Möbius–Kantor configuration

Example

The Möbius–Kantor configuration is the configuration $(8_3)$: $8$ points and $8$ lines, with $3$ points on each line and $3$ lines through each point. Up to isomorphism there is only one $(8_3)$ configuration. A model has the points $0, 1, \dots, 7$ in $\Z/8\Z$ and the lines $\set{i, i + 1, i + 3}$ for $i \in \Z/8\Z$.

Check: each point $j$ lies on the three lines with $i = j$, $i = j - 1$ and $i = j - 3$. The differences of elements of $\set{0, 1, 3}$ are $\pm 1, \pm 2, \pm 3$, each obtained once and all distinct modulo $8$. So two points whose difference is one of these lie on exactly one common line, and two points with difference $4$ lie on none.

Another model: delete one point from the affine plane of order $3$, together with the $4$ lines through it. The remaining $8$ points and $8$ lines have $3$ points on each line, and each point lies on $4 - 1 = 3$ of the remaining lines.

The configuration cannot be realized by points and lines of the real plane, but it can be realized in the complex projective plane. Its incidence graph is the Möbius–Kantor graph.