finite planeFinitePlane2002-10-07 05:29:242005-05-22 08:29:42DefinitionFano planefinite geometry combinatorics\usepackage{graphicx}
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A {\bf finite plane} (synonym {\bf \PMlinkname{linear space}{LinearSpace2}}) is the finite (discrete) analogue of planes in more familiar geometries. It is an {\bf incidence structure} where any two {\bf points} are incident with exactly one {\bf line} (the line is said to ``pass through'' those points, the points ``lie on'' the line), and any two {\bf lines} are incident with {\em at most\/} one {\bf point} --- just like in ordinary planes, lines can be {\em parallel\/} i.e.\ not intersect in any point.
A finite plane without parallel lines is known as a {\bf projective plane}. Another kind of finite plane is an {\bf affine plane}, which can be obtained from a projective plane by removing one line (and all the points on it).
\subsection*{Example}
An example of a projective plane, that of order $2$, known as the \emph{Fano plane} (for projective planes, {\em order\/} $q$ means $q+1$ points on each line, $q+1$ lines through each point):
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An edge here is represented by a straight line, and the inscribed circle is also an edge. In other words, for a vertex set $\{1, 2, 3, 4, 5, 6, 7 \}$, the edges of the Fano plane are
$$
\{1, 2, 4 \},
\{2, 3, 5 \},
\{3, 4, 6 \},
\{4, 5, 7 \},
\{5, 6, 1 \},
\{6, 7, 2 \},
\{7, 1, 3 \}
$$
Notice that the Fano plane is generated by the triple $\{1, 2, 4\}$
by repeatedly adding $1$ to each entry, modulo $7$. The generating triple has the property that the differences of any two elements, in either order, are all
pairwise different modulo $7$. In general, if we can find a set of $q+1$
of the integers (mod~$q^2 + q + 1$) with all pairwise differences distinct,
then this gives a cyclic representation of the finite plane of order $q$.