i.e. a bit string of lengthn + 1.PG(3, 2) can therefore be coordinatized with 4-bit strings.

In addition, the line joining points(a₁,a₂,a₃,a₄) and(b₁,b₂,b₃,b₄) can be naturally assignedPlücker coordinates(p₁₂,p₁₃,p₁₄,p₂₃,p₂₄,p₃₄) wherep_ij =a_ib_j −a_jb_i, and the line coordinates satisfyp₁₂p₃₄ +p₁₃p₂₄ +p₁₄p₂₃ = 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 surfacep₁₂p₃₄ +p₁₃p₂₄ +p₁₄p₂₃ = 0.

• Fano, G. (1892),"Sui postulati fondamentali della geometria proiettiva",Giornale di Matematiche,30:106–132
• Lohmus, Jaak; Paal, Eugene; Sorgsepp, Leo (1994).Nonassociative algebras in physics. Palm Harbor, Florida: Hadronic Press.ISBN 0-911767-71-1.
• Meserve, Bruce E. (1983) [1955].Fundamental Concepts of Geometry. Dover.ISBN 0-486-63415-9.
• Hirschfeld, J. W. P. (1985).Finite Projective Spaces of Three Dimensions.Oxford University Press.ISBN 0-19-853536-8.
• Polster, Burkard (1998).A Geometrical Picture Book. Springer.ISBN 978-0-387-98437-7.
• Saniga, Metod; Holweck, Frédéric; Pracna, Petr (2015)."From Cayley-Dickson Algebras to Combinatorial Grassmannians".Mathematics.3 (4). MDPI AG:1192–1221.arXiv:1405.6888.doi:10.3390/math3041192.ISSN 2227-7390. This article incorporates text from this source, which is available under theCC BY 4.0 license.

• Projective geometry
• Finite geometry
• Incidence geometry
• Sedenions
• Configurations (geometry)

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