Selected Graphics from SPIRASOLARIS: Form and Phyllotaxis
Figure 2.
Phyllotactic Resonant
Triples in
the Solar System
Thus in Table 6 below
the
mean sidereal periods of revolution and intervening mean synodic
periods are multiplied by the corresponding X-Factors (numbers of mean sidereal
and mean synodic periods) to produce the resonant period for each pair
of planets with the latter also providing the corresponding Resonant
(RZ) Triples. There is more that could be shown here, including
resonant triple variants for Earth and its bracketing synodic periods,
and also further investigations involving slightly improved
correspondances obtained from the use of aphelion and and
perihelion periods rather than mean values.
![Table 3. Resonant Phyllotaxic Triples, Mercury to Neptune (Ceres included, Earth Synodic)](rztp1d8.gif)
Table 6.
Resonant
Phyllotactic Triples, Mercury to Neptune (Mean values; Ceres included,
Earth Synodic)
Although not
identical, the
resonant
periods obtained from Solar System mean values
essentially follow the
fibonacci series from its beginning values out to the number
34 (i.e.,
1,
1, 2, 3, 5, 8, 13, 21, 34,..)
and even further (embracing the next fibonacci number 55)
if
IMO, the
Inter-Mercurial
Object
from Part II is included. Thus for the four superior planets Neptune, Uranus,
Saturn and Jupiter the
first fibonacci
resonant
sequence [ 1 : 1 : 2 ] is
followed by [ 1 : 2 : 3 ] then
[ 2
: 3 : 5 ] and so on
down
to the last resonant triple [ 13 : 21
: 34 ] between Venus
and Mercury
(Figure 2 above).
IV.1. 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.
Table 3.
Benjamin Peirce's
Phyllotactic Divisors
PARENT PAPER:
SPIRASOLARIS:
Form and Phyllotaxis
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February 13, 2007