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SPIRA SOLARIS: Form and Phyllotaxis Assigned to the main heading SPIRA SOLARIS this January 2007 essay 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 (1809-1880). His investigations and conclusions resurfaced in September 2006 after the elevation of the asteroid Ceres and simultaneous 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 stipulated on theoretical grounds over 150 years ago. First published in 1850 and later recorded by his friend Louis Agassiz (1807-1873) in the latter's famous Essay on Classification (1857:131) Peirce theorised that "there can be no planet exterior to Neptune, but there may be one interior to Mercury." Revisiting my own attempts to come to terms with the fundamental structure of the Solar System in the first three sections of Spira Solaris Archytas-Mirabilis similarities between the two approaches now suggested that while I had unknowingly followed Peirce by omitting Pluto and adding an inter-Mercurial object, I had missed the point completely when it came to the significance of Neptune. Not so Benjamin Peirce. Nevertheless, although Agassiz provided additional details in his Essay on Classification concerning the latter's Solar System research including "the ratios of the laws of phyllotaxis" in this same astronomical context, Peirce's major conclusion was cast aside despite its potential significance. Although not stated in these precise terms, it was nothing less than the perception that

The Solar System is Pheidian in Form¹ and Phyllotactic² in Nature

Just how right (or wrong) was Benjamin Peirce concerning the phyllotactic aspect of the Solar System? He was certainly one of the leading scholars of his day, and there is little doubt that he was also an influential scientist in his own right, as the following excerpts from his Scientific Biblography attest:

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 Mécanique céleste (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 Linear 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, 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 indeed correct, all ramifications and consequences notwithstanding. Having made this statement, however, there is something else that needs to be addressed.

Although a decade has now elapsed since these Web Pages first appeared on the Internet, the final conclusion to Spira Solaris (though still incomplete at the time) was placed on indefinite hold after the tragic events of September 11, 2001 and the downward spiral into barbarism that followed. Rather than enlightenment and hope in this new millenium we have witnessed instead pre-emptive invasions, international kidnappings, torture, incarceration without trial, military tribunals, illegal evesdropping, increasing chaos, death and devastation in the Middle East, none of which need further elaboration here. No doubt we are all wiser now after 911, but few could have imagined that so many hard-won freedoms and rights could have been so swiftly and easily eroded because of it. A senseless, needless war based on false premises and shallow rhetoric continues to take its toll on the already weakened country of Iraq, and as many commentators note, the World has not become a safer place because of the pre-emptive policies of the current administration in the United States. In fact the opposite has come to pass, and we now appear to be moving ever-closer towards a greater escalation of the so-called "War on Terror" to the extent that it may engulf the entire Middle East.

This said, following the U.S. Mid-Term Elections in November 2006 matters seem to have taken an encouraging turn for the better with the more moderate Democratic Party now controlling both the U.S. House of Representatives and the U.S. Senate. Hopefully this may ultimately result in saner and more humanistic agendas than those currently in force in the great Republic. In any event, since these changes to the U.S. body politic provide a window of opportunity it seems worthwhile to attempt to close out this series of essays with one final, difficult conclusion in what may hopefully be a more receptive intellectual climate.

In short, the long-delayed legacy of Benjamin Peirce and the policies laid out in his closing remarks to the members of the American Association for the Advancement of Science concerning the founding of the Smithsonian Institution, and not least of all, the honest, unhindered pursuit of knowledge:

"Let us profit by the example of Smithson, and, instructed by the wisdom of this high-minded son of England, learn to confide in our own rulers. Let us be aroused to an earnest and harmonious effort to accomplish the plan proposed by our President at Albany, for the building up of an "institution for science, supplementary to existing institutions, to guide public action in reference to scientific matters." With the details of the plan and the arguments in its favor you are familiar. You know how useful it would be as a protection from the wasteful expenditure upon abortive attempts to reverse the laws of nature. You know how much it is required to sustain the purity and independence of science, even within its own proper domain. You know that in no age or country was there ever a more urgent call for a scientific society, in which scientific influence should predominate, where it should not be smothered by excess of patronage, and whence it should not be liable to banishment through any spirit or form of ostracism. If American genius is not fettered by the chains of necessity, and helplessly exposed to the assaults of envious mediocrity, but is generously nourished in the bosom of liberty, it will joyfully expand its free wings, and soar with the eagle to the conquest of the skies."

