  Suppose that the dial starts at angle I. The initial angle I that the dial is at before it encounters its older self, and the set of all possible final angles that the dial can have when it emerges from the time machine is represented by the circle I on the torus (see figure 1). Given any possible angle of the emerging dial the dial initially at angle I will develop to some other angle. One can picture this development by rotating each point on I in the horizontal direction by the relevant amount. Since the rotation has to depend continuously on the angle of the emerging dial, ring I during this development will deform into some loop L on the torus. Loop L thus represents the angle x that the dial is at when it is thrown into the time machine, given that it started at angle I and then encountered a dial (its older self) which was at angle y when it emerged from the time machine. We therefore have consistency if x=y for some x and y on loop L. Now, let loop C be the loop which consists of all the points on the torus for which x=y. Ring I intersects C at point &lt;i,i&gt;. Obviously any continuous deformation of I must still intersect C somewhere. So L must intersect C somewhere, say at &lt;j,j&gt;. But that means that no matter how the development of the dial starting at I depends on the angle of the emerging dial, there will be some angle for the emerging dial such that the dial will develop exactly into that angle (by the time it enters the time machine) under the influence of that emerging dial. This is so no matter what angle one starts with, and no matter how the development depends on the angle of the emerging dial. Thus even for a circular state-space there are no constraints needed other than continuity.