bcrowell said:
The locus of points that forms the circumference of the disk is a circle, regardless of whether the disk is rotating. One could ask whether an optical observation such as a camera snapshot would show the various *parts* of the disk as distorted to more or less than their normal size. But this implicitly assumes that it's possible to observe the disk first without rotation and then later after it has been spun up while maintaining its rigidity. This is impossible by the Noether-Herglotz theorem, which is essentially the resolution of the Ehrenfest paradox. (There is also a theorem in relativistic optics that says that a sphere still appears to be a sphere, regardless of the motion of the observer relative to the sphere's center. For an animation showing this, see the end of this video:
http://youtube.com/watch?v=JQnHTKZBTI4 .)
Most of the visualizations show a disk that is not just rotating, but moving and rotating as if the disk were rolling, a combination of linear motion and rotation.
Reading back, it's not clear if this is what the OP wants. In the case where the disk is rolling (and thus rotating and also moving in a translational manner), the locus of points won't be a circle anymore (either for the camera or for an observer who corrects for light propagation delays). But it appears the OP originally asked about a disk that was rotating, rather than rolling, so the visualizations given may not be the ones they wanted, and Ben's remark about the motion being circular would then be what they asked for.
I would agree that the disk must deform, so a detailed analysis of how the disk looks would depend on the exact nature of the unobtanium used to construct it. My initial notion was that if the disk were originally round and constructed in a manner that had symmetry under rotation around the ##\phi## axis, it would be reasonable to assume that it retained said symmetry - so the distortion would be one of the radius of the disk, not it's circular shape.
However, thinking about this further, the presence of the spokes (mentioned in the original post) does spoils the perfect symmetry under rotation, so it's quite possible and perhaps even probable that the disk would deform to a non-round "flower" shape, that is symmetrical under rotation by the angle ##\phi## between the spokes but not round in the sense that it was symmetrical under ##\phi## for all values, unless the disk were designed and pre-stressed in some manner so that it would become round when it did rotate.
A few of the simulations had some fine print on this, mentioning that their disks as simulated were assumed to be designed to be round when rolling.
Talk about what a perfectly rigid disk would do is rightly pointed out as non-productive, as one can't ask what happens within a theory when one violates the theory without generating nonsense, and SR doesn't allow perfectly rigid materials.