Concluding remarks from "The Address of Professor Benjamin Peirce, President of the American Association for the Year 1853, on retiring from the duties of President"

"Printed by Order of the Association" (1853:16-17)

SPIRA SOLARIS: Form and Phyllotaxis

Form

1. Form: Pheidian growth per revolution, etc.,

e.g., Ammonites and Shells

phyllotaxis
phyllotaxy
noun
2. phyllotaxes, phyllotaxies

The arrangement of leaves on a plant stem.*

Derivative: phyllotactic adj . (Source: ALLWords.com)

* Phyllotactic fractions were applied by Benjamin Peirce to the mean periods of revolution alone; the present dynamic treatment is more complex.

Associated graphics: Solar System Phyllotactic Resonant Triples, Neptune to Mercury

The 1853 Address of Benjamin Peirce

THE EXPONENTIAL PLANETARY FRAMEWORK: METHODOLOGY AND DATA

Bode's Flaw 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 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. [ Previous version: http://www.spirasolaris.ca/sbb4b.html ]

III The Exponential Order 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. [ Previous version: http://www.spirasolaris.ca/sbb4c.html ]

IVd2 Spira Solaris and The Middle Ages Ostensively the translation of Aristotle's De Caelo from medieval latin to French, Nicole Oresme's Le Livre du ciel et du monde [ ca.1375 CE] was more than a translation and a commentary. The numerous references in this work to the insights of the Arab scholar Ibn Rushd [Averroes,1128-1198 CE] lead back to Plato's Republic, Archimedes, Pythagoras, and the Golden Section in early Alchemical contexts.

IVc Spira Solaris, the Fourth Planet and Fifth Element PROCLUS [ Commentary on the Timaeus of Plato ]: "For through analogy the universe is completely rendered one, this having the power of making things that are divided to be one, of congregating things that are multiplied, and connecting things that are dissipated. Hence theologists surveying the cause of these things in the Gods, enclose Venus with Mars, and surround them with Vulcanian bonds; the difference which is in the world being connected through harmony and friendship. All this complication and connection likewise has Vulcan for its cause, who through demiurgic bonds connects sameness with difference, harmony with discord, and communion with contrariety."

V Spira Solaris.The Chaldean Oracles. Proclus, and Johannes Kepler Examines the role played by the Chaldean Oracles in influencing the Neoplatonist Proclus [410 - 485 CE] and the connection between the latter, The Harmonies of the World, and the Harmonic Law "conceived on the eighth day of the third month in the year 1618" by Johannes Kepler [1571-1630 CE].

VI Spira Solaris and the Universal Ouroboros Explores the historical antecedents and the relationship between the Equiangular Period Spiral, the Ouroboros, Alchemy, and the Sun.

VII Spira Solaris and the Three Parts of the Wisdom of the World The spiral configuration is found on rock art in regions as distant as Nicaragua, Italy, Indonesia, and the United States. It is also found on the Plains of Nazca in Peru, on Malta, in France, in Britain, and especially in Ireland, where a complex triple spiral is illuminated on the winter solstice inside the Neolithic Site at Newgrange. What is the significance of this complex motif, what message was it intended to convey, and where does the inquiry lead?

For the Record "A spendid cast of the deeply carved face of Wi te Manewha was taken by Lindauer and Sir Walter Buller. His moko was perfect, the lines cut exceptionally deeply, even to the eyelids. It was said that the tattooing was done twice to make deeper markings. The cast was made during the lifetime of the chief. It is of considerable ethnological value because it is the only known instance of an old-time Maori permitting such a thing, since the head was considered sacred."

