600.02 A structure is a self-stabilizing energy-event complex. |
601.00 Pattern Conservation |
602.01 Structural systems are cosmically localized, closed, and finite. They embrace all geometric forms-symmetric and asymmetric, simple and complex. |
602.02 Structural systems can have only one insideness and only one out-sideness. |
603.01 All structuring can be topologically identified in terms of tetrahedra. (See Sec. 362.) |
604.00 Structural System |
606.02 Structures most frequently consist of the physical interrelationships of nonsimultaneous events. |
608.00 Stability: Necklace |
Fig. 608.01 |
608.01 A necklace is unstable. The beads of a necklace may be superficially dissimilar, but they all have similar tubes running through them with the closed tension string leading through all the tubes. The simplest necklace would be one made only of externally undecorated tubes and of tubes all of the same length. As the overall shape of the necklace changes to any and all polygonal shapes and wavy drapings, we discover that the lengths of the beads in a necklace do not change. Only the angles between the tubes change. Therefore, stable refers only to angular invariability. |
608.05 We may say that structure is a self-stabilizing, pattern-integrity complex. Only the triangle produces structure and structure means only triangle; and vice versa. |
608.06 Since tension and compression always and only coexist (See Sec. 640) with first one at high tide and the other at low tide, and then vice versa, the necklace tubes are rigid with compression at visible high tide and tension at invisible low tide; and each of the tension-connectors has compression at invisible low tide and tension at visible high tide; ergo, each triangle has both a positive and a negative triangle congruently coexistent and each visible triangle is two triangles: one visible and one invisible. |
608.20 Even- and Odd-Number Reduction of Necklace Polygons |
Fig. 608.21 |
608.21 We undertake experimental and progressive reduction of the tubularly beaded necklace's multipolygonal flexibility. The reduction is accomplished by progressive one-by-one elimination of tubes from the assembly. The progressive elimination alters the remaining necklace assemblage from a condition of extreme accommodation of contouring intimacies and drapability over complexedly irregular, multidimensional forms until the assembly gradually approaches a number of remaining tubes whose magnitude can be swiftly assessed without much conscious counting. As the multipolygonal assembly approaches a low-number magnitude of components of the polygons, it becomes recognizable that an even number of remaining tubes can be arranged in a symmetrical totality of inward-outward, inward-outward points, producing a corona or radiant starlike patterning, or the patterning of the extreme crests and troughs of a circular wave. When the number of tubular beads is odd, however, then the extra tube can only be accommodated by either a crest-to-crest or a trough-to-trough chord of the circle. This is the pattern of a gear with one odd double-space tooth in each circle. If the extra length is used to join two adjacent crests chordally, this tooth could mesh cyclically as a gear only with an equal number of similarly toothed gears of slightly larger diameter, where the extra length is used to interconnect the two adjacent troughs chordally. (See Fig. 608.21) |
608.22 Even-numbered, equilength, tubular-bead necklaces can be folded into parallel bundles by slightly stretching the interconnection tension cable on which they are strung. Odd numbers cannot be so bundled. |
Fig. 608.23 |
608.23 Congruence with Mariner's Compass Rose: As the number of remaining tubes per circle become less than 40, certain patterns seem mildly familiar^{__}as, for instance, that of the conventional draftsman's 360-degree, transparent-azimuth circle with its 36 main increments, each subdivided into 10 degrees. At the 32-tube level we have congruence with the mariner's compass rose, with its four cardinal points, each further subdivided by eight points (see Fig. 608.23). |
608.24 Next in familiarity of the reduced numbers of circular division increments comes the 12 hours of the clock. Then the decimal system's azimuthal circle of 10 with 10 secondary divisions. Circles of nine are unfamiliar. But the octagon's division is highly familiar and quickly recognized. Septagons are not. Powerfully familiar and instantly recognized are the remaining hexagon, pentagon, square, and triangle. There is no twogon. Triangle is the minimum polygon. Triangle is the minimum-limit case. |
608.25 All the necklace polygons prior to the triangle are flexibly drapable and omnidirectionally flexible with the sometimes-square-sometimes-diamond, four-tube necklace as the minimum-limit case of parallel bundling of the tubes. The triangle, being odd in number, cannot be bundled and thus remains not only the minimum polygon but the only inflexible, nonfoldable polygon. |
608.30 Triangle as Minimum-altitude Tetrahedron |
608.31 In Euclidean geometry triangles and other polygons were misinformedly thought of as occurring in two-dimensional planes. The substanceless, no-altitude, planar polygons were thought to hold their shape^{__}as did any polygonal shape traced on the Earth's surface^{__}ignoring the fact that the shape of any polygon of more than three edges is maintained only by the four-dimensional understructuring. Only the triangle has an inherent and integral structural integrity. |
608.32 The triangular necklace is not two-dimensional, however; like all experienceable structural entities it is four-dimensional, as must be all experienceably realized polygonal models even though the beads are of chalk held together by the tensile coherence of the blackboard. Triangles at their simplest consist experientially of one minimum-altitude tetrahedron. |
Next Section: 609.00 |