![](spiras6b.gif)
FORM AND PHYLLOTAXIS
It is well known that the
arrangement
of the leaves in plantsmay 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. 1
(Louis
Agassiz, ESSAY ON
CLASSIFICATION, Ed. E. Lurie, Belknap Press, Cambridge,
1962:127; emphases supplied)
I.
PRELIMINARY REMARKS
II. INTRODUCTION
III.
BENJAMIN PEIRCE,
PHYLLOTAXIS, AND THE
SOLAR
SYSTEM
IV. THE PHYLLOTACTIC SOLAR SYSTEM
IV.1.
PHYLLOTACTIC DIVISORS AND
SYNODIC PERIODS
IV.2. THE DUPLICATED DIVISORS
IV.3. THE EXTENDED PHI-SERIES; INNER AND OUTER LIMITS
IV.4. SOLAR SYSTEM RESONANT PHYLLOTACTIC
TRIPLES
V. PHYLLOTAXIS AND
THE PHI-SERIES
IV. IMPLICATIONS AND RAMIFICATIONS
I. PRELIMINARY REMARKS
Initially this matter is simple enough:
Either the Solar System
is phyllotactic in nature, as Benjamin Peirce intimated over one
hundred and fifty years ago, or it is not.
Whether Benjamin Peirce's conclusions should
have been accepted at that
time (or at least received a better airing) is a different matter
altogether. Then again, this paper is not an historical commentary per se, nor is it concerned with
the religious elements evident in Louis Agassiz' Essay on Classification,
or
similar concerns (if that is what they truly are) expressed in Benjamin
Peirce's retirement speech to the membership
of the AAAS
in 1853.
In short, it is the data and the methodology applied by Benjamin Peirce
that will be
re-examined here and ultimately re-worked for modern consumption.
II. INTRODUCTION
Assigned
to the title heading SPIRA
SOLARIS this January 2007 conclusion owes its
origins to a number of events and issues, but the subtitle
itself ( Form and Phyllotaxis ) arises from the neglected Solar
System researches of American
mathematician Benjamin Peirce (1809-1890). Perhaps serendipitously, his
investigation and conclusions resurfaced in September 2006 as a
consequence of the official
elevation of the asteroid Ceres and demotion of the planet
Pluto to the status of "Dwarf Planets." Essentially,with
Pluto
thus demoted Neptune once more becomes the outermost planet, as Peirce
had
stipulated on
theoretical grounds over 150 years
ago. First
published in 18502
and later recorded by his friend Louis Agassiz (1807-1873) in the
latter's
famousEssay on
Classification (1857:131)
Peirce had declared that "there
can be
no
planet exterior to Neptune, but there may be one interior to Mercury." 3 revisiting
my
own attempts to come to terms with the
structure of the Solar System in the first three sections of Spira Solaris
Archytas-Mirabilis the
similarities between the
two approaches now
suggested
that while I had
unknowingly followed Peirce by omitting Pluto and
also adding an
Inter Mercurial object,4 I had nevertheless
missed the point entirely
when it
came to the significance of Neptune. Not so Benjamin Peirce.
Nevertheless, although Agassiz
provided additional details in his Essay
on Classificationconcerning the latter's Solar
System research
including "the
ratios of the laws
of phyllotaxis" 5
in this same astronomical context, Peirce's major conclusion was cast
aside despite its implications. Although not stated in these precise
terms, it was
nothing less than the perception that
The
Solar System is
Pheidian in Form and Phyllotactic in Nature.
Apparently
unable
to promulgate the essence of
the
matter officially, Professor Peirce nevertheless labored to pass
it on in a
speech to the American
Association
for the Advancement
of Science on retiring from
his duties as
President in 1853. For this reason, it would seem that there
was far more to Peirce's speech than casual readers might
suspect,
for at times it
encompassed the past, the present and the future in a most learned and
cryptic manner.
Peirce asked his audience, for example, to
reflect on the following statement, auxiliary questions, and what he
clearly considered to be facts: 6
"Modern
science has
realized some of
the most fanciful of the Pythagorean
and Platonic doctrines, and
thereby justified the divinity of their spiritual instincts. Is it not
significant of the nature of the creative intellect, the
simplicity of
the great laws of force? the fact
that the same curious series of numbers is developed by the growing
plant which assisted in marshalling the order of the planets? and that
the marriage of the elements cannot be consummated except in strict
accordance with the laws of definite proportion? ” (Address
of
Professor Benjamin Peirce, President of the Association of the American
Association for
the Year 1853, on retiring from the duties of President. 1853:6–7;"Printed by Order of
the Association." (The
Cornell Library Historical Mathematics Monographs (JPG) emphases
supplied. Html version: Peirce 1853 ).
Professor
Peirce's retirement
speech
contained further cryptic comments, e.g.,
after mentioning:
"the same
strong, embracing, golden chain of inductive argument.” (1853:7)
mathematician Peirce revisited "consummation" by noting further
(1853:8) that: "the rings of Saturn
are connected with their primary by a force not less mysterious than
that which holds its golden representative upon the finger of the fair
betrothed.” For
those unaware of
Peirce's
phyllotactic treatment of the Solar System parts of his speech
would likely always remain cryptic, although even if his muted
classical
references may have been similarly misunderstood, they would be clear
enough to anyone
following the same route,
as would be his time-honored intent to preserve and pass the matter
on.
Just
how right (or wrong)was Benjamin Peirce concerning
the phyllotactic aspect ot the Solar
System? He was certainly
one of the leading scholars of his day, as was Louis Agassiz, though
both their stars seem to have waned somewhat in their later years. As
for the substance and quality of Benjamin Peirce's work - the rings of
Saturn included
- there is little doubt that he
was an accomplished and influential scientist in his own right, as the
following
excerpts from his Scientific
Bibliography attest:7
PEIRCE,
BENJAMIN
(b.
Salem,
Mass., 1809; d. Cambridge, Mass., 1880), mathematics, astronomy.
Graduated from
Harvard (1829;
M.A., 1833), where he was a tutor
(
1831–33) and professor (1832–80). In mathematics, he amended N.
Bowditch's translation of Laplace's (1829-39);
proved (1832) that there is no odd
perfect number with fewer than four prime factors; published popular
elementary textbooks; discussed possible systems of multiple algebras
in Mécanique
célesteLinear
Associative Algebra ... and
set forth, in A
System of Analytic
Mechanics (1855), the
principles and methods of that science as
a branch of mathematical theory, developed from the idea of the
"potential." In astronomy, he studied comets; worked on revision of
planetary theory and was the first to compute the perturbing influence
of other planets on Neptune; and worked on the mathematics of the rings
of Saturn, deducing that they were fluid. From 1852 he worked with the
U.S. Coast Survey on longitude determination, ... became head of the
survey (1867-74) (and) superintended measurement of the arc of the
thirty-ninth parallel in order to join the Atlantic and Pacific systems
of triangulation. Influential in founding the Smithsonian Institution
and the National Academy of Sciences. (Concise
Dictionary of
Scientific Bibliography,
Charles Scribner’s Sons, New York,
1981:540)
For
my own part, I intend to
show
in this title
essay that Benjamin Peirce's phyllotactic approach to the structure of
the Solar system was essentially correct, all
ramifications and consequences notwithstanding.
