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<meta name="DATE" content="11/11/1999">
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<meta name="Author" content="John N.Harris">
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<meta name="Description" content="SpiraSolaris: Form and Phyllotaxis, The Golden Section in Nature and the Structure of the Solar System">
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<meta name="KeyWords" content="Archytas, Equiangular, Spiral, Galileo, Phi, Benjamin Peirce, Golden Section, Kepler, Newton, Ouroborus, Plato, phyllotaxis, Pythagoras">
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<title>Spirasolaris: Form and Phyllotaxis</title>
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<br><big style="color: #666600; font-family: helvetica,arial,sans-serif"><big><big><big><span style="font-weight: bold">FORM AND PHYLLOTAXIS</span></big></big></big></big>
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<span style="color: #000000"></span><strong style="font-family: times new roman,times,serif; color: #000000"><font size="+2"> </font></strong><span style="color: #000000"></span><span style="color: #000000"></span><small style="color: #000000"><span style="font-family: times new roman,times,serif"></span></small><span style="color: #000000"></span><big style="font-family: times new roman,times,serif; color: #000000"><small><span style="font-style: italic">It is well known that the arrangement of the leaves in plants</span><small style="font-style: italic"><strong><sup></sup></strong></small><span style="font-style: italic">may be expressed by very simple series of fractions, all of which are gradual approximations to, or the natural means between</span><strong style="font-style: italic"> 1/2 </strong><span style="font-style: italic">or </span><strong style="font-style: italic">1/3</strong><span style="font-style: italic">, 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:</span><strong style="font-style: italic"> </strong><strong style="font-style: italic">1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34, 21/55</strong><span style="font-style: italic">, etc. Now upon comparing this arrangement of the leaves in plants with the revolutions of the members of our solar system, <span style="font-weight: bold">Peirce has discovered the most perfect identity between the fundamental laws which regulate both</span>.<span style="font-weight: bold"> </span></span><small style="font-weight: bold"><small><sup>1</sup></small></small><span style="font-style: italic"><br> <br></span></small></big><big><small><font style="color: #000000" face="Times New Roman,Times">(Louis Agassiz, <em style="font-weight: bold">ESSAY ON CLASSIFICATION</em>, Ed. E. Lurie, Belknap Press, Cambridge, 1962:127; emphases supplied)</font></small></big><big><small><font style="color: #000000" face="Times New Roman,Times"> </font><span style="color: #000000"> </span></small></big>
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</blockquote><font style="color: #000000" face="Arial,Helvetica"><span style="font-weight: bold"></span></font>
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<div style="text-align: justify"> <span style="color: #000000"></span><font style="color: #000000" face="Arial,Helvetica"><big><big><span style="font-weight: bold">I. PRELIMINARY REMARKS<br> </span><span style="font-weight: bold">II. INTRODUCTION<br> </span></big></big><span style="font-weight: bold"><big><big>III. BENJAMIN PEIRCE, PHYLLOTAXIS, AND THE SOLAR SYSTEM<br> IV. THE PHYLLOTACTIC SOLAR SYSTEM<br> </big></big></span></font><big><span style="font-weight: bold"> <span style="font-family: helvetica,arial,sans-serif">IV.1. PHYLLOTACTIC DIVISORS AND SYNODIC PERIODS</span><br style="font-family: helvetica,arial,sans-serif"> </span></big><big style="font-family: helvetica,arial,sans-serif"><span style="font-weight: bold"> IV.2. THE DUPLICATED DIVISORS<br> </span></big><span style="font-weight: bold; font-family: helvetica,arial,sans-serif"></span>
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<big><span style="font-weight: bold"> IV.3. THE EXTENDED PHI-SERIES; INNER AND OUTER LIMITS</span></big>
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</div><big style="font-family: helvetica,arial,sans-serif"><span style="font-weight: bold"> IV.4. SOLAR SYSTEM RESONANT PHYLLOTACTIC TRIPLES<br> </span><big><span style="font-weight: bold">V. PHYLLOTAXIS AND THE PHI-SERIES</span></big></big>
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<br> <font style="color: #000000" face="Arial,Helvetica"><span style="font-weight: bold"><big><big>IV. IMPLICATIONS AND RAMIFICATIONS</big></big><br> <br></span></font>
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<hr style="width: 100%; height: 2px"><font style="color: #000000" face="Arial,Helvetica"><span style="font-weight: bold"><br> </span></font><font style="color: #000000" face="Arial,Helvetica"><span style="font-weight: bold"><big><big>I. PRELIMINARY REMARKS</big></big><br> </span>Initially this matter is simple enough:<br> <br> Either the Solar System is phyllotactic in nature, as Benjamin Peirce intimated over one hundred and fifty years ago, or it is not.<br> <br> 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 <span style="font-style: italic">per se,</span> nor is it concerned with the religious elements evident in Louis Agassiz' <span style="font-style: italic; font-weight: bold">Essay on Classification</span>, or similar concerns (if that is what they truly are) expressed in Benjamin Peirce's retirement speech to the membership of the <span style="font-style: italic; font-weight: bold">AAAS</span> in 1853.<br> <br> 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.<br> <br></font><font style="color: #000000" face="Arial,Helvetica"><span style="font-weight: bold"></span></font>
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<hr style="width: 100%; height: 2px"><font style="color: #000000" face="Arial,Helvetica"><span style="font-weight: bold"><br> </span><big><big><span style="font-weight: bold">II. INTRODUCTION</span></big></big></font>
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<br> <font style="color: #000000" face="Arial,Helvetica"><span style="font-weight: bold"></span></font><font style="color: #000000" face="Arial,Helvetica">Assigned to the title heading</font><font style="color: #000000" face="Arial,Helvetica"><span style="font-weight: bold"> <span style="font-style: italic">SPIRA SOLARIS</span></span> t</font><font style="color: #000000" face="Arial,Helvetica">his January 2007 conclusion </font><font style="color: #000000" face="Arial,Helvetica">owes its origins to a number of events and issues, but t</font><font style="color: #000000" face="Arial,Helvetica">he subtitle itself <span style="font-style: italic">(</span><span style="font-style: italic; font-weight: bold"> Form and Phyllotaxis</span><span style="font-style: italic"> ) </span>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 t</font><font style="color: #000000" face="Arial,Helvetica">he official elevation of the asteroid Ceres and demotion of</font><font style="color: #000000" face="Arial,Helvetica"> the planet Pluto to the status of "Dwarf Planets."</font><font style="color: #000000" face="Arial,Helvetica"> Essentially,</font><font style="color: #000000" face="Arial,Helvetica">with Pluto thus demoted Neptune once more becomes the outermost planet, as Peirce had stipulated <span style="font-weight: bold; font-style: italic">on theoretical grounds</span> over 150 years ago. </font><font style="color: #000000" face="Arial,Helvetica">First published in 1850<small><sup>2</sup></small> and later recorded by his friend Louis Agassiz (1807-1873) in the latter's famous</font><font style="color: #000000" face="Arial,Helvetica"><span style="font-style: italic; font-weight: bold">Essay on Classification</span></font><font style="color: #cc9933" face="Arial,Helvetica"><span style="color: #000000"> (1857:131) Peirce had declared that <span style="font-family: times new roman,times,serif">"</span></span><span style="font-style: italic; color: #000000; font-family: times new roman,times,serif">there can be no planet exterior to Neptune, but there may be one interior to Mercury</span><span style="color: #000000"><span style="font-family: times new roman,times,serif">."</span><small> <sup>3</sup></small></span><span style="font-family: helvetica,arial,sans-serif"> </span></font><span style="color: #000000; font-family: helvetica,arial,sans-serif">revisiting my own attempts to come to terms with the structure of the Solar System in the first three sections of <span style="font-style: italic; font-weight: bold">Spira S</span></span><span style="font-style: italic; color: #000000; font-weight: bold; font-family: helvetica,arial,sans-serif">olaris Archytas-Mirabilis</span><span style="color: #000000"><span style="font-family: helvetica,arial,sans-serif"> 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,</span><small style="font-family: helvetica,arial,sans-serif"><sup>4</sup></small><span style="font-family: helvetica,arial,sans-serif"> I had nevertheless missed the point entirely when it came to the significance of Neptune. Not so Benjamin Peirce. Nevertheless,</span> although </span><font style="color: #ffffff" face="Arial,Helvetica"><span style="color: #000000">Agassiz provided additional details in his </span><span style="font-weight: bold; font-style: italic; color: #000000">Essay on Classification</span><span style="color: #000000">concerning the latter's Solar System research including </span><span style="font-weight: bold; color: #000000">"</span><span style="font-style: italic; font-weight: bold; color: #000000">the ratios of the laws of phyllotaxis</span><span style="font-weight: bold; color: #000000">"</span><span style="color: #000000"> <small><sup>5</sup></small> in this same astronomical context, Peirce's major conclusion was cast aside despite its implica
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<font style="color: #ffffff" face="Arial,Helvetica"><span style="color: #000000"> </span></font><font style="color: #000000" face="Arial,Helvetica"><span style="font-style: italic; font-weight: bold"><big>The Solar System is Pheidian in Form and Phyllotactic in Nature.</big></span></font>
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<br> <font style="color: #000000" face="Arial,Helvetica"><span style="font-style: italic; font-weight: bold"></span></font>
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</div><font style="color: #000000" face="Arial,Helvetica"><span style="font-style: italic; font-weight: bold"><br> </span></font><font style="color: #ffffff" face="Arial,Helvetica"><span style="color: #000000"></span></font>
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<span style="color: #000000">Apparently unable to promulgate the essence of the matter officially, Professor Peirce nevertheless labored to pass it on in a speech to the </span><span style="font-style: italic; font-weight: bold; color: #000000">American Association for the Advancement of Science</span><span style="color: #000000"> 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<span style="font-style: italic"> </span></span><span style="color: #000000; font-style: italic">f</span><span style="color: #000000; font-style: italic">acts</span><span style="color: #000000">:<small><sup> 6</sup></small></span>
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<p style="text-align: justify; margin-left: 0.5in; margin-right: 0.5in"><small><span style="font-family: 'Arial',sans-serif"></span></small><span style="color: #000000; font-family: times new roman,times,serif">"</span><span style="font-style: italic; color: #000000; font-family: times new roman,times,serif">Modern science has realized some of the most fanciful of the<span style="font-weight: bold"> Pythagorean</span> and <span style="font-weight: bold">Platonic doctrines</span>, and thereby justified the divinity of their spiritual instincts. <span style="font-weight: bold">Is it not significant</span> of the nature of the creative intellect, the simplicity of the great laws of force? <span style="font-weight: bold">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? </span></span><span style="font-weight: bold; color: #000000; font-family: times new roman,times,serif"></span><big style="color: #000000"><span style="font-family: times new roman,times,serif"> </span></big><span style="font-family: times new roman,times,serif; color: #000000">(</span><span style="font-family: times new roman,times,serif; color: #000000">Address of Professor Benjamin Peirce, President of the Association of the American Association for the Year 1853, on retiring from the duties of President. 1853:67;<span style="font-weight: bold">"Printed by Order of the Association."</span></span><span style="font-family: times new roman,times,serif"><span style="color: #000000"> (</span></span><a href="http://historical.library.cornell.edu/math/index.html"><span style="font-family: times new roman,times,serif">The Cornell Library Historical Mathematics Monographs</span> </a><span style="color: #000000"><span style="font-family: times new roman,times,serif">(JPG) emphases supplied. Html version: <a href="../../Peirce1853.html">Peirce 1853</a> ).</span></span></p>
