Planes and Designs

Projective and affine planes can be viewed as special kinds of designs. The following functions convert between designs and planes.

Design(P): Plane -> Dsgn, SetIncPt, SetIncBlk

The design corresponding to the points and lines of the plane \(P\).

FiniteAffinePlane(D): Inc -> Plane, PlanePtSet, PlaneLnSet

The affine plane corresponding to the incidence structure \(D\).

FiniteProjectivePlane(D): Inc -> Plane, PlanePtSet, PlaneLnSet

The projective plane corresponding to the incidence structure \(D\).

Example: designs (ex-55665c)

The development of a Singer difference set provides a design which satisfies the projective plane axioms, and thus can be converted to a projective plane in Magma.

> sds := SingerDifferenceSet(2, 3);
> sds;
{ 0, 1, 3, 9 }
> sdv := Development(sds);
> sdv;
2-(13, 4, 1) Design with 13 blocks
> spp := FiniteProjectivePlane(sdv);
> spp: Maximal;
Projective Plane of order 3
Points: {@ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 @}
Lines:
    {0, 1, 3, 9},
    {1, 2, 4, 10},
    {2, 3, 5, 11},
    {3, 4, 6, 12},
    {0, 4, 5, 7},
    {1, 5, 6, 8},
    {2, 6, 7, 9},
    {3, 7, 8, 10},
    {4, 8, 9, 11},
    {5, 9, 10, 12},
    {0, 6, 10, 11},
    {1, 7, 11, 12},
    {0, 2, 8, 12}
> Universe(Support(spp));
Residue class ring of integers modulo 13

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