The
full description of Benjamin Peirce's application of the Fibonacci
series to
the structure of the Solar System as published by Louis Agassiz is
provided
below; perhaps significantly, the words "Fibonacci" and "Golden
Section" are noticeably absent -- such words perhaps already
unacceptable
to the powers that be and also a perceived threat to the status quo.
Nevertheless, there can be no mistaking the sequence applied or the
major
premise, called here fittingly enough (for the moderns, at
least)
"the law of phyllotaxis". One may also note that Peirce had already
considered
the practical differences between his theoretical treatment and the
Solar
System itself and subsequently considered not only the position of
Earth,
but also discrepancies encountered for the positions of Mars, Uranus
and
Neptune.
Initially Pierce also applied a double form of Fibonacci series but
subsequently
reduced the set to arrive are a situation similar to that involving the
synodic
difference cycle between adjacent planets.
That he did not examine the
matter from this particular viewpoint in greater detail is the reason
why this present essay
exists.
III. BENJAMIN PEIRCE, PHYLLOTAXIS, AND THE
SOLAR
SYSTEM
ESSAY
ON CLASSIFICATION 8
Louis
Agassiz 1857
FUNDAMENTAL
RELATIONS OF
ANIMALS
SECTION XXXI
COMBINATIONS IN TIME
AND SPACE OF
VARIOUS KINDS OF RELATIONS AMONG
ANIMALS
It must
occur to
every
reflecting mind, that the mutual relation and respective parallelism of
so many structural, embryonic, geological, and geographical
characteristics of the animal kingdom are the most conclusive proof
that they were ordained by a reflective mind, while they present at the
same time the side of nature most accessible to our intelligence, when
seeking to penetrate the relations between finite beings
and the cause of their existence.
The phenomena of the inorganic world are all simple,
when compared to those of the organic world. There is not one of the
great physical agents, electricity, magnetism, heat, light, or chemical
affinity, which exhibits in its sphere as complicated phenomena as the
simplest organized beings; and we need not look for the highest among
the latter to find them presenting the same physical phenomena as are
manifested in the material world, besides those which are exclusively
peculiar to them. When then organized beings include everything the
material world contains and a great deal more that is peculiarly their
own, how could they be produced by physical causes, and how can the
physicists, acquainted with the laws of the material world and who
acknowledge that these laws must have been established at the
beginning, overlook that à fortiori the more
complicated laws which regulate the organic world, of the existence of
which there is no trace for a long period upon the surface of the
earth, must have been established later and successively at the time of
the creation of the successive types of animals and plants?
Thus far we have been considering chiefly the
contrasts existing between the organic and inorganic worlds. At this
stage of our investigation it may not be out of place to take a glance
at some of
the coincidences which may be traced between them, especially as they
afford
direct evidence that the physical world has been ordained in conformity
with
laws which obtain also among living beings, and disclose in both
spheres equally
plainly the workings of a reflective mind. It is well known that the
arrangement
of the leaves in plants148 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, as may be at once seen by the following
diagram,
in which the first column gives the names of the planets, the second
column
indicates the actual time of revolution of the successive planets,
expressed
in days; the third column, the successive times of revolution of the
planets,
which are derived from the hypothesis that each time of revolution
should
have a ratio to those upon each side of it, which shall be one of the
ratios
of the law of phyllotaxis; and the fourth column, finally, gives the
normal
series of fractions expressing the law of the phyllotaxis.149><>
![Table I (Peirce-Agassiz 1857, column titles added)](Agt3a.gif)
Table
I
(Peirce-Agassiz 1857; column titles added)
In this series the Earth
forms a break; but this apparent irregularity admits of an easy
explanation. The fractions: 1/2, 1/3, 2/5, 3/8, 5/13,
8/21, 13/34, etc.,
as expressing the position of successive leaves upon an axis, by the
short way of ascent along the spiral, are identical as far as their
meaning is concerned with the fractions expressing these same positions
by the long way, namely, 1/2, 2/3, 3/5, 8/13,
13/21, 21/34, etc.
Let us therefore repeat our diagram in another form, the third column
giving the theoretical time of revolution.
![Table II (Peirce-Agassiz 1857, column titles added)](Agt3c.gif)
Table
II (Peirce-Agassiz
1857; column titles added)
It
appears from this table that two intervals usually elapse between two
successive planets, so that the normal order of actual fractions,
1/2, 1/3, 2/5, 3/8,
5/13,
etc., or the fractions by the short way in phyllotaxis, from
which, however, the Earth is excluded, while it forms a member of the
series by the long way. The explanation of this, suggested by Peirce,
is that although the tendency to set off a planet is not sufficient at
the end of a single interval, it becomes so strong near the
end of the second interval that the planet is found exterior to the
limit of this second interval. Thus, Uranus
is rather too far from the Sun relatively to Neptune, Saturn relatively
to Uranus, and Jupiter relatively to Saturn; and the planets thus
formed engross too large a proportionate share of material, and this is
especially the case with Jupiter. Hence, when we come to the Asteroids,
the disposition is so strong at the end of a single interval, that the
outer Asteroid is but just within this interval, and the whole material
of the Asteroids is dispersed in separate masses over a wide space,
instead of being concentrated into a single planet. A consequence of
this dispersion of the forming agents is that a small proportionate
material is absorbed into the Asteroids. Hence, Mars is ready for
formation so far exterior to its true place, that when the next
interval elapses the residual force becomes strong enough to form the
Earth, after which the normal law is resumed without any further
disturbance. Under this law there can be no planet exterior to Neptune,
but there may be one interior to
Mercury.
Let us now look
back upon some of the leading
features alluded to before, omitting the simpler relations of organized
beings to the world around, or those of individuals to individuals, to
consider only the different parallel series we have been comparing when
showing that in their respective great types the phenomena of animal
life correspond to one another, whether we compare their rank as
determined by structural complication with the phases of their growth,
or with their succession in past geological ages; whether we compare
this succession with their embryonic growth, or all
these different relations with each other and with the geographical
distribution of animals upon earth. The same series everywhere! These
facts are true
of all the great divisions of the animal kingdom, so far as we have
pursued
the investigation; and though, for want of materials, the train of
evidence
is incomplete in some instances, yet we have proof enough for the
establishment of this law of a universal correspondence in all the
leading features which binds all organized beings of all times into one
great system, intellectually and intelligibly linked together, even
where some links of the chain are
missing. It requires considerable familiarity with the subject even to
keep
in mind the evidence, for, though yet imperfectly understood, it is the
most
brilliant result of the combined intellectual efforts of hundreds of
investigators
during half a century. The connection, however, between the facts, it
is
easily seen, is only intellectual; and implies therefore the agency of
Intellect as its first cause. 150
And if the power of thinking connectedly is
the privilege of cultivated minds only; if the power of combining
different thoughts and of drawing from them new thoughts is a still
rarer privilege of a few superior minds; if the ability to trace
simultaneously several trains of thought
is such an extraordinary gift, that the few cases in which evidence of
this
kind has been presented have become a [p.131] matter of historical
record
(Caesar dictating several letters at the time), though they
exhibit
only the capacity of passing rapidly, in quick succession, from one
topic
to another, while keeping the connecting thread of several parallel
thoughts:
if all this is only possible for the highest intellectual powers, shall
we
by any false argumentation allow ourselves to deny the intervention of
a
Supreme Intellect in calling into existence combinations in nature, by
the
side of which all human conceptions are child's play?
