All of these designs are built upon the involute spiral, which is essentially the unwinding of a circle.

Imagine a blue string wound counterclockwise on a spool of radius one.

As it unwinds, the end of the string traces out an involute path.

The distance from the end of the involute curve of a full circle of radius-1 to its origin is 2pi = 6.28..., the circumference of the circle.Â The successive unwinding can continue indefinitely, with the distance between successive spiral lines equal to the circumference of the inner circle being unwound.

Involute segments can also be duplicated in a circular pattern, creating pathways between spirals that are non-constricting.

This and the unique curvature geometry of the involute spiral allow ducts in turbines to capture the force of moving fluids with all the resistance imparted to rotation, not turbulence.

the simplest inflatable would be an inflated rim around the base disk, stiffening the base of each taught fabric sail with radial stiffeners, with a central mast and tension cables below tightening the leading edge of the fabric sail.

3 vanes and 5 vanes -- many variations possible

These pictures show a 15ft diameter by 21 ft high turbine stiffened with 12” diameter rim and spoke tubes and tapered tubes on leading edge of sails.

Tubes stiffened sails, as pictured here, will be much more work. The top will have to be continued with single thick fabric above where the inflated tubes become too small.

I think they will still need to be stiffened with compression spreaders and tension cables as described above.