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ZOOM, 10Kb

YE QiLan,
Stellar Bloom,

Most spirals, generated by students in past years in the Formative Design Studio, have been compound, geometric spirals with rotational periods that are factors of 360 and with radiants that, serving to mark off the periods, accommodate mirror-reflective symmetry. One of the objectives of this designer was to generate a geometric (i.e., logarithmic or equiangular) spiral that precludes the potential for the application of mirror-reflective symmetry and that winds in a dominant direction, characterized as either clockwise or counterclockwise in regard to the dilatative point of origin-though it must be granted that all geometric spirals that are regularly partitioned display multiple spiral vectors, which run both clockwise and counterclockwise.

There may be innumerable ways to construct such a spiral, but this is the way this designer has done it. The period of her spiral (67) falls between that of a regular hexagon (60) and of a regular pentagon (72). She started with a conveniently sized, kite-shaped diamond, derived from the partitioning of a regular hexagon. Increasing and decreasing them by a factor of 1.1, she then fanned an unending inward and outward continuum of diamonds at 67 intervals. Looking something like petals of a pine cone, these dilatating diamonds overlap. It is to be noted that the points of the diamonds nearest to the center of dilatation and the points farthest from it, as well as those between, all move commensurately in respect to the origin of dilation by a factor of 1.1, as they conform to a logarithmic sequence of similarities. The designer then singled out only the outermost points of the diamond petals; and connecting those points, she drew a line that appears to be more pentagonal than hexagonal. Finally, the designer reticulated the area between the contours of the crinkled spiral by connecting each of the outer vertices to the two closest inner vertices-thus triangulating the whole interstitial area that lies between.


The finalized image presented here is but one of several different gestalts that can be obtained by virtue of the way that colors are applied.

This student employed the power of the computer to do the heavyweight calculations of the coordinates of the petal elements, as they dilate by 1.1. The irksome (to non-mathematicians) calculating and the onerous approximating of incremental changes that are virtually unscalable can, however, be totally avoided through the construction of meticulous logarithmic progressions according to antique methods.


Without the computer scaling incremental measures that are more than likely unwieldy, the successive, dilatative transformation of any one point can be readily plotted in keeping with the exquisite limitations of the Platonic rules of geometrizing. A device for this job may be called a "dilatation truss." To construct this truss, draw to the right a horizontal base line from an origin of dilatation O; then draw to the right a second line from point O at any acute angle (preferably between 20 and 30). Locate a point A at any nonspecific distance and a point B at-in this case - a designated .1 of OA, (The sectioning of any unit into tenths can be easily performed without measurement. This is left to the reader's resourcefulness.) From point A draw (not by necessity, but for convenience) a line that is perpendicular to OA, and let it intersect the angled line at A'. Then draw a line from A' to B. From here on, draw a new perpendicular from point B to intersect the angled line at B'; next draw a line, parallel to A'B, from point B' to intersect at point C on the base line. Repeating these operations, a triangulation will accurately set up the sequence of geometrically spaced points (A, B, C, D,...). To facilitate the construction of a geometric spiral (based on a factor of 1.1) draw concentric circles, centered at the origin of dilatation, through points A, B, C, D... Then radiants are radiated from the origin of dilatation at any desired angle (in this case, 67).

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