If I have succeeded, even very imperfectly, in showing
that the various relations observed between animals and the physical
world, as well as between themselves, exhibit thought, it follows that
the whole has an Intelligent Author; and it may not be out of
place to attempt to
point out, as far as possible, the difference there may be between
Divine
thinking and human thought. Taking nature as exhibiting thought for my
guide,
it appears to me that while human thought is consecutive, Divine
thought
is simultaneous, embracing at the same time and forever, in the past,
the
present, and the future, the most diversified relations among hundreds
of
thousands of organized beings, each of which may present complications
again,
which, to study and understand even imperfectly, as for instance, Man
himself,
Mankind has already spent thousands of years. And yet, all this has
been
done by one Mind, must be the work of one Mind only, of Him before whom
Man can only bow in grateful acknowledgment of the prerogatives he is
allowed
to enjoy in this world, not to speak of the promises of a future life.
I have intentionally
dismissed many points in my argument with mere questions, in order not
to extend unduly a discussion which is after all only accessory to the
plan of my work. I
have felt justified in doing so because, from the point of view under
which
my subject is treated, those questions find a natural solution which
must
present itself to every reader. We know what the intellect of Man may
originate,
we know its creative power, its power of combination, of foresight, of
analysis, of concentration; we are, therefore, prepared to recognize a
similar action emanating from a Supreme Intelligence to a boundless
extent. We need therefore not even attempt to show that such an
Intellect may have originated all
the Universe contains; it is enough to demonstrate that the
constitution
of the physical world and, more particularly, the organization of
living
beings in their connection with the physical world, prove in general
the
existence of a Supreme Being as the Author of all things. The task of
science
is rather to investigate what has been done, to inquire if possible how
it has been done, than to ask what is possible for the Deity, as we can
know that only by what actually exists. To attack such a position,
those
who would deny the intervention in nature of a creative mind must show
that
the cause to which they refer the origin of finite beings is by its
nature
a possible cause, which cannot be denied of a being endowed with the
attributes
we recognize in God. Our task is therefore completed as soon as we have
proved His existence. It would nevertheless be highly desirable that
every
naturalist who has arrived at similar conclusions should go over the
subject
anew from his point of view and with particular reference to the
special
field of his investigations; for so only can the whole evidence be
brought
out. I foresee already that some of the most striking illustrations may
be drawn from the morphology of the vegetable kingdom, especially from
the
characteristic succession and systematical combination of different
kinds
of leaves in the formation of the foliage and the flowers of so many
plants,
all of which end their development by the production of an endless
variety
of fruits. The inorganic world, "considered in the same light, would
not
fail to exhibit also unexpected evidence of thought, in the character
of
the laws regulating the chemical combinations, the action of physical
forces,
the universal attraction, etc., etc. Even the history of human culture
ought
to be investigated from this point of view. But I must leave it to
abler
hands to discuss such topics.
SECTION
XXXI
RECAPITULATION
Last Section (31st)
<> 31st. The combination in time and
space
of all these thoughtful conceptions exhibits not only thought, it shows
also premeditation, power, wisdom, greatness, prescience, omniscience,
providence. In one word, all these facts in their natural connection
proclaim aloud
the One God, whom man may know, adore, and love; and Natural History
must
in good time become the analysis of the thoughts of the Creator of the
Universe,
as manifested in the animal and vegetable kingdoms, as well as in the
inorganic world.
<> It may appear
strange that I should have included
the preceding disquisition under the title of an "Essay on
Classification." Yet
it has been done deliberately. In the beginning of this chapter I have
already
stated that Classification seems to me to rest upon too narrow a
foundation
when it is chiefly based upon structure. Animals are linked together as
closely
by their mode of development, by their relative standing in their
respective
classes, by the order in which they have made their appearance upon
earth,
by their geographical distribution, and generally by their connection
with
the world in which they live, as by their anatomy. All these relations
should
therefore be fully expressed in a natural classification; and though
structure
furnishes the most direct indication of some of these relations, always
appreciable
under every circumstance, other considerations should not be neglected
which
may complete our insight into the general plan of creation. (Louis Agassiz, ESSAY ON
CLASSIFICATION, Ed. E. Lurie, Belknap Press, Cambridge,
1962:127-128)
IV.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.
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
IV. 2. THE DUPLICATED DIVISORS
We come now to the underlying
reason why the divisors applied by Benjamin Peirce occur in pairs,
though not from any a
priori or phyllotactic viewpoint. Although perhaps not
immediately obvious, the pairings
concern matters that have long been known, namely resonances
between
adjacent pairs of planets. For example, one of the better known
planetary resonances concerns
the relative motion of Jupiter with respect to Saturn, specifically, a 2:3:5 fibonacci resonance that
occurs after approximately 60 years when slower-moving Saturn (mean
period of revolution 29.45252 years)
completes 2 sidereal
revolutions around the centre of the Solar
System (for present purposes, simply the Sun) with swifter-moving Jupiter (mean period of revolution11.869237 years) completing 5
sidereal revolutions about the same
centre while also lapping Saturn 3
times (lap cycle:19.88813 years),
resulting in a resonant triple with the three
periods expressed in the form 2:3:5.
The same procedures can be carried out on the remaining superior
planets, but in view
of the elliptical nature of planetary
orbits the application of mean motion is a vast
over-simplification, as the real-time investigations carried out in Part
III of Spira Solaris
Archytas-Mirabilis showed (seePart
III, Figures
8-8b; and Figure 1 below for the real-time
Jupiter-Saturn and Uranus-Neptune Resonances from 1940-1990).
IV. 3. THE EXTENDED PHI-SERIES AND THE INNER LIMIT
Once
these resonant relationships are seen for
what they are, however,
it is clear that Benjamin Peirce was indeed correct about
the occurrence of Fibonacci fractions in the the Solar System, and it
is also clear why he took
Neptune to be the outermost limiting planet, i.e., the resonant triples
necessarily commence with the first three fibonacci numbers (1,1,2,3,5,8,13,21,34,55,89,
etc.) and increase accordingly. But there is far more to this matter
for the
three resonant triples 1:1:2, 1:2:3and 2:3:5 associated with the four superior planets Neptune, Uranus,
Saturn and Jupiter are not only
sequential, they are also successive Fibonacci triples
bound together in a most complex manner.
