Contemporary Reflections on Triangular Symmetry - A Cursory Interdisciplinary Analysis of Weber's m haiku (excerpt)
By Leslie Mösknvorr, Haiku Analyst, Vestibriüm University, 1997
Integrated within the deceptively stark, minimalist beauty of m haiku are pivotal implications demanding careful re-evaluation of both the nominal and the intrinsic syllabic forms. As noted by Dr. Lori Kördann, "the inherent irony of actually writing 'five, seven, five' (and nothing more) is its blatant defiance of the conventional '5-7-5' syllabic form."1 Indeed, the employed "1-2-1" alternative is arguably the epitomical manifestation of minimalist symmetric haiku; and the fact that it describes precisely what it is not queries the very essence of structure. Thus, the poet offers a blunt and bitterly poignant meta-satire. But further scrutiny reveals the topicality of said argument, as a veritable plethora of numerological interrelations comprise the spectral foundation underlying this curious facade.
The nominal form of m haiku is, of course, the quite literal translation of "five, seven, five." This sums to seventeen, which is the seventh prime number. The digits of seventeen sum to eight, which is the first non-unit perfect cube. But most significant is the difference of these digits. Six is not only the arithmetic mean of five and seven, but was also considered by the ancient Greeks to be the first "perfect" number, as the sum of its "proper" divisors -- which happen to be the first three natural numbers. Thus, six also represents the third "triangular" number (being of the Gaussian form [n(n+1)]/2 where n=3), paralleling haiku's triangular symmetry. As a side note, the second perfect number, twenty-eight, is also triangular, raising an intriguing postulate I'll defer to the reader.
Not surprisingly, the true beauty of the nominal form emerges when we depart from our base ten convention. In binary representation, 5-7-5 translates into the hauntingly symmetric 101-111-101. Immediately, we see that the three central digits (representing seven) sum to three, the five innermost digits sum to five, and the nine digits in their entirety sum to seven. Obviously, each individual quantity is also symmetric, thus emulating fractal sensibility.2 Echoing this pattern, the sum itself conveys a minimalist base two symmetry, as seventeen translates to 10001.
The intrinsic syllabic form is (ironically) derived from the words, "five," "seven," and "five." That is, 1-2-1, which sums to four. Interestingly, four is far from prime, being the first non-unit perfect square. It is also clear that four is not triangular, and this begs the question of whether any perfect square (other than one) could be triangular. That is, does there exist a natural number x such that x2 = [n(n+1)]/2 for some natural number n? An appeal to n=8 offers a resounding "yes" to this question, generating both perfect square and triangular number, thirty-six. At this juncture, we can't help but notice three things: First, that eight (as we've already seen) is the first non-unit perfect cube; second, that n=8 eerily implies that x=6 (the significance of which was discussed above); and third, that the digits of thirty-six (rather tellingly, three and six) sum to nine, which is the second non-unit perfect square, as the product of three (coincidentally the second prime) multiplied by itself. And thus (after acknowledging nine's base two symmetry of 1001), we have come full circle, since (as previously illustrated) nine is the total number of digits required for a binary representation of the nominal components, five, seven, and five.3
It is impossible to avoid focusing our attention on the binary representation of the sum of the sums of the nominal and the intrinsic forms. That is, seventeen plus four, which is twenty-one -- the triangular number given by n=6. (This uncanny relationship begs an exclamation point, but risks misinterpretation as a factorial -- which isn't entirely trivial, since the digits of that product sum to nine.) Represented in binary form, twenty-one is 10101; the symmetry and digital sum of which require no additional commentary. But furthermore (as the reader has no doubt derived), the base three representation of seven is 21; while in base five, these digits are transposed -- yielding a trans-order reflective symmetry.
Properly oriented, we can now authoritatively address the fundamental question of whether the "m" in m haiku represents "minimal," "meta," "mercury," or "mathematical." The answer, of course, is inherent in the demonstration; and the proof is left as an exercise.