A Sunburst of Forty-Eight Logical Garnets

The eight rows in the top half of Table I generate the 8-cell of Logical Garnets. A model for the 16 rows in all of Table I would have 16 garnets. This covers ordinary infix (A * B), so that the 16 garnets would appear (8, 8) on the front and back faces of a large 3-cube. A model that allows the asterisk to be placed in three positions generates a table that has 48 rows. Along with (A * B), the three positions include (* AB) and (AB *). In other words, the sunburst absorbs the product of all combinations and all permutations of the three elements in (A * B). This product is called a wreath product. Specifically, (8 x 6) activates 48 symmetry regions located eight each on the six faces of the large 3-cube, the same large 3-cube that is serving as a containing frame for the sunburst of 48 Logical Garnets.

The eight garnets on each of the six faces are easily collected into three pairs of opposite faces (3 x 16), so that the sunburst gives us a symmetry-loaded one-model view of ordinary infix (A * B), regular Polish prefix (* A B), and reverse Polish postfix (A B *). Loaded with both the beauty and the elegance of deep symmetry, it just so happens that the three pairs of opposite faces also belong to the one and only one perfect 3-coloring of a 3-cube. This carries us head-on into what we find in crystallography, where the same emphasis is placed not on the three elements in (A * B) but on the three elements in the Miller Index (hkl). Note, for example, that crystallographers call on the same 48-group to describe and classify a special piece of carbon, namely, the gemstone diamond. Consequently, this approach opens the door to the crystallography of logic.

The logic alphabet at work can also be generalized to obtain a sunburst fractal. First, start with the unending series: (A * B), (A * B), (A * B) . . . , when all of them are connected by equivalence signs. Next, add three over-dots and three under-arcs to each (A * B), thereby constructing an unending wreath product that is acting on each term in an unending Master Equivalence. The sunburst at the first (A * B) becomes the motif and then, taking the next step in the change of scale, each next higher-order self-similar sunburst is obtained from each previous (A * B). This generates an infinite fractal nesting of sunbursts. Simple, complex, transparent, now the unending Master Equivalence is standing in another rare moment. This moment displays the sunburst fractal as an elegent, hyper-dimensional, analytic polyform, one that is correct, exact, and beautiful.

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