Before moving on to the last issue a question naturally arises whether
an
inner limit exists, and whether this might be determined. Even so,
with Neptune "accounted for" there still remain two additional
anomalous regions to complicate matters, but once again the
elevation of Ceres to the status of Dwarf planet at least fills
the next region below Jupiter, though still leaving the
"anomalous"
location of Earth to be resolved. Fortunately, however, the
inclusion of Ceres provides two further sequential resonant
triples (3:5:8 and5:8:13 respectively) while
the
Phi-Series planetary framework also renders assistance below
Mercury, since (in
theoretical terms at least) it is only a matter of decreasing exponents
by unity
to move inwards as far as one wishes. In this manner it becomes
possible to
suggest
that the resonant triples not only extend sequentially downwards from
Neptune to
Mercury (resonant triple 13:21:34)
but also beyond Mercury to a specific limit
at the fifth inner position (final resonant triple of 144:233:377) as seen below in Table
4 for Phi-Series
periods Phi 5
through Phi -13
Table 4. The Phi-Series Planetary
Periods ( N = 5 to -13 ), Resonant Triples and the Inner Limit
In Table 4 it is the pairings that that are
duplicated, not
the divisors. Some columns are self-explanatory, e.g., the Phi-Series
exponents in Column
N, as are
the resulting sidereal (T) and
synodic periods (S) although in
the latter case they are generated by the
even-numbered exponents rather than the more usual general synodic
formula - just one more facet of an already complex dynamic matter. The
parameters in Column
X are the numbers of complete sidereal revolutions and
synodic cycles per pairing (i.e., sidereal and synodic X-Factors),
while
(omitting Column
FN for
the time being) the data in the Products
and the RZ
Triples are exactly that, with the latter also theX-Factors
from the X
Column.
The rounded periods in Column Fn
require some explanation, although the perceptive reader may have
already
seen that the closer the Products get to
the Inner Limit,
the more obvious it becomes that the former in every instance are
consistent approximations for sequential, fractional exponential values
of Phi, e.g., the lowest (0.723606)
approximates Phi -2/3
= 0.72556, the next (1.17085) Phi 1/3
= 1.17398,
followed by 1.8944
( Phi 4/3
= 1.89954
), then 3.065 ( Phi7/3 = 3.0735) and so
on, thus the exponents increase by unity with each
sidereal pairing. It is also clear from these lower products that for
integer values expressed in years, rounding alternates between each
pairing (i.e., 0.72360 rounded up = 1,
1.1708 rounded down = 1,
1.8944 rounded up = 2, 3.065
rounded down = 3, etc.,
producing yet again the Fibonacci Series from the initial commencement
point. Lastly, the period below Inner 5 would
be Phi -14 with the product obtained from central value of the
previous pairing (233) which
in turn rounds down to zero. Hence the
suggestion that the
Inner 5
position provides the innermost limit.
Applying the same techniques to Solar Systen mean values for Mars,
Ceres and the four superior
planets results in a similar, but not identical pattern, for although
the alternative rounding continues in sequence as far as the lower
products of the Jupiter -Saturn pairing, beyond this pair the products
increasing diverge from the rounded Fibonacci Periods, as do the last
pair of resonant triples:
Table
5. The Phi-Series Planetary
Periods (N = 1 to 11), Resonant Triples and Departures
But this is the Phi-Series planetary
framework which supplies the underlying, rigid Pheidian Form,
but not necessarily
the phyllotactic essence of the more complex, dynamic Solar System
itself. For this we need to examine the superior planet resonances in
real-time in terms of the Fibonacci Series and like multiples utilising
(as before in Part III)
the methods of Bretagnon and Simon (1986) 9
adapted to time-series
analysis:
Figure 1.
Jupiter-Saturn-Uranus-Neptune real-time Fibonacci Resonances I,
1890-1990
Figure
1 above is basically
similar to Figures 8 and 8b and related research
originally
introduced in Part III
with the addition of
blue lines
representing the two Fibonacci departures from Table 5 (89 and 144
years) that occur in consistently low
positions with respect to the waveform of
Jupiter. More importantly, however, is the fact that although
these graphics include fibonacci multiples and divisions, they also
show in one form or another the resonant triples in the sequential
forms under
discussion, i.e, 1:1:2, 1:2:3 and 2:3:5.
IV.4. SOLAR SYSTEM
RESONANT
PHYLLOTACTIC TRIPLES
At which point we return to the divisions obtained by
Benjamin Peirce and consider next the overall Solar System from Mercury
to Neptune with Pluto omitted, Ceres included and Earth in a synodic
location between Mars and Venus as applied in Tables 4 and 5.
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 number34 (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. Phyllotactic Resonant
Triples in
the Solar System
Although it is the synodic cycle that binds each individual
pair, it is clear from the resonant triples and the lower right insert
that the middle
number of the first triple (the number of Neptune-Uranus synodics) is
the same as the firstvalue of the next triple (the number of sidereal revolutions ofUranus), and that the last value of
the first triple (also the number of revolutions of Uranus in the first
set) is the middlevalue of the next (i.e.,
the numbers of Uranus-Saturn
synodics)
and that this is a consistent pattern that links the resonant
triples
throughout. Which in passing is more than faintly reminiscent of
details and relationships furnished in Plato'sTimaeus
31b-32c, though this is not our current concern here.
V. PHYLLOTAXIS AND THE
PHI-SERIES
What remains now is the relationship between
Benjamin Peirce's divisors, the
Phi-Series, and
the application of phyllotaxis to the
structure of the Solar System.
The above somewhat
limited
discussion necessarily concerns complex waveforms and motions for the
mean, varying
and extremal values dictated by elliptical orbits.
Although one could suggest that both the Fibonacci and
the Lucas Series
are embedded in the Solar System, it might be more accurate to say that
they are in fact pulsating through it, and perhaps have been since time
immemorial. That this
inquiry should eventually lead to planetary
resonances involving both the Lucas and the Fibonacci Series
is
hardly surprising given
the prominence of the mathematical
relationships known to exist between the two series (see, for example The
Lucas Numbers and Phi
- More Facts and Figures detailed by Dr. R. Knott). What is
different here,
however, is that the
known relations
that combine the two series occur in a specific and distinct
astronomical context -- not only with respect to the residual elements
of Solar System -- but also with respect to the theoretical Phi-Series
exponential
planetary framework such that (a)
the two major period constants ( PhiandPhi
2 ):
Relations
5a and 5b. The Fundamental
Period Constants
re-occur in the form of the double Fibonacci sequence, and (b), the proximity of theLucas
Series
to the Phi-Series
becomes increasingly apparent as thePhi-Series
planetary periods increase from Jupiter onwards and outwards (i.e.,Lucas 11 : Phi-Series
11.09016994; Lucas 29 :
Phi-Series 29.03444185, etc.):
Table
7. Lucas Series, Phi-Series Periods, Phi-Series Decomposition and
the Fibonacci Series
As for the Phi-Seriesplanetary
framework itself, it
should be noted that while the importance
of the reciprocals of phi and its square (0.618033989
and 0.381966011) needs no introduction to
those familiar with phyllotaxis (especially the latter with respect to
the "ideal angle") both numbers occur with surprising
frequency in
the Phi-Seriesplanetary framework.
As indeed does the primary
constant of linearity,Phi squared
(2.618033898) determined in Part II and applied in Part III and elsewhere -- the
latter
with respect to the the
inverse velocity (Vi) of
the Jupiter-Saturn
synodic (or lap) cycle, with its
reciprocal providing the relative
velocity Vr, which as it turns
out, is also
the Mercury-Venus
synodic period(T), and
additionally, the mean
distance (R) ofMercury.
