A Klemperer Rosette is defined as a system of an even number of sets of bodies. The bodies in each set possess equal mass. All bodies are situated at the corners of a regular polygon (with or without a central mass).
A Klemperer Rosette:
It is common to describe a similar configuration of identical mass bodies at the points of a regular polygon as a Klemperer Rosette but this is a misuse of the term. This configuration was known before Klemperer identified Klepmerer Rosette.
This picture from Cubist-Assassin64 provides a good view of a rosette, but it is NOT Klemperer Rosette
Such symmetry is also possessed by a peculiar family of geometrical
configurations which may be described as 'rosettes'. In these an even
number of 'planets' of two (or more) kinds, one (or some) heavier than
the other, but all of each set of equal mass, are placed at the
corners of two (or more) interdigitating regular polygons so that the
lighter and heavier ones alternate (or follow each other in a cyclic
Klemperer Rosettes are unstable.
More precisely, they are statically stable but dynamically unstable. Any tiny perturbation from this stable state leads to problems.
Simulations of this system2 (or a simple linear perturbation
analysis) demonstrate that such systems are definitely not stable: any
motion away from the perfect geometric configuration causes an
oscillation, eventually leading to the disruption of the system
(Klemperer's original article also states this fact). This is the case
whether the center of the Rosette is in free space, or itself in orbit
around a star.
Although still unstable, a hexagonal Rosette (of either type), should possess some additional stability because the adjacent bodies of the Rosette will sit in each other's L4 & L5 points.