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<span style="color: #000000">Professor Peirce's retirement speech contained further cryptic comments, <span style="font-style: italic">e.g.</span>, after mentioning: "</span><span style="font-style: italic; color: #000000">the same strong, embracing, golden chain of inductive argument</span><span style="color: #000000">. (1853:7) mathematician Peirce revisited "consummation" by noting further (1853:8) that: "</span><span style="font-style: italic; color: #000000">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.</span><span style="color: #000000"> 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. <br> <br></span>
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<span style="color: #000000">Just how right (or wrong)</span><span style="font-style: italic; color: #000000">was</span><span style="color: #000000"> 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 </span><span style="font-style: italic; font-weight: bold; color: #000000">Scientific Bibliography</span><span style="color: #000000"> attest:<small><sup>7</sup></small></span><span style="color: #000000"></span><small><span style="font-family: 'Arial',sans-serif"></span></small>
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<p style="margin-left: 0.5in; margin-right: 0.5in; text-align: justify"><span style="font-weight: bold; color: #000000; font-family: times new roman,times,serif">PEIRCE, BENJAMIN</span><span style="color: #000000; font-family: times new roman,times,serif"> (b. Salem, Mass., 1809; d. Cambridge, Mass., 1880), mathematics, astronomy. </span><br style="color: #000000"> <span style="font-family: times new roman,times,serif">Graduated from Harvard (1829; M.A., 1833), where he was a tutor ( 183133) and professor (183280). In mathematics, he amended N. Bowditch's translation of Laplace's </span> (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 <span style="font-style: italic; font-weight: bold; font-family: times new roman,times,serif">Mécanique célesteLinear Associative Algebra</span><span style="font-family: times new roman,times,serif"> ... and set forth, in </span><span style="font-weight: bold; font-style: italic; font-family: times new roman,times,serif">A System of Analytic Mechanics </span><span style="font-family: times new roman,times,serif">(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. (</span><span style="font-style: italic; font-weight: bold; font-family: times new roman,times,serif">Concise Dictionary of Scientific Bibliography</span><span style="font-family: times new roman,times,serif">, Charles Scribners Sons, New York, 1981:540)</span><br style="color: #000000"> </p>
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<span style="color: #000000"></span>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.
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<br><font style="color: #000000" face="Helvetica, Arial, sans-serif">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 <em>status quo</em>. 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.<br> That he did not examine the matter from this particular viewpoint in greater detail is the reason why this present essay exists.</font>
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<br><big><big><font style="color: #000000" face="Arial,Helvetica"><span style="font-weight: bold">III. BENJAMIN PEIRCE, PHYLLOTAXIS, AND THE SOLAR SYSTEM</span></font></big></big>
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<strong style="color: #000000; font-family: helvetica,arial,sans-serif"><font size="+2">ESSAY ON CLASSIFICATION <small><sup>8</sup></small></font></strong>
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<br style="color: #000000; font-family: helvetica,arial,sans-serif"> <span style="color: #000000; font-family: helvetica,arial,sans-serif"> </span><strong style="font-family: times new roman,times,serif; color: #000000"><font size="+2"><span style="font-family: helvetica,arial,sans-serif">Louis Agassiz 1857</span> </font></strong>
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<br style="color: #000000"> <span style="color: #000000"> </span><strong style="font-family: times new roman,times,serif; color: #000000"><font size="+2"> </font></strong><span style="font-family: times new roman,times,serif; color: #000000"></span>
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<br style="font-family: times new roman,times,serif; color: #000000"> <span style="color: #000000"></span><strong style="color: #000000; font-family: helvetica,arial,sans-serif"><font style="font-family: times new roman,times,serif"><big>FUNDAMENTAL RELATIONS OF ANIMALS</big></font></strong>
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<br style="color: #000000; font-family: helvetica,arial,sans-serif"> <span style="color: #000000; font-family: helvetica,arial,sans-serif"></span><strong style="color: #000000; font-family: helvetica,arial,sans-serif"><font style="font-family: times new roman,times,serif">SECTION XXXI</font></strong>
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<br style="color: #000000; font-family: helvetica,arial,sans-serif"> <span style="color: #000000; font-family: helvetica,arial,sans-serif"></span><strong style="color: #000000; font-family: helvetica,arial,sans-serif"><font style="font-family: times new roman,times,serif">COMBINATIONS IN TIME AND SPACE OF VARIOUS KINDS OF RELATIONS AMONG ANIMALS</font></strong>
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<br style="color: #000000"> <span style="color: #000000"></span><small style="color: #000000"> <span style="font-family: times new roman,times,serif"> </span></small><big style="font-family: times new roman,times,serif; color: #000000"><small> 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.</small></big>
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<br style="color: #000000"> <span style="color: #000000"></span><big style="font-family: times new roman,times,serif; color: #000000"><small> 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 <em>à fortiori </em>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?</small></big>
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<br style="color: #000000"> <span style="color: #000000"></span><big style="font-family: times new roman,times,serif; color: #000000"><small> 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 plants<small><strong><sup><small>148</small> </sup></strong></small><span style="font-weight: bold">m</span>ay be expressed by very simple series of fractions, all of which are gradual approximations to, or the natural means between<strong> 1/2 </strong>or <strong>1/3</strong>, 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:<strong> </strong><strong>1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34, 21/55</strong>, 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</small>.<small><small><sup>149</sup></small></small></big>><>
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<br><font style="color: #000000" face="Times New Roman, Times, serif"> </font><font style="color: #000000" face="Helvetica, Arial, sans-serif"><strong>Table I (Peirce-Agassiz 1857; column titles added)<br> <br><br> </strong></font>
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<font face="Times New Roman, Times, serif">In this series the Earth forms a break; but this apparent irregularity admits of an easy explanation. The fractions:</font><strong><tt> 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,</tt></strong><font face="Times New Roman, Times, serif"> 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, </font><tt><strong>1/2, 2/3, 3/5, 8/13, 13/21, 21/34</strong>, etc.</tt>
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<br> <tt> </tt><font face="Times New Roman, Times, serif"> Let us therefore repeat our diagram in another form, the third column giving the theoretical time of revolution.</font><small> </small>
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<p><img alt="Table II (Peirce-Agassiz 1857, column titles added)" src="Agt3c.gif" style="width: 431px; height: 388px"><br> </p>
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<p><font style="color: #000000" face="Helvetica, Arial, sans-serif"><strong>Table II (Peirce-Agassiz 1857; column titles added) <br> <br></strong></font></p>
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<p style="text-align: justify"><font face="Times New Roman, Times, serif"><big><span style="color: #000000; font-family: times new roman,times,serif"><small>It appears from this table that two intervals usually elapse between two successive planets, so that the normal order of actual fractions</small></span><small><strong style="color: #000000; font-family: times new roman,times,serif">, 1/2, 1/3, 2/5, 3/8, 5/13</strong><span style="color: #000000; font-family: times new roman,times,serif">, 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. </span><br style="color: #000000; font-family: times new roman,times,serif"> <span style="color: #000000; font-family: times new roman,times,serif"> 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.</span></small> </big><small><sup style="color: #000000; font-family: times new roman,times,serif"><small><small>150</small></small></sup></small><big style="color: #000000"><small><strong style="fo
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<font face="Times New Roman, Times, serif"><big style="color: #000000"><strong>SECTION XXXI<br> RECAPITULATION<br> </strong> Last Section (31st)</big></font>
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<><font face="Times New Roman, Times, serif"><big style="color: #000000"> <small>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.</small></big></font>
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<br> <><font face="Times New Roman, Times, serif"><big style="color: #000000"><small> 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. </small></big></font><big><small><font style="color: #000000" face="Times New Roman,Times">(Louis Agassiz, <em>ESSAY ON CLASSIFICATION</em>, Ed. E. Lurie, Belknap Press, Cambridge, 1962:127-128)<br> <br></font></small></big><big><small><font style="color: #000000" face="Times New Roman,Times"> </font><span style="color: #000000"> </span></small></big>
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<hr style="width: 100%; height: 2px"><font face="Helvetica, Arial, sans-serif"><br> <br><span style="font-weight: bold"></span></font><big><font style="color: #000000" face="Arial,Helvetica"><span style="font-weight: bold"></span></font><big><span style="font-weight: bold">IV.</span></big></big><font style="color: #000000" face="Arial,Helvetica"><span style="font-weight: bold"><big><big>THE PHYLLOTACTIC SOLAR SYSTEM</big></big></span></font>
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<big><span style="font-weight: bold"></span></big><big><span style="font-weight: bold"></span></big><span style="font-family: helvetica,arial,sans-serif">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 </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Time</span><span style="font-family: helvetica,arial,sans-serif"> (mean sidereal periods of revolution) rather than mean heliocentric</span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif"> Distances</span><span style="font-family: helvetica,arial,sans-serif">, 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 </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Spira Solaris Archytas-Mirabilis</span><span style="font-family: helvetica,arial,sans-serif"> -- 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 </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Phi </span><span style="font-family: helvetica,arial,sans-serif">= </span><span style="font-weight: bold; font-style: italic; font-family: helvetica,arial,sans-serif">1.6180339887949</span><span style="font-family: helvetica,arial,sans-serif"> and for planet-to-planet increases</span><span style="font-style: italic; font-family: helvetica,arial,sans-serif"><span style="font-weight: bold"> </span></span><span style="font-family: helvetica,arial,sans-serif">the square of this value, </span><span style="font-style: italic; font-family: helvetica,arial,sans-serif">i.e,</span><span style="font-family: helvetica,arial,sans-serif"> </span><span style="font-weight: bold; font-style: italic; font-family: helvetica,arial,sans-serif">Phi </span><small style="font-weight: bold; font-style: italic; font-family: helvetica,arial,sans-serif"><sup>2</sup></small><span style="font-family: helvetica,arial,sans-serif"> = </span><span style="font-weight: bold; font-style: italic; font-family: helvetica,arial,sans-serif">2.6180339887949</span><span style="font-family: helvetica,arial,sans-serif"> </span><span style="font-style: italic; font-family: helvetica,arial,sans-serif"><span style="font-weight: bold"></span></span><span style="font-family: helvetica,arial,sans-serif">(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).</span>