Thus we already
encounter in the Phi-seriesplanetary
framework complex
dynamic relationships between
the two
inner inferior planets and the two largest superior planets involving
constants known to be intimately related to phyllotaxis
Table
8. Phi-Series Planetary Data:
Periods, Distances, Velocities, and inherent Phyllotactic Constants
To put the
full
significance of the Phi-Seriesplanetary framework
and the constants in question in their full
perspective it may be useful to include here a discussion
pertaining to phyllotaxis from Part
III:11
" It has long been
recognized that although Phi and the Fibonacci Series are intimately
related
to the subject of natural growth that they are not limited to these
two fields alone. Remaining with the Phi-Series, Jay Kappraff points out that the French
architect Le Corbusier "developed
a linear scale of lengths based on the irrational number (phi), the
golden
mean, through the double geometric and Fibonacci (phi) series" for his
Modular
System. The latter's interest in the topic is explained further in the
following informative passage from Kappraff's CONNECTIONS: The Geometric
Bridge between Art and Science:
As a young man, Le
Coubusier
studied the elaborate spiral patterns of stalks, or paristiches as they
are called, on the surface of pine cones, sunflowers, pineapples, and
other
plants. This led him to make certain observations about plant growth
that
have been known to botanists for over a century.
Plants, such as sunflowers, grow
by laying down leaves or stalks on an approximately planar surface. The
stalks are placed successively around the periphery of the surface.
Other
plants such as pineapples or pinecones lay down their stalks on the
surface
of a distorted cylinder. Each stalk is displaced from the preceding
stalk
by a constant angle as measured from the base of the plant, coupled
with
a radial motion either inward or outward from the center for the case
of
the sunflower [see Figure 3.21 (b)] or up a spiral ramp as on the
surface
of the pineapple. The angular displacement is called the divergence
angle and is related to the golden mean. The radial or vertical
motion
is measured by the pitch h. The dynamics of plant growth
can be described by and h; we will explore this further
in
Section 6.9 [Coxeter, 1953].
Each stalk lies on two
nearly
orthogonally intersecting logarithmic spirals, one clockwise and the
other
counterclockwise. The numbers of counterclockwise and clockwise spirals
on the surface of the plants are generally successive numbers from the
F series, but for some species of plants they are successive numbers
from
other Fibonacci series such as the Lucas series. These successive
numbers
are called the phyllotaxis numbers of the plant.
For example, there are 55 clockwise and 89 counterclockwise
spirals
lying on the surface of the sunflower; thus sunflowers are said to have
55, 89 phyllotaxis. On the other hand, pineapples are examples of 5, 8
phyllotaxis (although, since 13 counterclockwise spirals are also
evident
on the surface of a pineapple, it is sometimes referred to as 5, 8, 13
phyllotaxis). We will analyze the surface structure of the
pineapple
in greater detail in Section 6.9.
3.7.2 Nature responds to
a
physical constraint After more than 100 years of study, just
what
causes plants to grow in accord with the dictates of Fibonacci series
and
the golden mean remains a mystery. However, recent studies suggest some
promising hypotheses as to why such patterns occur [Jean, 1984],
[Marzec
and Kappraff, 1983], [Erickson, 1983].
A model of plant growth developed
by Alan Turing states that the elaborate patterns observed on the
surface
of plants are the consequence of a simple growth principle, namely,
that
new growth occurs in places "where there is the most room," and some
kind
of as-yet undiscovered growth hormone orchestrates this process.
However,
Roger Jean suggests that a phenomenological explanation based on
diffusion
is not necessary to explain phyllotaxis. Rather, the particular
geometry
observed in plants may be the result of minimizing an entropy
function such
as he introduces in his paper [1990].
Actual measurements and
theoretical
considerations indicate that both Turing's diffusion model and Jean's
entropy
model are best satisfied when successive stalks are laid down at
regular
intervals of 2Pi /Phi 2 radians, or 137.5 degrees about a
growth center,
as Figure 3.22 illustrates for a celery plant. The centers of gravity
of
several stalks conform to this principle. One clockwise and one
counterclockwise
logarithmic spiral wind through the stalks giving an example of 1,1
phyllotaxis.
The points representing the
centers
of gravity are projected onto the circumference of a circle in Figure
3.23,
and points corresponding to the sequence of successive iterations of
the
divergence angle, 2Pi n/Phi 2, are
shown for values of n
from 1 to 10 placed in 10 equal sectors of the circle. Notice how the
corresponding
stalks are placed so that only one stalk occurs in each sector. This is
a consequence of the following spacing theorem that is used by computer
scientists for efficient parsing schemes [Knuth, 1980].
Theorem 3.3 Let x
be any irrational number. When the points [x] f, [2x] f,
[3x] f,..., [nx] f are placed on the
line
segment [0,1], the n + 1 resulting line segments have at most
three
different lengths.
Moreover, [(n + 1)x]f will
fall into one of the largest existing segments. ( [ ] f means
"fractional part of ").
Here clock arithmetic based on the
unit
interval, or mod 1 as mathematicians refer to it, is used, as shown in
Figure 3.24, in place of the interval mod 2pi around the plant stem. It
turns out that segments of various lengths are created and destroyed in
a first-in-first-out manner. Of course, some irrational numbers are
better
than others at spacing intervals evenly. For example, an irrational
that
is near 0 or I will start out with many small intervals and one large
one.
Marzec and Kappraff [1983] have shown that the two numbers 1/Phi and
1/Phi2
lead to the "most uniformly distributed" sequence among all numbers
between
Phi and 1. These numbers section the largest interval into the golden
mean
ratio,Phi :l, much as the blue series breaks the intervals of the red
series
in the golden ratio.
Thus nature provides a
system
for proportioning the growth of plants that satisfies the three canons
of architecture (see Section 1.1). All modules (stalks) are isotropic
(identical)
and they are related to the whole structure of the plant through
self-similar
spirals proportioned by the golden mean. As the plant responds to the
unpredictable
elements of wind, rain, etc., enough variation is built into the
patterns
to make the outward appearance aesthetically appealing (nonmonotonous).
This may also explain why Le Corbusier was inspired by plant growth to
recreate some of its aspects as part of the Modulor system.(Jay Kappraff, Chapter 3.7.
The Golden Mean and Patterns of Plant Growth, CONNECTIONS
: The Geometric Bridge between Art and Science, McGraw-Hill, Inc.,
New
York, 1991:89-96, bold emphases supplied.) For more on this
topic see also Dr. R.
