3.4 Making the correct choice for diagonal adjacency.

So we must choose an adjacency rule that makes $\bf {X}_{-f}^{> - \tau} = \bf {X}_{f}^{< \tau}$. Now the sets $\bf {X}_{f}^{\geq \tau}$ obviously satisfy monotonicity as $\tau$ is decreased, in the sense that once a point becomes a member of the set it remains a member. This is obviously true as $f(p) \geq c$ implies $f(p) > \tau$ for $\tau < c$. This implies that if two Highs of a 4-hit face F are adjacent for a given threshold c, then they must then be adjacent for all values $\tau < c$. For 3 dimensions and higher we have to decide the maximum iso-value c for which there is a path through the interior of a hypercube sharing F, between the two Highs. The different methods one can use to interpolate c effect the values of certain types of criticalities but will not change the essential character of our results. The disambiguation value will only be important if there is no path between the two vertices that passes through any other High vertex within the hypercubes that share the face, at the disambiguation value (see the definition of critical 4-hit faces below and figure 33 through figure 39).