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<span style="font-family: helvetica,arial,sans-serif">Even so, c</span><big style="font-family: helvetica,arial,sans-serif"><small>omparisons between the Phi-Series planetary frameworks and the present Solar System were fraught with difficulty, not least of all because of three</small></big><span style="color: #000000; font-family: helvetica,arial,sans-serif"> 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. <br> 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, one<span style="font-style: italic; font-weight: bold">commences </span>with the latter, and with the larger fibonacci fractions applied by Peirce, one also moves with increasing accuracy towards the constant of linearity, <span style="font-style: italic; font-weight: bold">Phi.</span> 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 </span><span style="font-family: helvetica,arial,sans-serif">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 </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">Table 1</span><span style="font-family: helvetica,arial,sans-serif"> below the fibonacci fractions employed by Peirce are applied firstly to the mean period of revolution of Neptune (given here as </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">164.62423 years</span><span style="font-family: helvetica,arial,sans-serif">, i.e., Peirce's </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">60,129 days</span><span style="font-family: helvetica,arial,sans-serif"> divided by</span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">365.25 days</span><span style="font-family: helvetica,arial,sans-serif">). Thereafter the divisions continue sequentially from the previous result, i.e., division by </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">1</span><span style="font-family: helvetica,arial,sans-serif"> again, by</span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">1/2, 1/2,</span><span style="font-family: helvetica,arial,sans-serif"> then </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">2/3, 2/3,</span><span style="font-family: helvetica,arial,sans-serif"> etc., down to the final divisor </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">21/34</span><span style="font-family: helvetica,arial,sans-serif"> to obtain the mean sidereal period of
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<br><big style="font-weight: bold"><small><font face="Helvetica, Arial, sans-serif">IV.1. THE PHYLLOTACTIC DIVISORS AND SYNODIC PERIODS</font></small></big>
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</div><big><small><font face="Helvetica, Arial, sans-serif">For purposes of comparison modern values for the mean sidereal periods and the calculated synodic periods for the planets are given in the <span style="font-weight: bold">Sol.System</span> 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 <span style="font-weight: bold; font-style: italic">21/13</span> and <span style="font-style: italic; font-weight: bold">55/34</span> with resulting pheidian approximations of</font></small></big><big><small><font face="Helvetica, Arial, sans-serif"> <span style="font-weight: bold; font-style: italic">1.61665353</span> and <span style="font-style: italic; font-weight: bold">1</span></font></small></big><big><small><font face="Helvetica, Arial, sans-serif"><span style="font-style: italic; font-weight: bold">.61737532</span></font></small></big><big><small><font face="Helvetica, Arial, sans-serif"> </font></small></big><big><small><font face="Helvetica, Arial, sans-serif">respectively </font></small></big><big><small><font face="Helvetica, Arial, sans-serif">despite the variance in the individual ratios.</font></small></big>
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<br><span style="font-weight: bold"><span style="font-family: helvetica,arial,sans-serif">Table 3. Benjamin Peirce's Phyllotactic Divisors</span><br> <br> <br></span>
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<big style="font-weight: bold"><small><font face="Helvetica, Arial, sans-serif"> IV. 2. THE DUPLICATED DIVISORS<br> </font></small></big><big><small><font face="Helvetica, Arial, sans-serif">We come now to the underlying reason why the divisors applied by Benjamin Peirce occur in pairs, though not from any <span style="font-style: italic">a priori</span> 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<span style="font-weight: bold"> 2:3:5</span> fibonacci resonance that occurs after approximately 60 years when </font></small></big><big><small><font face="Helvetica, Arial, sans-serif">slower-moving Saturn (mean period of revolution <span style="font-weight: bold">29.45252 years</span>) completes <span style="font-weight: bold">2</span> sidereal revolutions </font></small></big><big><small><font face="Helvetica, Arial, sans-serif">around the centre of the Solar System (for present purposes, simply the Sun) with</font></small></big><big><small><font face="Helvetica, Arial, sans-serif"> swifter-moving </font></small></big><big><small><font face="Helvetica, Arial, sans-serif">Jupiter (mean period of revolution<span style="font-weight: bold">11.869237 years</span>) completing <span style="font-weight: bold">5</span> sidereal revolutions about the same centre while also lapping Saturn<span style="font-weight: bold"> 3</span> times (lap cycle:<span style="font-weight: bold">19.88813 years</span>), resulting in a </font></small></big><big><small><font face="Helvetica, Arial, sans-serif">resonant triple with the three periods expressed in the form <span style="font-weight: bold">2:3:5</span>. The same procedures can be carried out on the remaining superior planets, but i</font></small></big>n <big><small><font face="Helvetica, Arial, sans-serif">view of the</font></small></big><big><small><font face="Helvetica, Arial, sans-serif"> elliptical nature of planetary orbits </font></small></big><big><small><font face="Helvetica, Arial, sans-serif">the application of mean motion is</font></small></big><big><small><font face="Helvetica, Arial, sans-serif"> a vast over-simplification, as the real-time investigations carried out in<span style="font-weight: bold"> Part III </span>of<span style="font-weight: bold"> Spira Solaris Archytas-Mirabilis</span> showed </font></small></big><big><small><font face="Helvetica, Arial, sans-serif">(see</font></small></big><big><small><font face="Helvetica, Arial, sans-serif"><span style="font-weight: bold"></span><span style="font-style: italic"></span></font></small></big><font face="Helvetica, Arial, sans-serif"><strong>Part III, Figures 8-8b;</strong> and<strong> Figure 1 </strong>below for the real-time Jupiter-Saturn and Uranus-Neptune Resonances from 1940-1990</font><big><small><font face="Helvetica, Arial, sans-serif">). </font></small></big><big><small><font face="Helvetica, Arial, sans-serif"><br> <br><br> </font></small></big><span style="font-weight: bold"> <span style="font-family: helvetica,arial,sans-serif">IV. 3. THE EXTENDED PHI-SERIES AND THE INNER LIMIT</span></span>
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<br style="font-family: helvetica,arial,sans-serif"> <span style="font-family: helvetica,arial,sans-serif"> 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 (</span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">1,1,2</span><span style="font-family: helvetica,arial,sans-serif">,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 </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">1:1:2,</span><span style="font-family: helvetica,arial,sans-serif"> </span><big style="font-family: helvetica,arial,sans-serif"><small><span style="font-weight: bold">1:2:3</span></small></big><span style="font-family: helvetica,arial,sans-serif">and </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">2:3:5</span><span style="font-family: helvetica,arial,sans-serif"> associated with the four superior planets Neptune, Uranus, Saturn and Jupiter are not only sequential, they are also successive </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Fibonacci </span><span style="font-family: helvetica,arial,sans-serif">triples bound together in a most complex manner.</span>
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<br style="font-family: helvetica,arial,sans-serif"> <span style="font-family: helvetica,arial,sans-serif"> 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 (</span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">3:5:8 </span><span style="font-family: helvetica,arial,sans-serif">and</span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">5:8:13 </span><span style="font-family: helvetica,arial,sans-serif">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 </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">13:21:34</span><span style="font-family: helvetica,arial,sans-serif">) but also beyond Mercury to a specific </span><span style="font-weight: bold; font-style: italic; font-family: helvetica,arial,sans-serif">limit</span><span style="font-family: helvetica,arial,sans-serif"> at the fifth inner position (final resonant triple of </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">144:233:377</span><span style="font-family: helvetica,arial,sans-serif">) as seen below in </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">Table 4</span><span style="font-family: helvetica,arial,sans-serif"> for </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Phi-Series</span><span style="font-family: helvetica,arial,sans-serif"> periods </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif"><span style="font-style: italic">Phi</span> <small><sup>5</sup></small></span><span style="font-family: helvetica,arial,sans-serif"> through </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif"><span style="font-style: italic">Phi </span><small><sup>-13</sup></small></span><span style="font-family: helvetica,arial,sans-serif"></span>
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</div><span style="color: #000000"></span><span style="font-weight: bold">Table 4. The Phi-Series Planetary Periods ( N = 5 to -13 ), Resonant Triples and the Inner Limit<br> <br></span>
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<br> <span style="font-family: helvetica,arial,sans-serif"> In</span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif"> Table 4</span><span style="font-family: helvetica,arial,sans-serif"> it is the</span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif"> pairings </span><span style="font-family: helvetica,arial,sans-serif">that that are duplicated, not the divisors. Some columns are self-explanatory, </span><span style="font-style: italic; font-family: helvetica,arial,sans-serif">e.g.,</span><span style="font-family: helvetica,arial,sans-serif"> the </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Phi-Series</span><span style="font-family: helvetica,arial,sans-serif"> exponents in </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Column N,</span><span style="font-family: helvetica,arial,sans-serif"> as are the resulting sidereal (</span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">T</span><span style="font-family: helvetica,arial,sans-serif">) and synodic periods (</span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">S</span><span style="font-family: helvetica,arial,sans-serif">) 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 </span><span style="font-weight: bold; font-style: italic; font-family: helvetica,arial,sans-serif">Column X</span><span style="font-family: helvetica,arial,sans-serif"> are the numbers of complete sidereal revolutions and synodic cycles per pairing (i.e., sidereal and synodic </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">X-Factors</span><span style="font-family: helvetica,arial,sans-serif">), while (omitting </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Column FN </span><span style="font-family: helvetica,arial,sans-serif">for the time being) the data in the </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Products</span><span style="font-family: helvetica,arial,sans-serif"> and the </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">RZ Triples </span><span style="font-family: helvetica,arial,sans-serif">are exactly that, with the latter also the</span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">X-Factors</span><span style="font-family: helvetica,arial,sans-serif"> from the </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">X Column</span><span style="font-family: helvetica,arial,sans-serif">. </span>
