Some Central Configurations of the Newtonian $n$-Body Problem

In this work we construct symmetric central configurations for the $n$-body problem, for $n = 4, 5, 6, 7, 8$. Geometric arguments are used in the proof of existence of these configurations. We observe that these examples of central configurations can be proved by other means, for instance, using arguments for the existence of symmetric solutions to variational problems as is done by J. Montaldi in [10]. However, we believe that our geometrical approach will be useful in tackling other questions of central configurations of the Newtonian $n$-body problem.