Knott's
extensive treatment The
Fibonacci Numbers and the Golden Section, the latter's related
links and the The
Phyllotaxis
Home Page of Smith University)
The observation made by Kappraff that: "After more than 100 years of study, just
what
causes plants to grow in accord with the dictates of Fibonacci series
and
the golden mean remains a mystery" may well describe the
modern situation -- the above mentioned on-going researches
notwithstanding -- but from a simpler viewpoint, i.e., more in terms of first (or
perhaps better stated, secondary) causes, the
primary pheidian constant Phi 2
= 2.618033898
can at least be examined in terms of the real-time motions of the two
superior planets Jupiter
and
Saturn that (in the case of the Phi-Series)
generate both this important parameter and its
reciprocal Phi
-2 = 0.381966011. This
second constant is
not only
intimately related to
the phyllotaxic
"ideal" growth
angle of 137.50776405 degrees (0.381966011 x 360O)
it is also a unusual repeat parameter in the planetary framework as
shown
in Table
8 above. As
explained in a
later section (Spira
Solaris and the Pheidian Planordidae):
These
multiple
occurrences
arise from the three-fold
nature of the Phi-Series Planetary Framework, which necessarily
incorporates identical values for periods, distances and velocities
according to their
exponential position in the framework, including
the Inverse
Velocities, which (as will be shown later) also play an
important role in the computation of angular momentum. From this
viewpoint the
well-known 60-year, 2 : 3 : 5
fibonacci resonance between
the two most massive planets in the Solar system (Jupiter and Saturn)
takes
on further significance as Figure 8 (21c] shows,
for
the arithmetic mean of the actual Jupiter and Saturn mean
velocities Vr (shown in the upper and lower
real-time waveforms) is
not only 0.381280708; the
daily
average function for this pair of planets for the 400-year
interval from 1600 -
2000
CE [ 146,000+ data points; Julian Day 2305447.5 through Julian Day
2451544.5 ] is in turn 0.381071579. The figure is
necessarily a
two-dimensional representation of the orbital motions of the two major
Fibonacci planets. Further
complications naturally arise from minor differences in the inclination
of the planetary planes, periodic changes in the lines of apsides, and
the fact
that the entire Solar System is (in a temporal sense) also spiraling
towards the constellation of Hercules.
Figure 3.
The Jupiter-Saturn Average Velocity Function (JS-Avg.Vr) and the
Phyllotactic
Constant k = Phi-2
Thus the plot of
orbital velocities in Figure 3
shows firstly (at the top)
the cyan velocity waveform and mean value for Jupiter and also the
dashed-linePhi-Series
mean value. With a shorter sidereal period (11.090169 versus
11.86924 years) the latter is swifter than that of Jupiter per se, and accordingly it appears
near the top of the waveform rather than the mean. At the
bottom (green waveform), because the difference between the Phi-Series mean
velocity and that of Saturn is relatively small the two mean values are
much closer. In between
lies firstly (in grey) the real-time velocity of the
Jupiter-Saturn
Synodic cycle SD1, and
(in orange) the waveform and mean value for the
arithmetic mean of the Jupiter-Saturn velocities over the test
interval.
The inclusion of real-time velocities
(both relative and inverse) is natural and necessary enough because of
the
complex dynamic motions involved and they are doubly significant when
it
is
realized that relative to unity angular momentum is in turn the product
of the mass and the inverse velocity. Moreover, the role and possible
influence of the four superior planets can hardly be ignored given that
together the latter provide 99% of the total angular momentum
with Jupiter and Saturn alone providing more than 65% of the total.
In passing, as far as the current lack of familiarity in dealing with
orbital
velocities in the above manner may be concerned, I can only say that
almost 18 years have elapsed since my restoration of the velocity
components of the laws of planetary motion 10
was published in the Journal of the Royal
Astronomical Society of Canada ( "Projectiles, Parabolas,
and Velocity Expansions of the Laws of Planetary Motion "JRASC,
Vol 83, No. 3, June 1989 ) and
that it is disappointing that they have not been put into more general
use, especially since the available orbital
relationships are
effectively quadrupled by their inclusion, i.e.,
Table 9.
Distance-Period-Velocity
Relationships
The use of the Inverse
Velocity in
this context may appear unusual at first acquaintance, but it is a
useful
device nevertheless, not least of with respect to the computation of
angular momentum, and (as seen in Part II) the inverse velocities also play
an important and unexpected role in the determination of the
fundamental
log-linear framework by linking the inferior and superior planets,
present gaps
and deficience notwithstanding.
IV.
IMPLICATIONS AND RAMIFICATIONS
The implications
and ramifications of a phyllotactic Solar
System
generate in turn complex questions. Some implications we
may not wish to address; others we may prefer to ignore (if not deny),
though I suggest that we can ill-afford to do so. For if the Solar
System is indeed
phyllotactic, then it follows that there are wider implications to
consider concerning the nature of not only "Life" as we understand it,
but the nature of
Humankind, our current behavioral traits and our role in the general
scheme of things, assuming that we have one.
We are at present, it
seems, far too
pleased with ourselves, and for far too little
reason.
But from a wider and larger
"organic" perspective we surely need to consider where we ourselves are
ultimately headed. And we also need to ask ourselves whether humankind
exists in benign and symbiotic relationships with other living
organisms (both near and far), or whether our behavior
more resembles that of a parasitic infestation, even
perhaps to the
point
of destroying our environment and ourselves. A harsh
assessment?
Perhaps so, but it is not un-called for in light of our
virtually unchecked population growth, our
ceaseless
depredation of the environment and our worsening cycles of violence.
So how indeed do we measure up
from this wider
and more complex perspective, and what might this augur for future
expansion beyond planet Earth? Perhaps more pertinently
still, when we
ask the perennial
question: "Are we Alone?" we might also be wise to ask ourselves whether we are always
likely to
remain so in view of
our unsavoury past, troublesome present, and far from
certain future.
Returning
to the researches of Benjamin Peirce, in
retrospect it is hard to say how far his lines of inquiry might
have
been extended, or
what might ultimately have resulted, but it must surely have been a far
more
useful endeavor than the circular, simplistic and ad hoc
diversions
introduced and perpetuated by "Bode's Law." How could something so momentous
and
far-reaching have been so easily driven into obscurity? According to
the
modern editor of Agassiz' Essay on
Classification, (E. Lurie)
it was
partly the work of Asa Gray and Chauncey Wright, as explained in the
following
footnote (the latter's No.149):
Agassiz tried to interest
Americans in this concept, an idea typical of German speculative
biology and one that he had been much impressed with since his student
days at the University of Munich. See Asa Gray, "On the Composition of
the Plant by Phytons, and Some Applications of Phyllotaxis," Proceedings,
AAAS, II (1850), 438-444, and Benjamin Peirce, "Mathematical
Investigations of the Fractions Which Occur in Phyllotaxis," in ibid.,
444-447. Gray was never entirely convinced of the validity of this
ideal conception. He subsequently encouraged Chauncey Wright to
examine the problem of leaf arrangement, with the result that such
facts were shown to be understandable in terms of the principle of
natural selection.
but it is still incredible
that it should have been driven down so swiftly, except, perhaps that
it was undoubtedly heliocentric as well as a major departure from
the views perpetuated by organized religion. Thus it may have come too
late, a century after Linneaus'
classifications, a little less with respect to Cook's voyages, and half
a century or more
of continued activity that was simply too much for those who wished to
maintain the status quo. Not that this was the only field
affected where the Golden Section was concerned; it was also difficult
for the likes of Canon Mosely and later others around the turn of the
last century engaged in the analysis of spiral forms (especially
applied to
shells) as
outlined in the closing excerpt from
The
Matter of lost Light:
".. There is a great deal more, of
course, that could be said
concerning the details and the methodology applied to the fitting of
spirals forms to shells and many other natural applications provided in
Sir d'Arcy Wentworth
Thompson's voluminous On Growth
and Form.12 And
indeed in other
works that for a brief time seem to have flourished around the
beginning
of the last century. The above is included here because it epitomizes
the darker, stumbling side of human progress. And also the realization
that when Thomas Taylor (Introduction toLife
and
Theology of Orpheus) speaks of social decline, loss of knowledge in
ancient times and the efforts to preserve it by those who, "though
they lived in a base
age" nevertheless"happily
fathomed the depth of their great master's works, luminously and
copiously developed their
recondite meaning, and benevolently communicated it in their writings
for the general good," that sadly, such times are still upon
us. Thus, just as Sir Theodore Andrea Cook,13who in the Curves of
Life (1914) was unable to define the "well known logarithmic
spiral"
equated in 1881 with the chemical elements (see the previous section: Spira
Solaris and the Three-fold number),
neither Canon Mosely14 nor
Thompson were
able write openly about the
either the
Golden Ratio or the Pheidian planorbidae. Nor unto the present day, it
seems have others, for if not a forbidden subject per se, it
long
seems to have been a poor career choice, so to speak. Moreover, even
after
Louis Agassiz introduced Benjamin Peirce's phyllotaxic approach to
structure of the Solar System in his Essay
on Classification
(1857) the matter was swiftly dispatched and rarely referred to again.
