17. Maps Larger Than Four-Variables - - - Part 2
The main problem with larger maps is - - - determining what is symmetrical? With four-variable maps, all we have to know about any two cells is - - - if they are adjacent they are symmetrical, and if they are symmetrical they are adjacent. With larger maps, it's not quite that simple. Obviously, any two adjacent cells are still symmetrical, but now, cells can be non-adjacent and also be symmetrical. On the other hand a lot of non-adjacent cells are not symmetrical to each other.
To answer this question in the simplest, most straightforward way, figure 28 is included. This figure defines which cells across any horizontal row are "symmetrical" to which other cells along that row. The first illustration (figure 28-a) shows us the obvious, that all cells are symmetrical to their immediate neighbors. Figure 28-b shows us which cell pairs (other than the adjacent ones) are symmetrical within each half - - the left or the right - - of the map. In this case, there are two possible symmetrical couplings (four, if the two adjacent ones in the middle of each half are counted, but these have already been considered as being adjacent). Finally, figure 28-c shows us which cells are symmetrical between the two horizontal halves of the map (again, the adjacent coupling possibilities have been omitted). We can verify these non-adjacent symmetries via either of two methods: 1) study of the "unfolding" process through which the map was formed, or 2) observation of the binary value of the cell numbers, and that only one bit is different between any two symmetrical cells. The same coupling considerations are true vertically.
In figures 29 and 30, we have four cases, in which we shall determine whether 16-cell groupings can be made . In figure 29-a, there are two groupings of eight cells each, with both arranged vertically in columns. Obviously, each column can itself be grouped, since all cells within each column are symmetrical to that column's other cells (because they are all adjacent). The two columns themselves, however are not symmetrical, and thus cannot be combined to form a 16-cell grouping.
In figure 29-b, we cannot create full groupings either vertically or horizontally. Taking the top row, the reason for this horizontally can be ascertained in two ways: 1) The first two cell minterms will couple, as will the last two, but then the remaining two terms will not couple, because they have different variable sets. 2) The simplest way to state it, however is - - that where the cells [19,18] will couple because they are symmetrical, the other cell pair [17,22] will not couple with the first pair because the two pairs are not symmetrical. The same problem exists vertically.
In figure 30-a, there are two 8-cell groupings, but these will not couple because neither the inside columns, nor the outside columns are symmetrical with each other.
In figure 30-b, the cell groupings will not couple either verticlly nor horizontally. In both of these cases, the reason is the same that it was in the figure 29-c case.
P.S: In the maps that have been supplied, the "axes" for any given variable have been 'color matched' to the header block area for that variable. Thus the "axes" between [A'-A] or [D'-D] are black. Likewise, the "axes" between [B'-B] or [E'-E] are blue. Likewise, the "axes" between [C'-C] or [F'-F] are red. This is done to make it a bit easier to determine which variable is being eliminated by a coupling.