Is everything moving at c in a c unit circle I was trying to explain special relativity to a few friends in a simple way and wound up with an analogy using a c unit circle.
I was using y as travelling in time, x moving in space; move in space and you are borrowing from your clock-speed.  E.g. the twin paradox: Your travelling twin has borrowed clock-speed. Twins clock was slower -> twin is younger. 
What is the problem with representing relativity using a (c) unit circle from an observers point of view?  Anything Lorentz or Minkowski would be out of the question of course, I'm looking for a layman friendly description that is close to true.
The circle discussed, was in my mind a half circle $y>=0$. Where $y=0$ would be a photon.
 A: Your model is not only useful for layman, but it does also have physical importance.
One thing needs to be clarified: Your diagram is not a Minkowski diagram (permitting Lorentz transforms). In particular, your y-axis is not coordinate time (as in the Minkowski diagram) but proper time. 
I proposed a similar scheme in Minkowski spacetime: Is there a signature (+,+,+,+)?
By the way, the advantage of such a diagram is that it permits an improved description of time (because any time derives from proper time). Currently we describe time only by the means of Minkowski diagrams – however, Minkowski diagrams were made for Lorentz transformation and not for a description of what time is.
A: Einstein (i think) used a simple example. Travelling upon a photon and looking himself in a mirror. Would he be able to look at his mirror-image or not? Einstein (and Lorentz et al) said yes, as such light (should) travels with same constant speed $c$ in all (inertial) frames of reference. 
Then read Einstein's original 1905 paper (On the electrodynamics of moving bodies) which derives all Relativistic/Lorentz transformations based on this principle (of constancy of speed of light) only.
i think this is good intuitive explanation and analogy.
