Basic demo 3
Real steps
Click one of the buttons:

What happens when we actually step the ring by some angle, on top of a slow rotation? By clicking on the "Small backward" and "Small forward" buttons, you can see that small steps are perceived correctly. However, beyond a certain limit, all steps are perceived as a large, backward jump (with respect to the preceding motion)—even if the step is actually forward. Thus, the "Large backward" and "Large forward" buttons seem to yield the same, backward jump; but if you closely follow one of the pattern elements as you press the "Large forward" button, you can convince yourself that it actually turns forwards. Note that the "huge" jumps do not seem to go any farther than the "large" jumps: both go about as far as ordinary high phi, obtained by randomizing the texture (click "Randomize").

The limit between the region of small steps perceived veridically, and large steps that are all perceived as the same backward jump, is at a value called dmax—the largest step across which motion can be coherently perceived. This value varies from one observer to the next, but the angle by which the ring seems to jump is closely correlated to individual values of dmax, being exactly at or just beyond this limit. In other words, when seeing the "high phi" jump, you are seeing movement that is closely correlated to your own dmax displacement limit.

When the preceding motion is very brief, as in Demonstration 2, the default jump perceived for all steps beyond a certain limit is in the forward direction with respect to the preceding motion.

Clicking "Randomize" replaces the texture by a new, uncorrelated random texture, as in Demonstrations 1 and 2. As you can see, the backward jump is about the same as you perceive for the large (and huge) steps.