A possibly momentous shift in awareness, shunted aside with greatest of
ease, as the editor of
Essay on Classification, (E. Lurie)15
explained in the short loaded
footnote discussed in the
previous section.
Nor it would seem, were the
works of Arthur
Harry Church16 (On the Relation of
Phyllotaxis to Mechanical Law,
1904) or Samuel Colman17-18 (Nature's
Harmonic Unity,
1911) allowed
to take root. Nor again were the lines of inquiry laid out in
Jay Hambidge's19 Dynamic Symmetry
(1920) permitted to have
much on effect on the status quo either, not to mention Sir
Theodore Andrea Cook's Curves of Life
(1914) and the general
the thrust of the many papers published during the previous century and
on into the present.20-26
Where does this
obfuscation and
stagnation leaves us now?
Wondering
perhaps where we might be today if the implications of the phyllotactic
side of the matter introduced in 1850 by Benjamin Pierce had at
least been allowed to filter into the mainstream of knowledge with
its wider, all-inclusive perspective concerning "life" as we currently
understand it. The realization, perhaps, that we may indeed belong to
something larger than ourselves, and that as an integral, living part
of the Solar System rather than an isolated destructive apex,
that we should conduct ourselves with more care and consideration
towards all forms of life...."
As indeed do many so-called
"primitive" cultures. At the end of a
documentary entitled The
Great Barrier Reef
(Science Museum of Minnesota, 2001) an Australian aboriginal commentary
states that:27
All
living things are one, like the blood
which unites one family. All
of life forms one great web; man did not weave the Web of Life, he is
only a part of it. He is only a strand within it. Whatever he does to
the Web of Life he does to himself. ( The
Great Barrier Reef,
dir. Dr. George Casey, Science museum of Minnesota, 2001)27
And
again, this time from the Northern Hemisphere:28
Native wisdom tends to assign
human
beings enormous responsibility for sustaining harmonious relations
within
the whole natural world rather than granting them unbridled license to
follow
personal or economic whim ... Native wisdom sees spirit, however one
defines
that term, as dispersed throughout the cosmos or embodied in an
inclusive,
cosmos- sanctifying divine being. Spirit is not concentrated in a
single,
monotheistic Supreme Being ... Native spiritual and ecological
knowledge
has intrinsic value and worth, regardless of its resonances with or
"confirmation"
by modern Western scientific values.. (Peter
Knudtson
and David Suzuki, Wisdom of the Elders, 1992)
But however
one looks at it, this matter seems to extend far, far back in time, and
I say this not at the beginning of my researches, nor near the middle,
but towards the very end.
Indeed,
it is
entirely possible that this complex subject lies
at the heart of many ancient writings, especially those of Plato,
Aristotle, and the Neoplatonists. It is also likely that it is inherent
in
the teachings of the Pythagoreans and in The
Chaldean Oracles
so venerated by Proclus. And not least of all, this
concept of a
natural living entity may also be obscurely addressed in alchemical
works received from the Arab World and thus preserved for
us throughout
the Darkness and the Middle
Ages. There
is
little doubt, for example, how much lost knowledge was regained by the
medieval French scholar Nicole Oresme (1323-1382 CE) from his analyses
and
exposition of the work of Averroes (Ibn Rushdie, ca. 1128 CE) even if
the latter work29
remains under-appreciated to the present day, and there are undoubtedly
other Arab sources likely to be informative in their original state. This is not to
suggest that the Western World has not already acknowledged the
preservation and furtherance of ancient
learning from this source, but there long seems to have been
a grudging (if not racist) element to it that continues to impede both
our
growth and our development.
Here
we surely need to come of age, and quickly.
Needless to say, there is much work to be done and many fences to
be mended, but with a
distinct
focus and increased cooperation between Western
and Arab scholars perhaps we
may yet still build a few solid bridges towards a
better and a more tolerant world.
NOTES AND REFERENCES
- Agassiz,
Louis. ESSAY
ON
CLASSIFICATION, Ed. E. Lurie, Belknap Press, Cambridge,
1962:131.
- Peirce, Benjamin. "Mathematical Investigations of
the Fractions Which Occur in Phyllotaxis," Proceedings, AAAS, II
1850:444-447.
- Agassiz,
Louis. ESSAY
ON
CLASSIFICATION, Ed. E. Lurie, Belknap Press, Cambridge,
1962:127.
- Called here IMO from Leverrier, M, "The
Intra-Mercurial Planet Question," Nature 14
(1876)
533. [Anon.]
- Agassiz,
Louis. ESSAY
ON
CLASSIFICATION, Ed. E.
Lurie, Belknap Press, Cambridge,
1962:127.
- Peirce, Benjamin. Address
of
Professor Benjamin Peirce, President of the American
Association for
the Year 1853, on retiring from the duties of President. AAAS, 1853:6–7.
Source (JPG): The
Cornell Library Historical Mathematics Monographs. [ html version: Peirce1853 ]
- Concise
Dictionary of
Scientific Bibliography,
Charles Scribner’s Sons, New York,
1981:540.
- Agassiz,
Louis. (Louis
Agassiz, ESSAY ON
CLASSIFICATION, Ed. E. Lurie, Belknap Press, Cambridge,
1962:127-128.
- Bretagnon,
P and Jean-Louis Simon, Planetary Programs
and
Tables
from
-4000 to +2800, Willman-Bell, Inc.
Richmond, 1986.
- Harris, John N. "Projectiles, Parabolas,
and Velocity Expansions of the Laws of Planetary Motion " JRASC,
Vol 83, No. 3, June 1989:207-218.
- Kappraff, Jay. CONNECTIONS
: The Geometric Bridge between Art and Science, McGraw-Hill, Inc.,
New
York, 1991:89-96.
- Thomson, Sir D'Arcy
Wentworth. On Growth and Form, Cambridge University Press,
Cambridge 1942; Dover Books, Minneola 1992.
- Cook, Sir Theodore
Andrea. The Curves of Life, Dover,
New York 1978;
republication of the London (1914) edition.
- Mosely, Rev. H. "On
the geometrical forms of turbinated and discoid shells," Phil. trans.
Pt. 1. 1838:351-370.
- Lurie, E. (Ed.) Essay
On Classification, Belknap Press, Cambridge
1962:128.
