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In
finite geometry A finite geometry is any geometry, geometric system that has only a finite set, finite number of point (geometry), points. The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based ...
, PG(3, 2) is the smallest three-dimensional
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
. It can be thought of as an extension of the
Fano plane In finite geometry, the Fano plane (named after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and ...
, ''PG(2, 2)''.


Elements

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. These can be summarized in a rank 3 configuration 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. :\left begin15&7&7\\3&35&3\\7&7&15\end\right /math> It has the following properties: * Each plane is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
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. has 20160
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphism ...
s. 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.


Related affine spaces

If one plane is removed (and its 7 points and 7 lines), we create the
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...
AG(3,2), composed of 7 sets of 2 parallel planes (each K4 graphs). The 8 points and 28 lines alone make a
complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices i ...
K8 graph. It has 20160/15 = 1344 automorphisms. :\left begin8&7&7\\2&28&3\\4&6&14\end\right /math> 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). :\left begin7&6&4\\2&21&2\\4&6&7\end\right /math>


Constructions


Construction from ''K''6

Take a
complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices i ...
''K''6. 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 In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the Point (geometry), points and Line (geometry), lines of the Euclidean plane as t ...
between each triangle or matching (line) and its three constituent edges (points) induces a .


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 In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the Point (geometry), points and Line (geometry), lines of the Euclidean plane as t ...
between the Fano planes and the 35 triplets they mutually cover induces a .


Representations


Tetrahedral depiction

can be represented as a
tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
. 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 by Burkard Polster. The tetrahedral depiction has the same structure as the visual representation of the multiplication table for the
sedenion In abstract algebra, the sedenions form a 16-dimension of a vector space, dimensional commutative property, noncommutative and associative property, nonassociative algebra over a field, algebra over the real numbers, usually represented by the cap ...
s. 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 the
configuration Configuration or configurations may refer to: Computing * Computer configuration or system configuration * Configuration file, a software file used to configure the initial settings for a computer program * Configurator, also known as choice board ...
can be grouped by symmetry positions in the tetrahedron:


Square representation

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 that XOR together to 0000. With certain arrangements of the vertices in the 4×4 grid, such as the "natural" row-major ordering or the
Karnaugh map A Karnaugh map (KM or K-map) is a diagram that can be used to simplify a Boolean algebra expression. Maurice Karnaugh introduced the technique in 1953 as a refinement of Edward W. Veitch's 1952 Veitch chart, which itself was a rediscovery of ...
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 in the section on Automorphisms.) This representation is possible because geometrically the 35 lines are represented as a
bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
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.


Sedenion representation


Doily depiction

The Doily diagram often used to represent the
generalized quadrangle In geometry, a generalized quadrangle is an incidence structure whose main feature is the lack of any triangles yet containing many quadrangles. A generalized quadrangle is by definition a polar space of rank two. They are the with ''n'' = 4 a ...
is also used to represent . This was described by Richard Doily.


Kirkman's schoolgirl problem

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 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 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 , that fall into two conjugacy classes of 120 under the action of (the collineation group of the space); a correlation interchanges these two classes.


Coordinates

It is known that a can be coordinatized with (GF(2))''n''+1, i.e. a bit string of length . can therefore be coordinatized with 4-bit strings. In addition, the line joining points and can be naturally assigned Plücker coordinates where , and the line coordinates satisfy . 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 .


Notes


References

* * * * * * {{Creative Commons text attribution notice, cc=by4, from this source=yes Projective geometry Finite geometry Incidence geometry Sedenions Configurations (geometry)