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<br style="font-family: helvetica,arial,sans-serif"><span style="font-family: helvetica,arial,sans-serif"> The rounded periods in </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Column Fn</span><span style="font-family: helvetica,arial,sans-serif"> require some explanation, although the perceptive reader may have already seen that the closer the </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Products</span><span style="font-family: helvetica,arial,sans-serif"> get to the</span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif"> Inner Limit,</span><span style="font-family: helvetica,arial,sans-serif"> the more obvious it becomes that the former in every instance are consistent approximations for sequential, fractional exponential values of </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Phi,</span><span style="font-style: italic; font-family: helvetica,arial,sans-serif"> e.g.,</span><span style="font-family: helvetica,arial,sans-serif"> the lowest (</span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">0.723606</span><span style="font-family: helvetica,arial,sans-serif">) approximates </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Phi </span><small style="font-family: helvetica,arial,sans-serif"><sup>-2/3</sup></small><span style="font-family: helvetica,arial,sans-serif"> = </span><span style="font-weight: bold; font-style: italic; font-family: helvetica,arial,sans-serif">0.72556, </span><span style="font-style: italic; font-family: helvetica,arial,sans-serif"></span><span style="font-family: helvetica,arial,sans-serif">the next (</span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">1.17085</span><span style="font-family: helvetica,arial,sans-serif">) </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Phi </span><small style="font-family: helvetica,arial,sans-serif"><sup>1/3</sup></small><span style="font-family: helvetica,arial,sans-serif"> =</span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif"> 1.17398</span><span style="font-family: helvetica,arial,sans-serif">, followed by </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">1.8944</span><span style="font-family: helvetica,arial,sans-serif"> ( </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Phi </span><small style="font-family: helvetica,arial,sans-serif"><sup>4/3</sup></small><span style="font-family: helvetica,arial,sans-serif"> = </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">1.89954</span><span style="font-family: helvetica,arial,sans-serif"> ), then 3.065 ( </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Phi</span><small style="font-family: helvetica,arial,sans-serif"><sup>7/3</sup></small><span style="font-family: helvetica,arial,sans-serif"> =</span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif"> 3.0735</span><span style="font-family: helvetica,arial,sans-serif">) and so on, thus the exponents increase by </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">unity</span><span style="font-family: helvetica,arial,sans-serif"> 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 = </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">1</span><span style="font-family: helvetica,arial,sans-serif">, 1.1708 rounded down = </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">1</span><s
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<br style="font-family: helvetica,arial,sans-serif"><span style="font-family: helvetica,arial,sans-serif"> 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</span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">: </span>
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<span style="font-weight: bold; font-family: helvetica,arial,sans-serif">Table 5. The Phi-Series Planetary Periods (N = 1 to 11), Resonant Triples and Departures<br> <br><br> </span>
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<span style="font-family: helvetica,arial,sans-serif">But this is the </span><span style="font-weight: bold; font-style: italic; font-family: helvetica,arial,sans-serif">Phi-Series planetary framework</span><span style="font-family: helvetica,arial,sans-serif"> 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) </span><small style="font-family: helvetica,arial,sans-serif"><sup>9</sup></small><span style="font-family: helvetica,arial,sans-serif"> adapted to </span><a style="font-family: helvetica,arial,sans-serif" href="../../time1.html">time-series analysis</a><span style="font-family: helvetica,arial,sans-serif">:</span>
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<big><small><font face="Helvetica, Arial, sans-serif"><img alt="Figure 1. Jupiter-Saturn-Uranus-Neptune real-time Fibonacci Resonances I, 1890-1990" src="nusaj7a.gif" style="width: 801px; height: 655px"></font></small></big>
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</div><big><small><font face="Helvetica, Arial, sans-serif"><br> <span style="font-weight: bold">Figure 1. Jupiter-Saturn-Uranus-Neptune real-time Fibonacci Resonances I, 1890-1990<br> <br></span></font></small></big>
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<span style="font-weight: bold">Figure 1</span> above is<span style="font-weight: bold"></span> basically similar to <span style="font-weight: bold">Figures 8</span> and <span style="font-weight: bold">8b</span> and related research originally introduced in <span style="font-weight: bold">Part III </span> with the addition of blue lines representing the two Fibonacci departures from <span style="font-weight: bold">Table 5</span> (<span style="font-weight: bold">89</span><span style="font-weight: bold"> </span>and<span style="font-weight: bold"> 144 years</span>) 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, <span style="font-weight: bold">1:1:2</span>, <span style="font-weight: bold">1:2:3</span> and <span style="font-weight: bold">2:3:5</span>.
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<br style="font-family: helvetica,arial,sans-serif"> <big style="font-family: helvetica,arial,sans-serif"><span style="font-weight: bold">I<small>V.4. SOLAR SYSTEM RESONANT PHYLLOTACTIC TRIPLES</small><br> </span></big><span style="font-family: helvetica,arial,sans-serif">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 </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">Tables 4</span><span style="font-family: helvetica,arial,sans-serif"> and </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">5</span><span style="font-family: helvetica,arial,sans-serif">.</span>
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<br style="font-family: helvetica,arial,sans-serif"> <span style="font-family: helvetica,arial,sans-serif"> Thus in </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">Table 6</span><span style="font-family: helvetica,arial,sans-serif"> below the mean sidereal periods of revolution and intervening mean synodic periods are multiplied by the corresponding </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">X-Factors</span><span style="font-family: helvetica,arial,sans-serif"> (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.</span>
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<span style="color: #cc9933"><img alt="Table 3. Resonant Phyllotaxic Triples, Mercury to Neptune (Ceres included, Earth Synodic)" src="rztp1d8.gif" style="width: 603px; height: 659px"><br> </span><span style="color: #cc9933"></span>
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<span style="color: #cc9933"><span style="color: #000000"><span style="font-weight: bold"></span></span></span><span style="color: #cc9933"><br> </span><span style="color: #cc9933; font-family: helvetica,arial,sans-serif"><span style="color: #000000"><span style="font-weight: bold">Table 6.</span></span><span style="color: #000000"><span style="font-weight: bold">Resonant Phyllotactic Triples, Mercury to Neptune (Mean values; Ceres included, Earth Synodic)<br> </span></span><span style="color: #000000"><span style="font-weight: bold"></span><br> </span></span>
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<span style="font-family: helvetica,arial,sans-serif">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</span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">34</span><span style="font-family: helvetica,arial,sans-serif"> (</span><span style="font-style: italic; font-family: helvetica,arial,sans-serif">i.e</span><span style="font-family: helvetica,arial,sans-serif">.,</span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">1, 1, 2, 3, 5, 8, 13, 21, 34,.</span><span style="font-family: helvetica,arial,sans-serif">.) and even further (embracing the next fibonacci number </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">55</span><span style="font-family: helvetica,arial,sans-serif">) if</span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif"> IMO</span><span style="font-family: helvetica,arial,sans-serif"> the Inter-Mercurial Object from Part II is included. Thus for the four superior planets </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">Neptune, Uranus, Saturn</span><span style="font-family: helvetica,arial,sans-serif"> and </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">Jupiter </span><span style="font-family: helvetica,arial,sans-serif">the first fibonacci resonant sequence [ </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">1 : 1 : 2 </span><span style="font-family: helvetica,arial,sans-serif">] is followed by [ </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">1 : 2 : 3</span><span style="font-family: helvetica,arial,sans-serif"> ] then [ </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">2 : 3 : 5</span><span style="font-family: helvetica,arial,sans-serif"> ] and so on down to the last resonant triple [ </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">13 : 21 : 34</span><span style="font-family: helvetica,arial,sans-serif"> ] between </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">Venus</span><span style="font-family: helvetica,arial,sans-serif"> and </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">Mercury</span><span style="font-family: helvetica,arial,sans-serif">:</span>
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<br> <span style="font-weight: bold; font-family: helvetica,arial,sans-serif">Figure 2. Phyllotactic Resonant Triples in the Solar System</span>
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<span style="color: #000000; font-family: helvetica,arial,sans-serif"><br> </span><span style="font-family: helvetica,arial,sans-serif">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 </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">middle</span><span style="font-family: helvetica,arial,sans-serif"> number of the first triple (the number of </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">Neptune-Uranus</span><span style="font-family: helvetica,arial,sans-serif"> synodics) is the same as the </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">first</span><span style="font-family: helvetica,arial,sans-serif">value of the next triple (the number of sidereal revolutions of</span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">Uranus</span><span style="font-family: helvetica,arial,sans-serif">), and that the </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">last </span><span style="font-family: helvetica,arial,sans-serif">value of the first triple (also the number of revolutions of Uranus in the first set) is the </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">middle</span><span style="font-family: helvetica,arial,sans-serif">value of the next (</span><span style="font-style: italic; font-family: helvetica,arial,sans-serif">i.e</span><span style="font-family: helvetica,arial,sans-serif">., the numbers of </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">Uranus-Saturn</span><span style="font-family: helvetica,arial,sans-serif"> 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 </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">Plato's</span><span style="font-family: helvetica,arial,sans-serif"></span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Timaeus 31b-32c,</span><span style="font-style: italic; font-family: helvetica,arial,sans-serif"></span><span style="font-family: helvetica,arial,sans-serif"> though this is</span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif"> </span><span style="font-style: italic; font-family: helvetica,arial,sans-serif"></span><span style="font-family: helvetica,arial,sans-serif">not our current concern here. </span>
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<br> <big><big><span style="font-weight: bold">V. PHYLLOTAXIS AND THE PHI-SERIES</span></big></big>
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<br> What remains now is the relationship between Benjamin Peirce's divisors, the <span style="font-style: italic; font-weight: bold">Phi-Series</span>, and the application of phyllotaxis to the structure of the Solar System.
