The Möbius-Kantor configuration is the unique configuration. It is transitive
and self-dual. While it is realizable over the complex
numbers, it cannot be realized over the real or
rational numbers (Gropp 1997). Its incidence structure
is illustrated above using a circle in addition to seven
lines. Note that while some authors (e.g., Grünbaum
2009, p. 68) draw a full circle connecting the three points that cannot be drawn
on a line, others (e.g., Gropp 1997) draw only a circular arc with a gap in the middle.
Aigner, M. Combinatorial Theory. Berlin: Springer-Verlag, p. 335, 1979.Coxeter, H. S. M. "Self-Dual Configurations
and Regular Graphs." Bull. Amer. Math. Soc.56, 413-455, 1950.Gropp,
H. "Configurations and Their Realization." Discr. Math.174,
137-151, 1997.Grünbaum, B. Configurations
of Points and Lines. Providence, RI: Amer. Math. Soc., pp. 67-68, 2009.Pisanski,
T. and Randić, M. "Bridges between Geometry and Graph Theory." In
Geometry
at Work: A Collection of Papers Showing Applications of Geometry (Ed. C. A. Gorini).
Washington, DC: Math. Assoc. Amer., pp. 174-194, 2000.
configuration. It is transitive
and self-dual. While it is realizable over the complex
numbers, it cannot be realized over the real or
rational numbers (Gropp 1997). Its incidence structure
is illustrated above using a circle in addition to seven
lines. Note that while some authors (e.g., Grünbaum
2009, p. 68) draw a full circle connecting the three points that cannot be drawn
on a line, others (e.g., Gropp 1997) draw only a circular arc with a gap in the middle.
