J. Geom. 86 (2006) 11 – 20
0047–2468/06/020011 – 10
© Birkh¨ auser Verlag, Basel, 2006
DOI 10.1007/s00022-006-1871-x
M¨ obius-Kantor configurations in the affine plane of order 7
Luigia Berardi and Tiziana Masini
Abstract. We partition the affine plane of order 7 into a set of M¨ obius-Kantor configurations 8
3
plus a set consisting
only of one point.
Mathematics Subject Classification (2000): 51E20, 05B25.
Key words: Configurations, finite projective planes, subplanes.
1. Introduction
Many wonderful properties of the geometries of small order can be turned into attractive
pictures that do not have counterparts for higher orders, see [6]. Our goal in writing this
paper is to provide an aesthetically pleasing model of the affine plane of order 7.
A (v
r
,b
k
) configuration C = (X, B) is a pair where X is a set of v points and B is a family
of b subsets called lines in a plane, with k points on each line and r lines through each
point. Two different lines intersect each other at most once and two different points are
connected by a line at most once. By definition it easily follows that the parameters of a
configuration (v
r
,b
k
) must satisfy vr = bk and v ≥ r(k - 1) + 1, see [2]. If v = b and
hence r = k, then configuration is symmetric and denoted by v
k
. Configuration is one of
the oldest combinatorial structure since it appeared for the first time in 1876 in the second
edition of Theodor Reye’s book Geometrie der Lage [7]. Symmetric configurations v
2
are
regular graphs with equally many vertex and edges and so v-cycles. Configurations v
3
exist
if and only if v ≥ 7. The Fano configuration, i.e. the projective plane of order two, is the
unique configuration 7
3
. In 1881 S. Kantor, see [4], classified all configurations v
3
with
v ∈{8, 9, 10}. He proved that the number of configuration 8
3
,9
3
, 10
3
are 1, 3 and 10
respectively. The deficiency d = v - r(k - 1) - 1 is the number of points which are not
connected to a given point. Let (X, B) denote a (v
r
,b
k
) configuration with deficiency 0,
and x a point of X. We may form its residual (at x ) by taking (X -{x }, {B : x/ ∈ B ∈ B}).
An extension of a configuration C is a configuration E such that C is a residual of E. The
M¨ obius-Kantor configuration 8
3
is unique, see [8]. We can construct it by removing a point
together with all the lines through this point from the affine plane of order three, i.e. it is
the residual of a (9
4
, 12
3
) configuration, see [1].
A (vertexless) projective triangle P
0
P
1
P
2
of side n in P G(2, q) is a set T of 3n points such
that
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