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<br> <font face="Helvetica, Arial, sans-serif">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 <span style="font-style: italic; font-weight: bold">Fibonacci</span> and the <span style="font-style: italic; font-weight: bold">Lucas Series</span> 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. </font><font face="Helvetica, Arial, sans-serif">That this inquiry should eventually lead to planetary resonances involving both the<span style="font-style: italic; font-weight: bold"> Lucas </span>and the <span style="font-style: italic; font-weight: bold">Fibonacci Series</span> is hardly surprising </font><font face="Helvetica, Arial, sans-serif">given the prominence of the mathematical relationships known to exist between the two series (see, for example <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/lucasNbs.html#LucasFibFormula">The Lucas Numbers</a> and <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/propsOfPhi.html#numprops">Phi - More Facts and Figures</a> 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 <span style="font-style: italic; font-weight: bold">Phi-Series</span> exponential planetary framework such that (<span style="font-weight: bold">a</span>) the two major period constants </font><font face="Arial">( <span style="font-weight: bold; font-style: italic">Phi</span>and<span style="font-style: italic; font-weight: bold">Phi</span><small><sup> 2</sup></small> )</font><font face="Helvetica, Arial, sans-serif">:<br> </font>
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<span style="font-weight: bold"><img alt="Relations 5a and 5b. The Fundamental Period Constants" src="rel5s.gif" style="width: 348px; height: 49px"></span>
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<span style="font-weight: bold">Relations 5a and 5b. The Fundamental Period Constants</span>
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</div><font face="Helvetica, Arial, sans-serif"><br> re-occur in the form of the double Fibonacci sequence, and (<span style="font-weight: bold">b</span>), the proximity of the<span style="font-style: italic; font-weight: bold">Lucas Series</span> to the <span style="font-style: italic; font-weight: bold">Phi-Series</span> becomes increasingly apparent as the<span style="font-style: italic; font-weight: bold">Phi-Series planetary periods</span> increase from Jupiter onwards and outwards</font><big><small><font face="Helvetica, Arial, sans-serif"> (<span style="font-style: italic">i.e.</span>,<span style="font-weight: bold">Lucas 11 : Phi-Series 11.09016994</span>; <span style="font-weight: bold">Lucas 29 : Phi-Series 29.03444185</span>, etc.):</font></small></big>
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<span style="font-weight: bold"></span><span style="font-weight: bold"><img alt="Table 7. Lucas Series, Phi-Series Periods, Phi-Series Decomposition and the Fibonacci Series" src="lucasdec9d.gif" style="width: 586px; height: 336px"><br> </span>
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<span style="font-weight: bold">Table 7. Lucas Series, Phi-Series Periods, Phi-Series Decomposition and the Fibonacci Series</span>
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<span style="font-family: helvetica,arial,sans-serif">As for the </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Phi-Series</span><span style="font-family: helvetica,arial,sans-serif"></span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">planetary framework</span><span style="font-family: helvetica,arial,sans-serif"> itself, </span><big style="font-family: helvetica,arial,sans-serif"><small>it should be noted that while the </small></big><font style="font-family: helvetica,arial,sans-serif" face="Helvetica, Arial, sans-serif">importance of the</font><font style="font-family: helvetica,arial,sans-serif" face="Times New Roman,Times"> reciprocals of <span style="font-style: italic; font-weight: bold">phi</span> and its <span style="font-style: italic; font-weight: bold">square </span><span style="font-style: italic"></span>(<span style="font-weight: bold">0.618033989</span> and <span style="font-weight: bold">0.381966011</span>)<span style="font-weight: bold"> </span></font><font style="font-family: helvetica,arial,sans-serif" face="Times New Roman,Times">needs no introduction to those familiar with phyllotaxis </font><font style="font-family: helvetica,arial,sans-serif" face="Times New Roman,Times">(especially the latter with respect to the "ideal angle") both numbers occur with surprising frequency in the<span style="font-style: italic; font-weight: bold"> Phi-Series</span><span style="font-style: italic; font-weight: bold">planetary framework</span>. As indeed does the primary constant of </font><span style="font-family: helvetica,arial,sans-serif">linearity,</span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Phi squared</span><span style="font-family: helvetica,arial,sans-serif"> (</span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">2.618033898</span><span style="font-family: helvetica,arial,sans-serif">) determined in </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">Part II </span><span style="font-family: helvetica,arial,sans-serif">and applied in </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">Part III </span><span style="font-family: helvetica,arial,sans-serif">and elsewhere -- the latter with respect to the the inverse velocity (</span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">V</span><span style="font-family: helvetica,arial,sans-serif">i) of the </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Jupiter-Saturn</span><span style="font-family: helvetica,arial,sans-serif"> synodic (or lap) cycle</span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif"></span><span style="font-family: helvetica,arial,sans-serif">, with its reciprocal providing the relative velocity </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">Vr</span><span style="font-family: helvetica,arial,sans-serif">, which as it turns out, is also the </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Mercury-Venus</span><span style="font-family: helvetica,arial,sans-serif"> synodic period(</span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">T</span><span style="font-family: helvetica,arial,sans-serif">), and additionally, the mean distance </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">(R</span><span style="font-family: helvetica,arial,sans-serif">) of</span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Mercury</span><span style="font-family: helvetica,arial,sans-serif">.</span>
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<br> <big><small><font face="Helvetica, Arial, sans-serif">Thus we already encounter </font></small></big><big><small><font face="Helvetica, Arial, sans-serif">in the <span style="font-style: italic; font-weight: bold">Phi-series</span><span style="font-style: italic; font-weight: bold">planetary framework</span> </font></small></big><big><small><font face="Helvetica, Arial, sans-serif">complex dynamic relationships between the two inner inferior planets and the two largest superior planets involving constants known to be intimately related to phyllotaxis</font></small></big>
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<img alt="Table 3. Phi-Series Planetary Data: Periods, Distances and Velocities plus inherent Phyllotactic Constants" src="phiseries3g6.gif" style="width: 641px; height: 411px">
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<span style="font-weight: bold; font-family: helvetica,arial,sans-serif">Table 8. <span style="font-style: italic">Phi-Series</span> Planetary Data: Periods, Distances, Velocities, and inherent Phyllotactic Constants</span>
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<br> <span style="font-weight: bold"></span>
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</div><span style="font-weight: bold"><br> </span><big><small><font face="Helvetica, Arial, sans-serif">To put the full significance of the <span style="font-style: italic; font-weight: bold">Phi-Series</span><span style="font-style: italic; font-weight: bold">planetary framework</span> and the constants in question in their full perspective it may be useful to include here a discussion pertaining to phyllotaxis from <span style="font-weight: bold">Part III</span>:<small><sup>11</sup></small></font></small></big>
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<p style="text-align: justify"><font style="font-family: times new roman,times,serif" face="Helvetica, Arial, sans-serif">" 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 </font><font style="font-family: times new roman,times,serif" face="Helvetica, Arial, sans-serif">points out that the French architect Le Corbusier "</font><font style="font-family: times new roman,times,serif" face="Helvetica, Arial, sans-serif">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 C</font><font style="font-family: times new roman,times,serif" face="Helvetica, Arial, sans-serif"><em>ONNECTIONS: The Geometric Bridge between Art and Science</em>:</font></p>
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<font face="Times New Roman,Times">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.</font>
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<br> <font face="Times New Roman,Times">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 <em>divergence angle</em> and is related to the golden mean. The radial or vertical motion is measured by the pitch <strong><em>h</em></strong>. The dynamics of plant growth can be described by and <strong><em>h</em></strong>; we will explore this further in Section 6.9 [Coxeter, 1953].</font>
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<br> <strong><em><font face="Times New Roman,Times">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</font></em></strong><font face="Times New Roman,Times">. For example, <strong><em>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</em></strong>). We will analyze the surface structure of the pineapple in greater detail in Section 6.9.</font>
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<p><font face="Times New Roman,Times">3.7.2 <strong><em>Nature responds to a physical constraint </em></strong>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].</font> <br> <font face="Times New Roman,Times">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 <em>entropy function</em> such as he introduces in his paper [1990].</font> <br> <font face="Times New Roman,Times">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<small><sup> 2</sup></small> radians, or<span style="font-style: italic; font-weight: bold"> 137.5 degrees about a growth cente</span><strong><em>r</em></strong>, 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.</font> <br> <font face="Times New Roman,Times">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 <em>n</em>/Phi<small><sup> 2</sup></small>, are shown for values of <em>n</em> 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].</font> </p>
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<font face="Times New Roman,Times">Theorem 3.3 Let <em>x</em> be any irrational number. When the points [x] <sub>f,</sub> [2x] <sub>f,</sub> [3x] <sub>f,</sub>..., [<em>nx</em>] <sub>f </sub>are placed on the line segment [0,1], the<em> n</em> + 1 resulting line segments have at most three different lengths.</font>
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<br> <font face="Times New Roman,Times">Moreover, [(<em>n</em> + 1)<em>x</em>]<sub>f </sub>will fall into one of the largest existing segments. ( [ ] <sub>f </sub>means "fractional part of ").</font>
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</blockquote><font face="Times New Roman,Times">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/Phi<small><sup>2</sup></small> 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.</font>
