Even- and Odd-Number Reduction of Necklace Polygons

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 star-like 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.

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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).

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Fig. 608.23 Mariner’s Compass Rose: This pattern accommodates 32 necklace tubes per circle.

608.24 Next in familiarity 6f 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.