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18. Maps Larger Than Four Variables - - Part 3 - - Recognizing The Patterns

In the previous two insertions, we have tried to show where and why symmetries exist. Ths simplest and most atraightforward consideration, however is to simply remember the various patterns of symmetry. To do this, figure 31 is included, which shows the predominant cases.

Full-column symmetry corresponding to that of figure 28-c is given in figure 31-a. In each of these cases, 32-cell symmetries are shown. Obvious subsets of these would be symmetrical (2^N) partial rows or partial columns.. Thus, for example, from figure 31-a, we could have just the first two rows, or the second and third - along with the sixth and seventh rows, or maybe the middle four rows. (Where there are symmetries, the intersection of these symmetries are also valid.) Similar, full row symmetry is shown in figure 31-b.

Full-row symmetry corresponding to that of figure 28-b is given in figure 31-c. The same variations hold as in the paragraph above. Similarly, for that shown in figure 31-d.

A couple of the many adjacent-cell symmetries, as laid out in figure 28-a, are shown in figures 31-e and 31-f. Finally, it is pointed out again that any intersection of these symmetries is valid. Thus, for example we could intersect the symmetries of figure 31-a with those of figure 31-d, and get a pattern of cells [1,3,7,5,17,19,23,21,49,51,55,53,33,35,39,37]. The trick here is to see all of the myriad possibilities.
(One possibility here is to make a set of transparent overlays.)

A few example cases are shown in figure 35. Fir each of the four cases, which, if any of these groups can be combined to form a 16-bit cell group?

KM
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 Figure 31.pdf (109.7 KB, 14 views)