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<br> <strong><em><font face="Times New Roman,Times">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.</font></em></strong><font face="Times New Roman,Times">(Jay Kappraff, Chapter </font><em>3.7. The<strong> </strong>Golden Mean and Patterns of Plant Growth</em>, <font face="Times New Roman,Times"><em style="font-weight: bold">CONNECTIONS : The Geometric Bridge between Art and Science</em><span style="font-weight: bold">,</span> McGraw-Hill, Inc., New York, 1991:89-96, bold emphases supplied.) </font>For more on this topic see also Dr. R. Knott's extensive treatment <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html">The Fibonacci Numbers and the Golden Section</a>, the latter's <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#links">related links</a> and the <a href="http://math.smith.edu/%7Ephyllo/">The Phyllotaxis Home Page of Smith University)</a>
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</div><span style="font-family: helvetica,arial,sans-serif"> The observation made by Kappraff that: "</span><font style="font-family: helvetica,arial,sans-serif" face="Times New Roman,Times">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, <span style="font-style: italic">i.e.</span>, more in terms of first (or perhaps better stated, secondary) causes, the primary pheidian constant <span style="font-weight: bold; font-style: italic">Phi</span> <small><sup>2</sup></small> = </font><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">2.618033898</span><span style="font-family: helvetica,arial,sans-serif"> 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 </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Phi-Series</span><span style="font-family: helvetica,arial,sans-serif">) generate both this important parameter </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">and </span><span style="font-family: helvetica,arial,sans-serif">its reciprocal </span><font style="font-family: helvetica,arial,sans-serif" face="Arial"><strong><em>Phi</em></strong><strong><sup><font size="-1"><small> -2</small></font></sup></strong> = </font><font style="font-family: helvetica,arial,sans-serif" face="Arial"><strong><em>0.381966011.</em></strong></font><font style="font-family: helvetica,arial,sans-serif" face="Arial"><span style="font-weight: bold"> </span>This</font><font style="font-family: helvetica,arial,sans-serif" face="Arial"> second constant </font><font style="font-family: helvetica,arial,sans-serif" face="Helvetica, Arial, sans-serif">is not only intimately related to the phyllotaxic "ideal" growth angle of 137.50776405 degrees (0.381966011 x 360<small><sup>O</sup></small>) it is also a unusual repeat parameter in the planetary framework as shown in <span style="font-weight: bold">Table 8</span> above. As</font><font face="Helvetica, Arial, sans-serif"><span style="font-family: helvetica,arial,sans-serif"> explained in a later section (</span><a style="font-family: helvetica,arial,sans-serif" href="../../sbb4d2c.html">Spira Solaris and the Pheidian Planordidae</a>): </font><font face="Helvetica, Arial, sans-serif"></font>
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<small>These multiple occurrences arise from the three-fold nature of the <em>Phi-Series Planetary Framework</em>, which necessarily incorporates identical values for periods, distances and velocities according to their exponential position in the framework, including the<em> <strong>Inverse Velocities</strong></em>, which (as will be shown later) also play an important role in the computation of angular momentum. From this viewpoint the well-known <strong><em>60-year,</em></strong><strong><em> 2 : 3 : 5</em></strong> fibonacci resonance between the two most massive planets in the Solar system (Jupiter and Saturn) takes on further significance as <strong>Figure 8 (21c]</strong> shows, for the <em>arithmetic mean</em> of the actual Jupiter and Saturn mean velocities <strong><em>V</em><em>r</em></strong> (shown in the upper and lower real-time waveforms) is not only <strong><em>0.381280708</em></strong>; <em>the daily average function</em> 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<strong><em> 0.381071579</em></strong>. 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.</small>
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<p align="center"><font face="Helvetica, Arial, sans-serif"><img src="sdvelk9.gif" alt="Figure 3. The Jupiter-Saturn Average Velocity Function (JS-Avg.Vr) and the Phyllotactic Constant k = Phi -2" style="width: 702px; height: 530px"> <br> </font></p>
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<font face="Helvetica, Arial, sans-serif"><strong>Figure 3. The Jupiter-Saturn Average Velocity Function (JS-Avg.Vr) and the Phyllotactic Constant</strong></font><font face="Helvetica, Arial, sans-serif"><strong> k = <em>Phi<sup>-2</sup></em></strong></font>
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</div><span style="font-weight: bold"></span><span style="color: #000000"></span><span style="color: #000000"></span><span style="color: #000000"></span><span style="color: #000000"></span><span style="color: #000000"></span><span style="color: #cc9933"><span style="color: #000000"><span style="font-weight: bold"></span></span></span><span style="font-family: helvetica,arial,sans-serif">Thus the plot of orbital velocities in </span><span style="font-weight: bold; font-family: helvetica,arial,sans-serif">Figure 3</span><span style="font-family: helvetica,arial,sans-serif"> shows firstly (at the top) the cyan velocity waveform and mean value for Jupiter and also the dashed-line</span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Phi-Series</span><span style="font-family: helvetica,arial,sans-serif"> mean value. With a shorter sidereal period (11.090169 versus 11.86924 years) the latter is swifter than that of Jupiter </span><span style="font-style: italic; font-family: helvetica,arial,sans-serif">per se,</span><span style="font-family: helvetica,arial,sans-serif"> and accordingly it appears near the top of the waveform rather than the mean. At the bottom (green waveform), because the difference between the </span><span style="font-style: italic; font-weight: bold; font-family: helvetica,arial,sans-serif">Phi-Series</span><span style="font-family: helvetica,arial,sans-serif"> 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. </span>
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<br style="font-family: helvetica,arial,sans-serif"> <span style="color: #cc9933; font-family: helvetica,arial,sans-serif"><span style="color: #000000">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. <br> <br> 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<small><small><sup> 10</sup></small></small> was published in the <span style="font-weight: bold; font-style: italic">Journal of the Royal Astronomical Society of Canada </span><span style="font-weight: bold"></span>(</span></span><font style="font-family: helvetica,arial,sans-serif" face="Times New Roman, Times, serif"> "<a href="../../sbb7a.html">Projectiles, Parabolas, and Velocity Expansions of the Laws of Planetary Motion</a> "</font><font style="font-family: helvetica,arial,sans-serif" face="Times New Roman, Times, serif"><span style="font-style: italic"></span><em>JRASC</em>, Vol 83, No. 3, June 1989 ) </font><span style="color: #cc9933; font-family: helvetica,arial,sans-serif"><span style="color: #000000">and that it is disappointing that they have not been put into more general use, especially since </span></span><font style="font-family: helvetica,arial,sans-serif" face="Helvetica, Arial, sans-serif">the available orbital relationships are effectively quadrupled by their inclusion, i.e.,</font>
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<h4 style="text-align: center"><font><strong><font face="Helvetica, Arial, sans-serif"><strong>Table 9. Distance-Period-Velocity Relationships</strong></font></strong></font></h4>
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</div><font face="Helvetica, Arial, sans-serif">The use of the <em>Inverse Velocity</em> 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) </font><font face="Helvetica, Arial, sans-serif">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. <br> </font>
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<br> <span style="color: #cc9933"><span style="color: #000000"><span style="font-weight: bold"><br> <br></span></span></span>
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<font style="color: #000000" face="Arial,Helvetica"><span style="font-weight: bold"><big><big>IV. IMPLICATIONS AND RAMIFICATIONS</big></big></span></font>
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<br> <span style="color: #cc9933"><span style="color: #000000"><span style="font-weight: bold"></span></span></span>
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</div><span style="color: #cc9933"><span style="color: #000000"><span style="font-weight: bold"></span></span></span><font style="color: #000000" face="Arial,Helvetica">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 i</font><font style="color: #000000" face="Arial,Helvetica">f 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.<br> <br> We are at present, it seems, far too pleased with ourselves, and for far too little reason.<br> <br> 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 </font><font style="color: #000000" face="Arial,Helvetica">whether </font><font style="color: #000000" face="Arial,Helvetica">we are always likely to remain so in view of our unsavoury past, troublesome present, and far from certain future. <br> <br></font><font style="color: #000000" face="Arial,Helvetica">Returning to the researches of Benjamin Peirce, i</font><font face="Helvetica, Arial, sans-serif">n 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 <em>ad hoc</em> diversions introduced and perpetuated by "Bode's Law." H</font><font face="Helvetica, Arial, sans-serif">ow could something so momentous and far-reaching have been so easily driven into obscurity? According to the modern editor of Agassiz' <em style="font-weight: bold">Essay on Classification</em>, (E. Lurie) it was partly the work of Asa Gray and Chauncey Wright, as explained in the following footnote (the latter's No.149):</font>
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<p><font face="Times New Roman,Times">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," <em>Proceedings, AAAS,</em> II (<strong>1850</strong>), 438-444, and Benjamin Peirce, "Mathematical Investigations of the Fractions Which Occur in Phyllotaxis," in <strong><em>ibid</em></strong>., 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.</font> </p>
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</div><font face="Helvetica, Arial, sans-serif">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<em> status qu</em>o. 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 </font>excerpt from <a href="../../llight.html">The Matter of lost Light:</a>
