You could construct a reference frame to make it any shape you like, but it would be very artificial. I think that the simplest ways of defining or enforcing simultaneity will always give a symmetrical result. [thread=98071]That thread[/thread] also deals with the state of the wheel when it is in constant motion, and related reference frames. rpenner's post, for example, explains that in the instantaneous inertial reference frame of any part of the wheel (such as Bottom, for example), the wheel is neither rigid nor moving simply. Further down in [post=2510505]posts 15 and 16[/post], I explicitly attempted to explore how the wheel's non-inertial rest frame could be defined. The links in rpenner's first post to that thread are also very relevant, if difficult to follow through (I get lost fairly quickly): Born coordinates define a coordinate system in which the wheel is stationary, which I think is derived from the Langevin observers like Bottom. There are some interesting issues with this system that I don't fully grasp. For example, see the discontinuity in a Langevin-observer's simultaneous slice of space-time, and the various distinct notions of distance. The Ehrenfest paradox is about both the problem of spinning up the wheel from rest, and the problem that the circumference of the wheel is more than pi times its diameter. Examination of the paradox seems to converge onto the difficulty of synchronizing clocks in a rotating reference frame. The equivalence principle of general relativity might provide some clues. I think that the wheel's rest frame has similarities to an inverted gravity field, a field that repels objects from the wheel's axel. This tells us that in our rotating reference frame, the axel is a special place, and that any natural coordinate system must be based on the axel, and symmetrical about the axel. For example, if all clocks on the wheel are artificially synchronized to a signal from the axel (similarly to GPS synchronization), and one spatial coordinate is the distance from the axel, then any system in which these definitions are uniform will describe a uniform circular wheel. Resources: Einstein's general theory of relativity: with modern applications in cosmology Section 5, Non-Inertial reference frames. Relativity in rotating frames (Guido Rizzi, Matteo Luca Ruggiero, 2004. Preview on Google Books). This looks like an excellent, readable, and complete coverage on exactly this topic. It includes in full the key historical and contemporary papers, and a transcript of a round table discussion between experts as they explore difficulties and disagreements over things like defining simultaneity and length in rotating frames. I'd like to find the time to get it from a library and read it properly.