It is well known that the arrangement of the leaves in plants may be expressed by very simple series of fractions, all of which are gradual approximations to, or the natural means between 1/2 or 1/3, which two fractions are themselves the maximum and the minimum divergence between two single successive leaves. The normal series of fractions which expresses the various combinations most frequently observed among the leaves of plants is as follows: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34, 21/55, etc. Now upon comparing this arrangement of the leaves in plants with the revolutions of the members of our solar system, Peirce has discovered the most perfect identity between the fundamental laws which regulate both.
Louis Agassiz, ESSAY ON CLASSIFICATION,
Ed. E. Lurie, Belknap Press, Cambridge, 1962:127;
Wikipedia provides the following definition:
In botany, phyllotaxis or phyllotaxy is the arrangement of leaves on a plant stem (from Ancient Greek phýllon “leaf” and táxis “arrangement”).
Phyllotactic spirals form a distinctive class of patterns in nature.
John N. Harris put it all back together in 2007:
THE PHYLLOTACTIC SOLAR SYSTEM
The essential question to be investigated here is whether Benjamin Peirce was correct concerning the phyllotactic structure of the Solar System. I suggest that the answer is undoubtedly yes, but nevertheless there is a difference between the approaches adopted by Peirce and myself. Both utilized Time (mean sidereal periods of revolution) rather than mean heliocentric Distances, but my also own included the successive intermediate synodic periods (i.e., lap times) between adjacent planets. It was this step that earlier — as described in Part II of Spira Solaris Archytas-Mirabilis — resulted in the determination of the underlying constant of linearity for the Solar System, which for successive periods (synodic lap cycles included) turned out to be the ubiquitous constant Phi = 1.6180339887949 and for planet-to-planet increases the square of this value, i.e, Phi 2 = 2.6180339887949 (see Part II). This produced in turn to a number of similar Phi-based planetary frameworks including a variant that owed its origin to the use of mean orbital velocities and complex inverse velocity relationships that linked the superior and inferior planets (again, see Part II and Part III for details).
Even so, comparisons between the Phi-Series planetary frameworks and the present Solar System were fraught with difficulty, not least of all because of three apparent anomalies: 1) the location of Earth in a synodic (i.e., intermediate) position between Venus and Mars; 2) the well-known “gap” between Mars and Jupiter, and 3), an “abnormal” location for Neptune, which produced an atypical synodic lap time for its inner neighbor, Uranus. Some may find the suggestion that Earth is occupying a synodic location uncomfortable, but at least the newly announced Dwarf planet status of Ceres accounts for the Mars-Jupiter gap reasonably enough, though Ceres remains but one asteroid among thousands in this region.
But in any event it is the location of Neptune that provides the key to Benjamin Peirce’s far-reaching understanding of the matter. This I failed to comprehend when I first added the latter’s material to Part VI owing to a basic difference in methodology. The inner starting point adopted by myself was perhaps always likely to produce the Pheidian Framework, but might not necessarily shed light on the final phyllotactic aspect. Starting from the opposite end, however, onecommences with the latter, and with the larger fibonacci fractions applied by Peirce, one also moves with increasing accuracy towards the constant of linearity, Phi. Thus starting from this direction was (in retrospect) always likely to be more productive. In one sense, however, Peirce’s approach may have appeared almost too simplistic with divisions involving periods of revolution expressed in days and the unexpected constant duplication of his divisors. The reason behind this latter occurrence is more complex than one might suspect since it is intimately related to intermediate synodic periods between adjacent pairs of planets. However, since the latter are simply obtained from the general synodic formula (the product of the two periods of revolution divided by their difference; see Part II), one can readily test the paired divisors applied Peirce from this particular viewpoint. Thus in Table 1 below the fibonacci fractions employed by Peirce are applied firstly to the mean period of revolution of Neptune (given here as 164.62423 years, i.e., Peirce’s 60,129 days divided by365.25 days). Thereafter the divisions continue sequentially from the previous result, i.e., division by 1 again, by1/2, 1/2, then 2/3, 2/3, etc., down to the final divisor 21/34 to obtain the mean sidereal period of Mercury. Further division could, of course, follow with another division by 21/34 and the application of the next divisor (34/55), etc., as far as one might wish.
THE PHYLLOTACTIC DIVISORS AND SYNODIC PERIODS
For purposes of comparison modern values for the mean sidereal periods and the calculated synodic periods for the planets are given in the Sol.System column. A second comparison lists the ratios for each step followed by the average values obtained from each column. As can be seen, the latter are close to fibonacci ratios of 21/13 and 55/34 with resulting pheidian approximations of 1.61665353 and 1.61737532 respectively despite the variance in the individual ratios.
The entire John N. Harris article is at: http://www.spirasolaris.ca/spirasolaris.html