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<p style="margin-left: 0.5in; margin-right: 0.5in"><small><big style="font-family: times new roman,times,serif">".. <span style="font-style: italic"></span> 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<em> <span style="font-weight: bold">On Growth and Form</span></em>.<small><small><sup>12</sup></small></small> 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 to<span style="font-weight: bold"></span><em style="font-weight: bold">Life and Theology of Orpheus</em>) speaks of social decline, loss of knowledge in ancient times and the efforts to preserve it by those who, <em>"though they lived in a base age"</em> nevertheless<em>"</em><em>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,"</em> that sadly, such times are still upon us. Thus, just as Sir Theodore Andrea Cook,<small><small><sup>13</sup></small></small>who in the <em style="font-weight: bold">Curves of Life</em> (1914) was unable to define the "well known logarithmic spiral" equated in 1881 with the chemical elements (see the previous section:<span style="font-weight: bold; font-style: italic"> Spira Solaris and the Three-fold number</span>), neither Canon Mosely<small><sup><small>14</small> </sup></small>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 <em>per se</em>, 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 <em style="font-weight: bold">Essay on Classification</em> (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<em> Essay on Classification</em>, (E. Lurie)<small><small><sup>15</sup></small></small> explained in the short loaded footnote<small><small><sup> </sup></small></small>discussed in the previous section. Nor it would seem, were the works of Arthur Harry Church<small><small><sup>16</sup></small></small> (<em style="font-weight: bold">On the Relation of Phyllotaxis to Mechanical Law</em>, 1904) or Samuel Colman<sup><small><small>17-18</small></small> </sup>(<em style="font-weight: bold">Nature's Harmonic Unity</em><span style="font-weight: bold">,</span> 1911)</big><small style="font-family: times new roman,times,serif"><sup> </sup></small><big style="font-family: times new roman,times,serif">allowed to take root. Nor again were the lines of inquiry laid out in Jay Hambidge's<small><small><sup>19</sup></small></small> <em style="font-weight: bold">Dynamic Symmetry</em> (1920) permitted to have much on effect on the <em>status quo</em> either, not to mention Sir Theodore Andrea Cook's<small><small><sup> </sup></small></small><em style="font-weight: bold">Curves of Life</em> (1914) and the general the thrust of the many papers published during the previous century and on into the present.</big></small><span style="color: #000000"></span><small><big><span style="font-family: times new roman,times,serif"><small><small><sup>20-26</sup></small></small></span></big></small><br> <small><big style="font-family: times new roman,times,serif"><font style="color: #000000"> </font><font style="color: #000000">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
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<p style="text-align: justify; margin-left: 0.5in; margin-right: 0.5in; color: #000000"><small><big><span style="font-style: italic; font-family: times new roman,times,serif">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. </span><span style="font-family: times new roman,times,serif">( </span><span style="font-style: italic; font-weight: bold; font-family: times new roman,times,serif">The Great Barrier Reef</span><span style="font-family: times new roman,times,serif">, dir. Dr. George Casey, Science museum of Minnesota, 2001)<small><small><sup>27</sup></small></small></span><br> </big></small></p>
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<small><big>And again, this time from the Northern Hemisphere</big></small>:<small><big><small><small><sup>28</sup></small></small></big></small>
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<em>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..</em><strong><em> </em></strong>(Peter Knudtson and David Suzuki<em>,</em> <strong><em>Wisdom of the Elders,</em></strong> 1992)
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</div><font style="color: #000000" face="Arial,Helvetica">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 <span style="font-style: italic">The Chaldean Oracles</span> 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. </font><font style="color: #000000" face="Arial,Helvetica">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 work</font><small><big><span style="font-family: times new roman,times,serif"><small><small><sup>29</sup></small></small></span></big></small><font style="color: #000000" face="Arial,Helvetica"> remains under-appreciated to the present day, and there are undoubtedly other Arab sources likely to be informative in their original state.</font><font style="color: #000000" face="Arial,Helvetica"> 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. <br> <br></font><font style="color: #000000" face="Arial,Helvetica">Here we surely need to come of age, and quickly.<br> <br> Needless to say, t</font><font style="color: #000000" face="Arial,Helvetica">here 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.</font><small> <br> </small><small><big><span style="color: #000000"><br> </span></big></small>
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<span style="color: #000000">John N. Harris<br> <big><a href="../../index.html">spirasolaris</a></big><br> January 19, 2007<br style="color: #000000"> </span><span style="color: #000000"></span>
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<p style="text-align: justify"><font face="Helvetica, Arial, sans-serif"><strong><font color="#666600"><font size="+2">NOTES AND REFERENCES</font></font></strong></font><br> </p>
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<li><span style="font-family: times new roman,times,serif">Agassiz, Louis. </span><big><small><font style="color: #000000" face="Times New Roman,Times"><em>ESSAY ON CLASSIFICATION</em>, Ed. E. Lurie, Belknap Press, Cambridge, 1962:131.</font></small></big></li>
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<li><font face="Times New Roman, Times, serif">Peirce, Benjamin.</font><font face="Times New Roman, Times, serif"> </font><font face="Times New Roman, Times, serif">"Mathematical Investigations of the Fractions Which Occur in Phyllotaxis," </font><font face="Times New Roman, Times, serif"><em>Proceedings, AAAS,</em> II 1850:444-447.</font></li>
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<li><span style="font-family: times new roman,times,serif">Agassiz, Louis. </span><big><small><font style="color: #000000" face="Times New Roman,Times"><em>ESSAY ON CLASSIFICATION</em>, Ed. E. Lurie, Belknap Press, Cambridge, 1962:127.</font></small></big></li>
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<li><big><small><font style="color: #000000" face="Times New Roman,Times">Called here IMO from <span style="font-family: times new roman,times,serif"> </span></font></small></big><span style="font-family: times new roman,times,serif">Leverrier, M, "The Intra-Mercurial Planet Question," </span><em style="font-family: times new roman,times,serif">Nature 14</em><span style="font-family: times new roman,times,serif"> (1876) 533. [Anon.]</span><span style="font-family: times new roman,times,serif"></span></li>
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<li><span style="font-family: times new roman,times,serif">Agassiz, Louis. </span><em style="font-family: times new roman,times,serif">ESSAY ON CLASSIFICATION</em><span style="font-family: times new roman,times,serif">, Ed. E. Lurie, Belknap Press, Cambridge, 1962:127.</span></li>
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<li><big><small><font style="color: #000000" face="Times New Roman,Times">Peirce, Benjamin.<span style="font-style: italic; font-weight: bold"> </span></font></small></big><span style="font-family: times new roman,times,serif; color: #000000"><span style="font-style: italic">Address of Professor Benjamin Peirce, President of the American Association for the Year 1853, on retiring from the duties of President.</span> <span style="font-style: italic">AAAS</span>, 1853:67. Source (JPG): <a href="http://historical.library.cornell.edu/math/index.html">The</a></span><span style="font-family: times new roman,times,serif"><a href="http://historical.library.cornell.edu/math/index.html"> Cornell Library Historical Mathematics Monographs</a>. [ html version: <a href="../../Peirce1853.html" title="">Peirce1853</a> ]<br> </span></li>
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<li><span style="font-family: times new roman,times,serif"></span><span style="font-style: italic; font-family: times new roman,times,serif">Concise Dictionary of Scientific Bibliography</span><span style="font-family: times new roman,times,serif">, Charles Scribners Sons, New York, 1981:540.</span></li>
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<li><span style="font-family: times new roman,times,serif">Agassiz, Louis. </span><font face="Times New Roman, Times, serif"><big style="color: #000000"><small> </small></big> </font><big><small><font style="color: #000000" face="Times New Roman,Times">(Louis Agassiz, <em>ESSAY ON CLASSIFICATION</em>, Ed. E. Lurie, Belknap Press, Cambridge, 1962:127-128. <br> </font></small></big></li>
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<li><span style="font-family: times new roman,times,serif">Bretagnon, P and Jean-Louis Simon, </span><em style="font-family: times new roman,times,serif">Planetary Programs and Tables from -4000 to +2800</em><span style="font-family: times new roman,times,serif">, Willman-Bell, Inc. Richmond, 1986.</span></li>
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<li><font face="Times New Roman, Times, serif">Harris, John N. </font><font face="Times New Roman, Times, serif"> "<a href="../../sbb7a.html">Projectiles, Parabolas, and Velocity Expansions of the Laws of Planetary Motion</a> " <em>JRASC</em>, Vol 83, No. 3, June 1989:207-218.</font></li>
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<li><big><small><font style="color: #000000" face="Times New Roman,Times">Kappraff, Jay. </font></small></big><font face="Times New Roman,Times"><em>CONNECTIONS : The Geometric Bridge between Art and Science</em><span style="font-weight: bold">,</span> McGraw-Hill, Inc., New York, 1991:89-96.</font></li>
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<li><font face="Times New Roman, Times, serif">Thomson, Sir D'Arcy Wentworth. <em>On Growth and Form</em>, Cambridge University Press, Cambridge 1942; Dover Books, Minneola 1992.</font></li>
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<li><font face="Times New Roman, Times, serif">Cook, Sir Theodore Andrea.<em style="font-weight: bold"> </em><em>The Curves of Life</em>, Dover, New York 1978; republication of the London (1914) edition.</font></li>
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<li><font face="Times New Roman, Times, serif">Mosely, Rev. H. "On the geometrical forms of turbinated and discoid shells,"<em> Phil. trans</em>. Pt. 1. 1838:351-370.</font></li>
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<li><font face="Times New Roman, Times, serif">Lurie, E. (Ed.) <em>Essay On Classification</em>, Belknap Press, Cambridge 1962:128.</font></li>
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<li><font face="Times New Roman, Times, serif">Church, Arthur Harry. <em>On The Relation of Phyllotaxis to Mechanical Law,</em> </font><font face="Times New Roman, Times, serif">Williams and Norgate,<em> </em>London 1904; also: <a href="http://www.sacredscience.com/">http://www.sacredscience.com</a> (cat #154).</font></li>
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<li><font face="Times New Roman, Times, serif">Colman, Samuel. <em>Nature's Harmonic Unity,</em> Benjamin Blom, New York 1971. Also:<br> </font></li>
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<li><font face="Times New Roman, Times, serif">___________ <span style="font-style: italic">Harmonic Proportion and Form in Nature, Art and Architecture</span>, Dover, Mineola, 2003.<br> </font></li>
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<li><font face="Times New Roman, Times, serif">Hambidge, Jay.<span style="font-weight: bold"> </span><em>Dynamic Symmetry</em>,</font><font face="Times New Roman, Times, serif" size="+0"><em></em>Yale University Press, New Haven 1920:16-18.</font></li>
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<li><font face="Times New Roman, Times, serif">It is necessary to acknowledge the many positive strides made during the last 25 years, especially Roger V. Jean's <span style="font-style: italic">Phyllotaxis: A systemic study in plant morphogenesis</span> (1994) which by virtue of its scale and scope invokes the same admiration reserved for Sir D'Arcy Wentworth Thompson's <span style="font-style: italic; font-weight: bold">On Growth and Form</span> (1917) and similar major works.<br> </font></li>
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<li><font face="Times New Roman, Times, serif">Jean, Roger V. <span style="font-style: italic">Phyllotaxis: A systemic study in plant morphogenesis</span>, Cambridge University Press, Cambridge 1994.<span style="font-style: italic"></span></font></li>
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<li><font face="Times New Roman, Times, serif">__________ </font><font face="Times New Roman, Times, serif"> <span style="font-style: italic">Mathematical Approach to Pattern and Form in Plant Growth</span>. Wiley, New York (1984).</font></li>
