SYNERGETICS

OCTAPHASE EVERYDAY MIND

In one segment of his TV show Cosmos, Carl Sagan tells a story about how science became differentiated from the metaphysical hocus pocus of Pythagorean numero-magic. One day, the Pythagoreans, playing with their new Law about the sum of the squares giving the square of the hypotenuse, discovered irrational numbers. The right triangle formed by the diagonal of a square of edge-length one has a “square root of two” length (in Synergetics we say “second root” since squares have been booted from center stage) and this strange length does not commensurate with any ordinary fraction derived from simple multiplication only by division of the original unit control length’s segmented “number line.” In other words, a thermometer that uses a root-of-2 control length is inherently incommensurable with one using a standard unit divider. No conversion formula using only rational ratios is extant. Then, the story goes, the Pythagoreans retreated into their caves in perplexity and hid their mathematics out of shame. They were ashamed to admit that nature permitted the irrational, the incommensurable, the eternal turmoil that appears to set up camp in the mind when all hope of a crystal clear omni-rational simple number mathematics is vanquished. The evil principle has won. Pythagoreans could not stomach this evil and so retreated into mysticism. Scientists, on the other hand, bravely set forth to explore the empirical domain of butterflies and atoms, star systems and computers. Scientists were not phased by the prospect of eternal turmoil. Life on Earth looked promising and still does to this day. Mystics still wrestling with suppressed experiences of mortification afraid to venture into the light of day grew up to become religious fanatics and hair-shirts, archetypally opposed to scientific reason. Their scholastic phase reached its peak during the Dark Ages when the light of Science was all but extinguished. A good story. Carl Sagan tells it well.

In Synergetics, this same eternal turmoil is expressed in a war scenario with the Vector Equilibrium and the Tetrahedron on one side and the Icosahedron on the other, with the Octahedron as a kind of go-between double agent. Octa 2 can mediate owing to its transformability into either the VE or the icosa. Tetra l is commensurable with the VE but the VE square diagonal must contract from the second-root-of-2 to 1 to become one more of the 30 (as opposed to 24) edge-lengths of unit length. When this happens, the 20-volumed VE contracts to give the Icosa with tetravolume…