Inversive plane

An inversive plane is a class of incidence structure in mathematics.

It may be axiomatised by taking two classes, "points" and "circles" (or "blocks") with the properties
* any three points lie on exactly one circle;
* if "P" and "Q" are points and "c" a circle with "P" on "c" and "Q" not, then there is exactly one circle "e" containing "P" and "Q" and intersecting "c" only in "P";
* there are four points not all on the same circle.

The finite inversive planes are precisely the $3-(n^2+1,n+1,1)-$designs. Such a design is always a Steiner system.

Ovoids

When one takes as points the points of an ovoid in "PG(3,q)", with "q" a prime power, and as blocks the planes that are not tangent to the ovoid, one finds a $3-(q^2+1,q+1,1)-$design.

Inversive planes that arise in this way are said to be egglike. Dembowksi proved that when "n" is even, every inversive plane is egglike (and thus "n" is a power of 2). It is not known to be true when "n" is odd.

References

* E.F. Assmus Jr and J.D. Key, "Designs and their codes", Cambridge University Press, ISBN 0-521-45839-0. Pp.309-312.
* P. Dembowski, "Finite geometries", Springer Verlag, 1968, repr.1996, ISBN 3540617868.
* D.R. Hughes and F.C. Piper, "Design theory", Cambridge University Press, ISBN 0-521-35872-8. Pp.133-136.