traveling past the speed of light
This would be more appropiate at Physics.s.c but here is the answer nonetheless.

Your object may be perfectly rigid or not.

If the object is perfectly rigid, you would not be able to accelerate it to the point at which the extreme points of the spires reach the speed of light, because when they are approaching it, their mass will increment towards infinity and the energy you will need to continue accelerating will increment to infinity as well.

So you will simply not be able to accelerate the whole thing to a rotation speed where the extreme points will reach the speed of light.

If the object is not rigid, well, it will simply bend, plus the previous explanation.