- Church, Arthur Harry. On
The Relation of Phyllotaxis to Mechanical Law, Williams and
Norgate, London 1904; also: http://www.sacredscience.com
(cat #154).
- Colman, Samuel. Nature's
Harmonic
Unity, Benjamin Blom, New York 1971. Also:
- ___________ Harmonic Proportion and Form in Nature,
Art and Architecture, Dover, Mineola, 2003.
- Hambidge, Jay. Dynamic
Symmetry,Yale University Press, New Haven 1920:16-18.
- It is
necessary to acknowledge the many positive strides made during the last
25 years, especially Roger V. Jean's Phyllotaxis:
A systemic study in plant
morphogenesis (1994) which by virtue of its scale and scope
invokes
the same admiration reserved for Sir D'Arcy Wentworth
Thompson's On
Growth and Form
(1917) and similar major works.
- Jean, Roger V. Phyllotaxis: A systemic study in plant
morphogenesis, Cambridge University Press, Cambridge 1994.
- __________ Mathematical Approach to Pattern and Form
in Plant Growth. Wiley, New York (1984).
- Kappraff, Jay.
"The Spiral in Nature, Myth, and Mathematics" in Spiral Symmetry, Eds.
István Hargittai and Clifford A. Pickover,
World Scientific, Singapore, 1992.
- __________"The
relations between mathematics and mysticism of the golden mean through
history." In Fivefold Symmetry,
ed. I. Hargittai. World Scientific, Singapore, 1992: 33-65.
- Stewart, Ian. What Shape is a Snowflake,
Weidenfeld & Nicholson, London, 2001.
- Ghyka, Matila C. The Geometry of Art and Life,
Dover Publications, New York, 1977.
- Casey, George. The
Great Barrier Reef,
dir. Dr. George Casey, Science museum of Minnesota, 2001.
- Knudson,
Peter and David Suzuki, Wisdom of the
Elders, University of British Columbia Press, Vancouver, 1992.
- Menut, Albert D. and
Alexander J. Denomy, Le Livre
du ciel et du monde, University of Wisconsin Press, Madison 1968.
Copyright ©
January 21, 2007. Last
updated: March 13, 2007. John N. Harris,
M.A.(CMNS).
RETURN TO SPIRA SOLARIS.CA
ASTRONOMICAL
FRAMEWORK
QUANTIFICATION
AND QUALIFICATION I
THE GOLDEN SECTION AND THE STRUCTURE OF THE SOLAR SYSTEM
Spira
Solaris: Form and Phyllotaxis
I Bode's Flaw
http://www.spirasolaris.ca/sbb4a.html
Bode's "Law" - more correctly the Titius-Bode relationship - was an
ad hoc scheme for approximating mean planetary distances that was
originated by Johann Titius in 1866 and popularized by Johann Bode in
1871. The " law " later failed in the cases of the outermost
planets Neptune and Pluto, but it was flawed from the outset with
respect to distances of both MERCURY and EARTH, as Titius was perhaps
aware.
II The Alternative
http://www.spirasolaris.ca/sbb4b_07.html
Describes an alternative approach to the structure of the Solar System
that employs logarithmic data, orbital velocity, synodic motion, and
mean
planetary periods in contrast to ad hoc methodology and the use
of mean heliocentric distances alone.
III The Exponential
Order
http://www.spirasolaris.ca/sbb4c_07.html
The constant of linearity for the resulting planetary framework
is the
ubiquitous constant Phi known since antiquity. Major departures
from the theoretical norm are the ASTEROID BELT, NEPTUNE, and EARTH in
a resonant synodic position between VENUS and MARS.
Fibonacci/Golden Section Resonances in the Solar System.
Spira Solaris and
the plan-view of the Milky Way.
http://www.spirasolaris.ca/dfmilkyway.html
Last updated: March 13, 2007
QUANTIFICATION
AND QUALIFICATION II
GOLDEN SECTION SPIRALS IN NATURE, TIME AND PLACE
THE
THREE-FOLD NUMBER
IVd2b Spira Solaris and
the 3-Fold Number
http://www.spirasolaris.ca/sbb4d2b.html
The Spiral of Pheidias; Pheidian/Golden Spirals
Defined.
Pheidian Spirals and the Chemical Elements.
Notes on the
Logarithmic Spiral (Jay Hambidge; R. C. Archibald)
http://www.spirasolaris.ca/hambidge1a.html
R.C. Archibald's Golden
Bibliography.
The Whirlpool Galaxy (M51) (BW: 100kb)
http://www.spirasolaris.ca/m51abw.html
The Whirlpool Galaxy (M51)(Colour: 200kb)
http://www.spirasolaris.ca/m51.html
The Phyllotaxic
approach to the structure of the Solar System of Benjamin Pierce
(1750)
THE PHEIDIAN PLANORBIDAE
IVd2c Spira Solaris and
the Pheidian Planorbidae.
http://www.spirasolaris.ca/sbb4d2c.html
Applied to Nautiloid spirals, Ammonites, Snails and Seashells.
The
Phedian Planorbidae in Astronomical context; Orbital velocity, Mass and
Angular Momentum.
Ammonites and Seashells (Beginning excerpt).
http://www.spirasolaris.ca/sbb4d2cs.html
Whirling Rectangles
and The Golden Section (Animation I)
http://www.spirasolaris.ca/animation1.html
Whirling Rectangles
and The Golden Section (Animation II)
http://www.spirasolaris.ca/animation2.html
Appendix: The Matter of Lost
Light.
The understanding of Canon Mosely and Sir D'Arcy Wentworth Thompson.
RELATED PAPERS AND
TOOLS
Velocity
Expansions of
the
Laws of Planetary Motion.
http://www.spirasolaris.ca/sbb7a.html
Abstract
Kepler's Third Law of planetary motion: T2 = R3
(T = period in years, R = mean distance in astronomical
units) may be extended to include the inverse of the mean speed Vi (in
units
of the inverse of the Earth's mean orbital speed) such that: R = Vi2
and T2 = R3 = Vi 6. The first relation
- found in Galileo's last major work, the Dialogues Concerning Two
New
Sciences (1638) - may also be restated and expanded to include
relative
speed Vr (in units of Earth's mean orbital speed k) and absolute speed
Va
= kVr. This paper explains the context of Galileo's velocity expansions
of
the laws of planetary motion and applies these relationships to the
parameters
of the Solar System. A related "percussive origins" theory of planetary
formation is also discussed.
Note: This paper (which deals with
the resurrection of the Fourth Law of Planetary Motion, i.e.,
the velocity component) was written north of the 70th parallel during
the Summer of 1988.
It was subsequently published in the Journal of the Royal
Astronomical Society of Canada (JRASC) the following year. It is
reproduced here with the permission of the Editor of the Journal.
Times
Series Analysis.
http://www.spirasolaris.ca/time1.html
The advent of modern computers permits the investigation of planetary
motion
on an unprecedented scale. It is now feasible to treat single events
sequentially and apply detailed time-series analyses to the results.
Time Series Graphics
http://www.spirasolaris.ca/times2.html
Examples of chaotic
and resonant planetary relationships in the Solar System and
a possible link with Solar Activity.
Copyright ©
1997. John N. Harris, M.A.
(CMNS). This section last updated on July 28, 2004.
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to spirasolaris.ca