It will depend upon how many flats you require & if the are a preset revolve or a mobile one.
For a 3 flat triangle structure, just place the 3 flats face down on the floor, next to each other with the side ?Stile? edges butted together.,square off all the tail rails, so it has a flush base when it stands up.
I use flap/butt hinge to join the flats together. Place 3 or 4 hinges on each of the two joints between the adjoining flats, created on each side of the central flat - Hinges at the top, bottom & centre of the these joints. You will now have a folding triple book flat. Don?t worry if the top rails of the flats don?t line up, this can be fixed later, as long as the bottom/tail rails line up. If you can get all the flats the same width & height, it does look better ascetically.
Now fold over the left & right flats to make a triangular structure, these edges can be fixed together either by using tek screws or drop pin hinges & /or what ever method you wish.
You can now attach a base board, this can be cut from plywood or other flat board. For a fixed revolving medicci, fix the central spigot to it, which can be as simple as a bolt or whatever, which drops into a floor plate. Choose fixed direction castors with wheel axles that have decent bearings & no sideway slop & not the type which hold the wheels on with a bolt & plastic bearings, also rubber or soft synthetic wheel treads, stay clear of the cheap hard plastic wheel type castors. Mount the castors so the wheel runs in the direction of the flats rotation.
You can use a masking ?Dutchman? over the stile joints on the flats. Depending if they are soft skin [canvas/calico/muslin] or the studio hard skin flats [ply/mdf/masonite or whatever] you can use roller bandage on the soft or tape on the hard. With soft flats, either run the roller bandage thru a container of wallpaper size or the paint you are covering the flats with or just paint the unrolled bandage on, from the top to the bottom.
Cut some black tat material - head cloth or hessian in a strip & attach to the base board as a skirt.
You now have a set of Medici or Periaktoi flats - Honestly it is just as easily as boiling water in a paper bag - even warm props can do that!