A Rosette is three or more objects of comparable mass spaced equidistantly about the same circular orbit. The objects will continue in orbit about the systemís center of gravity, even if no object is occupying the center of the system. In other words, the objects in an ideal rosette will circle endlessly in the same circular path, bound together but balanced equidistantly by each otherís gravity.
This arrangement of bodies was first discussed by W.B. Klemperer in The Astronomical Journal, vol. 67, number 3 (April, 1962), on pages 162-7, "Some Properties of Rosette Configurations of Gravitating Bodies in Homographic Equilibrium". Hence, they are sometimes known as Klemperer Rosettes. The most famous example in science fiction of such a rosette would be the Puppeteersí Fleet of Worlds from the Ringworld novels by Larry Niven (who mis-spelled Klemperer as "Kemplerer.")
Rosettes are prone to instability through gravitational perturbation over time. Depending on how well the rosette is set up, these instabilities may not become apparent for centuries or millennia, but they will eventually have to be dealt with. Thus, rosettes, like most megastructures, need active maintenance and periodic adjustment.
Tech Level: 15
The earliest and perhaps most practical application of the Rosette phenomenon would be by setting up a Rosette of redirected asteroids around a space-industrialized world, most likely in a geosynchronous orbit. The asteroids would serve as sources for both raw materials and as bases to set up colonies in orbit. Placing the asteroids in a rosette configuration would not only contribute to their orbital stability but would also facilitate travel and trade between the hypothetical colonies placed on or in the planetoids.
Advanced factories set up on the asteroids could mine and then manufacture the raw metals within it to produce space elevator cables, allowing a world to quickly and practically set up multiple space elevators for large-scale orbital cargo hauling. If the civilization has the intention of constructing a special-purpose Satellite Halo or Planetary Halo (q.v.) placing multiple asteroids in a Rosette may be a necessary first step. The asteroids would then be disassembled and the Halo constructed, the raw materials needed for its manufacture already in the required orbit.
Tech Level: 18
A Moon Rosette uses the same idea for an asteroid rosette, kicked up an order or two of magnitude. Moon rosettes seem much more likely to be constructed around gas giants than around terrestrial worlds, if for nothing else the preponderance of moons to be found around such bodies.
Like with an Asteroid Rosette, a moon Rosette could be used to facilitate manufacturing and resource exploitation (such as mining Helium-3 and isotopes of hydrogen from the gas giantís atmosphere) as well as trade and travel among any colonies set up on the moons. Setting up a rosette could also be a necessary first step in constructing an orbital megastructure in around the gas giant.
Tech Level: 22
A rosette involving planetary bodies might have a very different purpose than the previous two rosette concepts. A very advanced civilization may redirect most of the terrestrial planets in its solar system into a rosette configuration around its central star, in order to maximize the amount of real estate they can have in the life-supporting zone around the sun. The means to move such immense masses without rendering them completely uninhabitable would require incredibly advanced gravity manipulation or similar technology.
A rosette of life-bearing planets with no central star could also function much as described in the novel Ringworld, serving as a very advanced (and admittedly very comfortable) type of generation ship for a race looking to cross interstellar distances. If a race is advanced enough to create and move a planet rosette, its also assumed they could very easily create and maintain artificial orbital light sources in place of a sun.
Tech Level: 25
The purpose of creating an artificial rosette involving entire stars is open for speculation, but is a logical progression of the previous rosette ideas. Stellar rosettes might function as beacons for intergalactic travel, or as centralized power sources for immense trans-Dyson sphere structures.
Moving a star could be accomplished by constructing one or more immense rings around it, which could be used to manipulate or constrict the sunís magnetic field to produce jets of high-speed plasma around one of its poles. If done with precision (which in itself would be a mind-boggling technical feat), the star can be very slowly maneuvered into position over centuries or millennia.
Tech Level: 25
It is also possible to create rosettes of megastructures, both smaller ones and star-centric ones. Dyson Spheres, Ringworlds, and other immense structures could very slowly be maneuvered into a rosette arrangement by super-advanced technologies. The purpose of such an act would be open to speculation. Perhaps the rosette might exist for something as a simple as facilitating trade and travel among the cultures of the structures; perhaps the megastructures were created to be vast fortresses, or to power incomprehensible weapons of some kind against some intergalactic threat. Perhaps the building civilizations needs the immense power the megastructures can tap for some unknowable experiment.