Math 2.0

Geometry Projective geometry and linear incidence geometry Theorem

Coordinatization of projective planes

Builds on Projective planePlanar ternary ringIncidence structure

Statement

Let (R,T)(R, T) be a planar ternary ring, with distinguished elements 00 and 11. Let Π(R,T)\Pi(R, T) be the incidence structure defined as follows. Its points are the pairs (x,y)(x, y) with x,y∈Rx, y \in R, the symbols (m)(m) with m∈Rm \in R, and one more symbol (∞)(\infty). Its lines are the symbols [m,k][m, k] with m,k∈Rm, k \in R, the symbols [c][c] with c∈Rc \in R, and one more symbol [∞][\infty]. The line [m,k][m, k] consists of the point (m)(m) and the points (x,y)(x, y) with y=T(x,m,k)y = T(x, m, k). The line [c][c] consists of the point (∞)(\infty) and the points (c,y)(c, y) with y∈Ry \in R. The line [∞][\infty] consists of the points (m)(m) with m∈Rm \in R and the point (∞)(\infty).

  1. For every planar ternary ring (R,T)(R, T), the structure Π(R,T)\Pi(R, T) is a projective plane.
  2. For every projective plane Π\Pi, there is a planar ternary ring (R,T)(R, T) such that Π\Pi is isomorphic to Π(R,T)\Pi(R, T).

Remarks

For the second part, choose four points O,X,Y,IO, X, Y, I of Π\Pi with no three on a line. The set RR is the set of points of the line OIOI other than its point on the line XYXY. The point OO plays the role of 00 and II the role of 11. The points off XYXY get coordinates (x,y)(x, y) by projecting from XX and YY onto OIOI, and T(x,m,k)T(x, m, k) is read off from the line of slope mm through the point (0,k)(0, k). The planar ternary ring depends on the choice of the four points, and different choices can give rings that are not isomorphic. Marshall Hall Jr. introduced planar ternary rings for this purpose.

Example: for a field FF and T(x,m,k)=xm+kT(x, m, k) = xm + k, the plane Π(F,T)\Pi(F, T) is the projective plane over FF, with [m,k][m, k] the line y=mx+ky = mx + k and [∞][\infty] the line at infinity. The plane Π\Pi satisfies Desargues's theorem if and only if, for some (equivalently every) choice of the four points, T(x,m,k)=xm+kT(x, m, k) = xm + k for a division ring structure on RR. If Π\Pi is finite of order nn, then RR has exactly nn elements.

Going deeper

For a student

The theorem turns questions about projective planes into questions about algebraic systems. Geometric properties of the plane match algebraic properties of the ring: Desargues's theorem matches a division ring, and Pappus's theorem matches a field. Planes coordinatized by quasifields, semifields and near-fields give the main families of non-Desarguesian planes. In the finite case it gives a way to search for planes of a given order by searching for ternary operations on a set of that size.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph
Incidence structure, concept, no evidence levelIncidence structureconceptPlanar ternary ring, concept, no evidence levelPlanar ternary ringconceptProjective plane, concept, no evidence levelProjective planeconceptCoordinatization of projective planes, statement, no evidence levelCoordinatization of p…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Incidence structure concept
  2. Planar ternary ring concept
  3. Projective plane concept
  4. Coordinatization of projective planes statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Planar_ternary_ring
About this page
TagMAM2MDG0XG
URIm2://MAM2MDG0XG
Kindstatement, theorem
WritingW0 stub
MSC51A25, 51A35
StatusThis copy of the site has no build data. Run m2 status MAM2MDG0XG.
AboutProjective plane, Planar ternary ring
UsesProjective plane, Planar ternary ring, Incidence structure

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