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Projective and affine planes can be viewed as special
kinds of designs. The following functions convert between
designs and planes.
The design corresponding to the points and lines of the plane P.
The affine plane corresponding to the incidence structure D.
The projective plane corresponding to the incidence structure D.
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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