PART IVB2C: THE PHEIDIAN PLANORBIDAE

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A4.2.  WHIRLING RECTANGLES, SQUARES, AND EQUIANGULAR SPIRALS
In some respects the subject of "Whirling" rectangles represents a modern two-part puzzle--not so much the topic per se as the apparent stagnation and lack of understanding that (for whatever reason) currently attends it. This rectangle--"Golden" in the sense that the ratio between the length and the width is 1.61803398874 : 1 (i.e., Phi : 1 )--is more often than not shown in association with the side view of Nautilus pompilus, which is the first part of the puzzle, since the spiral assignment for Nautilus has long been known and the spiral in question has a growth factor more than twice that of the latter. The second part of the puzzle concerns why the matter is rarely taken further; it is surely a natural step when a spiral is shown in relation to a rectangle with attendant squares, etc., to investigate the details and if possible determine what lies behind the observed effect. By way of explanation, maintaining the same ratio between rectangle and square throughout, the original rectangle may be successively partitioned into firstly a square with both sides equal to the previous width, and secondly, into another similar golden rectangle, and so on, rotating 90 degrees with each successive partition. As it so happens, the combination of the quarter-perimeters inscribed in the resulting squares turn out to closely approximate an equiangular spiral, as is often demonstrated in discussions concerning this topic, though the spiral itself (actually k = Phi ⁴ ) is rarely identified. Nor is the representation a true spiral, as most commentators point out, though few tend to elaborate much further. An exception is Jay Hambidge,³⁶ who also describes a similar (though not identical) treatment of "Whirling Rectangles" with respect to root-5 rectangles in Dynamic Symmetry, (1920).

As for the "Golden Rectangle" and the observed spiralling effect, it is perhaps useful to remain with the astronomical side of the matter for a while and consider again what was stated in Section IV (Spira Solaris Archytas-Mirabilis), i.e.,

With respect to the present astronomical application and the exponential planetary framework it may be noted that all mean periods (planet-synodic-planet) increase by phi itself while all planetary periods per se increase by phi squared. Therefore the required period function should increase by the square root of phi per 90-degree segment and by phi squared per revolution. Thus for explanatory purposes, commencing with unity, the first 90-degree segment would have the value 1.27201965, the second (the half-cycle, or 180 degrees) 1.618033989 (phi itself), the third 2.058171027, and at the full cycle, phi squared = 2.618033989.

Not that this is new, though the above application is somewhat specialized. In fact Jay Kappraff ³⁷shows quadrantal growth for the equiangular spiral in this exact manner in a schematic diagram of the logarithmic spiral, replete with attendant rotating and expanding rectangles (Figure 2.11,1991:46). Here it may be noted that in general terms the fixed increase per quadrant is the fourth root of the growth factor per revolution, as Sir D'Arcy Wentworth Thompson was obviously aware in citing the square root for the half-cycle and the square root again for the quarter.³⁸ The fourth root in this context applies to all pheidian spirals and as such it is also inherent in the "Golden Rectangle," though this may not be immediately apparent for a number of reasons. Firstly, the associated spiral is in a sense incomplete with respect to the full rectangle and largest square, whereas it is always "complete" with respect to the smaller rectangles, etc.(see Figure 14 above). Secondly, quadrantal growth in this application is more naturally understood with respect to what may be termed the "outer" spiral as opposed to that enclosed by the rectangle itself. Thirdly, though identical in form, in addition to being external the outer spirals are also inclined at specific angles to both the rectangle and the inner spiral. This should become more apparent from Figure 15, which incorporates the above data for Spiral Solaris and also shows the orientation of the double spirals and the parent rectangle:

Fig. 15.  The Rectangle with Inner and outer Spirals for Spira Solaris, k = Phi ²

Thus while Figure 15 is a representation of quadrantal growth that results in an increase of Phi ²per revolution, the same delineation is also traced out by both the inner and outer spirals associated with the rectangle. However, it is the outer spiral that shows the fourth root increases more clearly, and both are perhaps best demonstrated by animated graphics, firstly with respect to the original "Golden" rectangle and the spiral k = Phi ⁴ and secondly with respect to the associated quadrantal radii vectores for the same spiral; see Quarteranimation I (60kb) and Quarteranimation II  (117kb) respectively.
    As for the "Golden Rectangle" in this particular context, one of its main values would appear to be that it provides a natural lead-in to the above because of the close fit between the quarter-circumferences and the  spiral k = Phi ⁴. But as the animations show, it is neither the rectangles nor the squares that are rotating, but the effects of spiral growth that the latter approximates so well in this particular example. Once the concept is understood, however, it may be extended to any and all pheidian spirals, which was in fact how the double spirals shown in Figures 1a and 1b were initially generated.

The above excerpt is from The Pheidian Planobidae: http://www.spirasolaris.ca/sbb4d2c.html
John N. Harris, M.A. (CMNS).
September 10, 2002