THE COAST SALISH / SQUAMISH WELCOME FIGURE

Squamish Welcome Figure Should you wander down an increasingly urban pathway along the western bank of British Columbia’s Capilano River – the natural boundary separating North and West Vancouver – you may come across a small inscription on a rock by the wayside. Carved with care, it is clearly modern, inscription and text alike. It says sadly and simply:

Strength in your weeping,
Tears that come seeping,
Down the old canyons,
Back to the sea.
From The River by Jean Gowland

Passing under a bridge the path continues along the pavement of a road that leads around a large Mall, where (now separated from the river by a wire and split-board fence) the path curves south again into a small wooded area. Finally, one eventually reaches the rivermouth and following the shoreline to the west, a small bay bounded by a pair of rocky outcroppings. Augmented and strengthened, these too are modern, and often visited for they extend outwards from the beach area of West Vancouver’s Ambleside Park. At the end of the largest outcropping, however, there is another recent inscription, this time on a bronze bi-lingual plaque. But here the text is accompanied by something else, namely a large wooden figure looking out over the Inlet towards the southwest with both arms outstretched. This is the Coast Salish “Welcome Figure” erected by the Squamish First Nation to commemorate the first “Gathering of Ocean Canoes” in the summer of 2001. For this is their land; this is their will, and this is their generous welcome...

GEORGE WOODCOCK : PEOPLES OF THE COAST

A Witnessing "Yet it was in one way the most dramatic dance of the whole night, for the drummers came down out of the bleachers to join those on the floor and make an avenue of sound through which the old dancer progressed, with hundreds of voices shouting out her song, and her attendants scattering handfuls of coins among the drummers and the singers. It was obviously a farewell, for we felt no doubt that this was the old woman's last dance, and that she and everyone else knew it. But it was also the kind of assertion of continuity, for here was a person who had been a child in the last flourishing of the old native culture, and by supporting her in her dance the rest of the people were not only proclaiming their continuity with the past but also celebrating the revival of the old ways." ( HTML; also in PDF )

Pythagorean Fragments Selections from the Doxographers, Fragments of Philolaus, Fragments of Archytas, Metaphysical and Political Fragments.

Man and the Sun " In following the solar cycle of this book, I have honoured those who worshipped the Sun God in his many forms. Yet I have also honoured those scientists whose probing minds have dispelled the simple divinity of the star. The members of the Holy Office were right to be fearful of the ideas of Copernicus to see that they would lead to the destruction of many of the old religious forms. They were wrong as well as ridiculous trying to turn back the tide of science, of man's efforts to comprehend the physical universe, for that pursuit is a part of what is divine in humanity. We have to honour both the King of Heaven and Prometheus. The present peril and despair of humanity show that we cannot live without religious meaning although we may do without religious institutions. (The time may come when even those few who still follow them turn against priests who in gem-encrusted copes and mitres, serve Him who taught poverty and humility, who betray Him who taught love of one enemy by raising no murmur against a holocaust of hate.) If we cannot find god in the world, we lose Him in ourselves and become contemptible in our own eyes. We become mere statistics. For this is the greatest evil coming from the unbalanced Apollonian mind. Science has won power over the universe of matter by breaking down and down, by numbering and measuring. So at last everything that cannot be broken down, numbered and measured must be deemed not to exist. Science is uniting man with the sun in a totality of energy and matter. That is communion at the lowest level of being. But we have always been right to seek it also at the highest." [Quotation from the concluding chapter of Man and the Sun by Jacquetta Hawkes, 1962:239-241 ]

Scholium: Newton's closing paragraph " And now we might add something concerning a certain most subtle spirit which pervades and lies hid in all gross bodies; by the force and action of which spirit the particles of bodies attract one another at near distances, and cohere, if contiguous; and electric bodies operate to greater distances, as well repelling as attracting the neighboring corpuscles; and light is emitted, reflected, refracted, inflected, and heats bodies; and all sensation is excited, and the members of animal bodies move at the command of the will, namely, by the vibrations of this spirit, mutually propagated along the solid filaments of the nerves, from the outward organs of sense to the brain, and from the brain into the muscles. But these are things that cannot be explained in few words, nor are we furnished with that sufficiency of experiments which is required to an accurate determination and demonstration of the laws by which this electric and elastic spirit operates."

Why did Sir Isaac Newton end his opus magnum in this way? A simple statement of faith? Or was it something more encompassing?

TIME SERIES ANALYSIS

TIME SERIES GRAPHICS

ALCHEMICAL GOLD