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<li><font face="Times New Roman, Times, serif">Kappraff, Jay. "The Spiral in Nature, Myth, and Mathematics" in <span style="font-style: italic"> Spiral Symmetry, </span>Eds. István Hargittai and Clifford A. Pickover, World Scientific, Singapore, 1992.</font></li>
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<li><font face="Times New Roman, Times, serif">__________"The relations between mathematics and mysticism of the golden mean through history." In <span style="font-style: italic">Fivefold Symmetry</span>, ed. I. Hargittai. World Scientific,</font><font face="Times New Roman, Times, serif"> Singapore,</font><font face="Times New Roman, Times, serif"> 1992: 33-65.<br> </font></li>
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<li><font face="Times New Roman, Times, serif">Stewart, Ian. <span style="font-style: italic">What Shape is a Snowflake</span>, Weidenfeld & Nicholson, London, 2001.<br> </font></li>
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<li><font face="Times New Roman, Times, serif">Ghyka, Matila C.<span style="font-style: italic"> The Geometry of Art and Life</span>, Dover Publications, New York, 1977.<br> </font></li>
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<li><font face="Times New Roman, Times, serif">Casey, George. </font><small><big><span style="font-family: times new roman,times,serif"> </span><span style="font-style: italic; font-family: times new roman,times,serif">The Great Barrier Reef</span><span style="font-family: times new roman,times,serif">, dir. Dr. George Casey, Science museum of Minnesota, 2001. </span></big></small><small><big><span style="font-family: times new roman,times,serif"></span></big></small></li>
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<li><small><big><span style="font-family: times new roman,times,serif">Knudson, Peter</span></big></small><span style="font-family: times new roman,times,serif"> and David Suzuki</span><em style="font-family: times new roman,times,serif">,</em><span style="font-family: times new roman,times,serif"> </span><span style="font-family: times new roman,times,serif"><em>Wisdom of the Elders, </em>University of British Columbia Press, Vancouver<em>,</em></span><span style="font-family: times new roman,times,serif"> 1992.</span></li>
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<li><font face="Times New Roman, Times, serif">Menut, Albert D. and Alexander J. Denomy, <em>Le Livre du ciel et du monde</em>, University of Wisconsin Press, Madison 1968.</font></li><span style="font-family: times new roman,times,serif"><br> </span>
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<font face="Arial">Copyright © January 21, 2007. </font><font face="Helvetica, Arial, sans-serif">Last updated: March 13, 2007. </font><font face="Arial">John N. Harris, M.A.(CMNS). </font>
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<p style="text-align: justify"><font face="Arial"><font size="-1"><a href="../../index.html">RETURN TO SPIRA SOLARIS.CA</a><br> <br></font></font></p>
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<p style="text-align: justify"><font color="#000000" face="Arial,Helvetica"><font color="#000000"><small><font face="Helvetica, Arial, sans-serif"><big><big><big><big>ASTRONOMICAL FRAMEWORK</big></big><strong> </strong></big></big></font></small></font></font></p>
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<p style="text-align: justify"><font color="#808000"><font color="#000000"><small><font face="Helvetica, Arial, sans-serif"><big><big><big><big><small>QUANTIFICATION AND QUALIFICATION I</small></big></big></big></big></font></small></font></font><br> <font color="#000000" face="Arial,Helvetica"><font color="#000000"><small><font face="Helvetica, Arial, sans-serif"><big><big><strong> THE GOLDEN SECTION AND THE STRUCTURE OF THE SOLAR SYSTEM</strong></big></big></font></small></font></font></p>
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<p style="text-align: justify"><font color="#000000" face="Arial,Helvetica"><font color="#000000"><small><font face="Helvetica, Arial, sans-serif"><big><big><a href="../../spirasolaris.html"><small>Spira Solaris: Form and Phyllotaxis</small></a><strong><br> <br></strong></big></big></font></small></font><a href="../../sbb4a.html"><font face="Arial,Helvetica">I Bode's Flaw</font></a><font face="Arial,Helvetica"> <br> </font></font><font face="Helvetica, Arial, sans-serif">http://www.spirasolaris.ca/sbb4a.html</font><font color="#000000" face="Arial,Helvetica"><font face="Arial,Helvetica"><font color="#000000"><font face="Helvetica, Arial, sans-serif"> </font><br> Bode's "Law" - more correctly the Titius-Bode relationship - was an<em> ad hoc</em> 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.<br> </font></font><br> <a href="../../sbb4b_07.html">II The Alternative</a><font face="Helvetica, Arial, sans-serif"><br> </font></font><font face="Helvetica, Arial, sans-serif">http://www.spirasolaris.ca/sbb4b_07.html</font><font color="#000000" face="Arial,Helvetica"><br> 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 <em>ad hoc</em> methodology and the use of mean heliocentric distances alone.</font></p>
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<div style="text-align: justify"></div>
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<p style="text-align: justify"><font color="#000000" face="Arial,Helvetica"><a href="../../sbb4c_07.html">III The Exponential Order</a><font face="Helvetica, Arial, sans-serif"> <br> </font></font><font face="Helvetica, Arial, sans-serif">http://www.spirasolaris.ca/sbb4c_07.html</font><font color="#000000" face="Arial,Helvetica"><font face="Helvetica, Arial, sans-serif"><br> </font>The constant of linearity for the resulting planetary framework is the ubiquitous constant <em>Phi</em> known since antiquity. Major departures from the theoretical norm are the ASTEROID BELT, NEPTUNE, and EARTH in a resonant <em>synodic</em> position between VENUS and MARS. Fibonacci/Golden Section Resonances in the Solar System. <br> </font></p>
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<p style="text-align: justify"><font face="Helvetica, Arial, sans-serif"><a href="../../dfmilkyway.html">Spira Solaris and the plan-view of the Milky Way</a></font><em><font face="Helvetica, Arial, sans-serif">.<br> </font></em><font face="Helvetica, Arial, sans-serif">http://www.spirasolaris.ca/dfmilkyway.html<br> </font></p>
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<p style="text-align: justify"><font face="Helvetica, Arial, sans-serif">Last updated: March 13, 2007<br> </font></p>
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<p style="text-align: justify"><font color="#808000"><font color="#000000"><small><font face="Helvetica, Arial, sans-serif"><big><big><big><big><small>QUANTIFICATION AND QUALIFICATION II</small></big></big><strong><big><big> <br> </big></big>GOLDEN SECTION SPIRALS IN NATURE, TIME AND PLACE<br> </strong></big></big></font></small></font></font></p>
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<p style="text-align: justify"><font color="#808000"><font color="#000000"><small><font face="Helvetica, Arial, sans-serif"><big><big><big>THE THREE-FOLD NUMBER</big><strong><br> </strong></big></big></font></small></font></font></p>
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<div style="text-align: justify"></div>
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<p style="text-align: justify"> <font color="#000000" face="Helvetica, Arial, sans-serif"><big><a href="../../sbb4d2b.html">IVd2b Spira Solaris and the 3-Fold Number</a></big></font><br> <font face="Helvetica, Arial, sans-serif">http://www.spirasolaris.ca/sbb4d2b.html</font></p>
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<div style="text-align: justify"></div>
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<p style="text-align: justify"><font color="#000000" face="Arial,Helvetica">The Spiral of Pheidias; Pheidian/Golden Spirals Defined.<br> Pheidian Spirals and the Chemical Elements.</font></p>
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<div style="text-align: justify"></div>
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<p style="text-align: justify"><font color="#000000" face="Arial,Helvetica"><a href="../../hambidge1a.html">Notes on the Logarithmic Spiral (Jay Hambidge; R. C. Archibald)</a> </font><font face="Helvetica, Arial, sans-serif"><br> http://www.spirasolaris.ca/hambidge1a.html</font><font color="#000000" face="Arial,Helvetica"> <br> R.C. Archibald's <a href="../../rcarchibald.html">Golden Bibliography</a>.</font></p>
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<div style="text-align: justify"></div>
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<p style="text-align: justify"><font color="#000000" face="Arial,Helvetica">The Whirlpool Galaxy (M51) (<a href="../../m51abw.html">BW: 100kb</a>)<font face="Helvetica, Arial, sans-serif"> <br> </font></font><font face="Helvetica, Arial, sans-serif">http://www.spirasolaris.ca/m51abw.html </font></p>
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<div style="text-align: justify"></div>
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<p style="text-align: justify"><font color="#000000" face="Arial,Helvetica">The Whirlpool Galaxy (M51)<a href="../../m51a.html"><u>(Colour: 200kb</u>)</a><font face="Helvetica, Arial, sans-serif"><br> </font></font><font face="Helvetica, Arial, sans-serif">http://www.spirasolaris.ca/m51.html<br> </font><br> <font color="#000000" face="Arial,Helvetica"><a href="../../sbb4d2bs.html">The Phyllotaxic approach to the structure of the Solar System of Benjamin Pierce</a> (1750)</font></p>
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<br> <font face="Helvetica, Arial, sans-serif"><big><big> THE PHEIDIAN PLANORBIDAE</big></big></font>
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<p style="text-align: justify"><font color="#000000" face="Helvetica, Arial, sans-serif"><big><a href="../../sbb4d2c.html">IVd2c Spira Solaris and the Pheidian Planorbidae</a>. </big><br> </font><font face="Helvetica, Arial, sans-serif">http://www.spirasolaris.ca/sbb4d2c.html</font><br> <font color="#000000" face="Arial,Helvetica"> Applied to Nautiloid spirals, Ammonites, Snails and Seashells.<br> </font><font color="#000000" face="Helvetica, Arial, sans-serif">The Phedian Planorbidae in Astronomical context; Orbital velocity, Mass and Angular Momentum.</font><br> </p>
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<div style="text-align: justify"></div>
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<p style="text-align: justify"><font color="#000000" face="Arial,Helvetica"><a href="../../sbb4d2cs.html"> Ammonites and Seashells</a> (Beginning excerpt).<br> </font><font face="Helvetica, Arial, sans-serif">http://www.spirasolaris.ca/sbb4d2cs.html</font></p>
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<div style="text-align: justify"></div>
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<p style="text-align: justify"><font color="#000000" face="Arial,Helvetica"><a href="../../animation1.html">Whirling Rectangles and The Golden Section (Animation I)</a><br> </font><font face="Helvetica, Arial, sans-serif">http://www.spirasolaris.ca/animation1.html</font></p>
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<div style="text-align: justify"></div>
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<p style="text-align: justify"><font color="#000000" face="Arial,Helvetica"><a href="../../animation2.html">Whirling Rectangles and The Golden Section (Animation II)</a><br> </font><font face="Helvetica, Arial, sans-serif">http://www.spirasolaris.ca/animation2.html</font></p>
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<p style="text-align: justify"><font color="#000000" face="Helvetica, Arial, sans-serif"></font><font face="Helvetica, Arial, sans-serif"><strong>Appendix</strong>: <a href="../../sbb4d2ca.html">The Matter of Lost Light</a>.</font><font color="#000000" face="Arial,Helvetica"><br> The understanding of Canon Mosely and Sir D'Arcy Wentworth Thompson.<br> </font></p>
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<font color="#808000"><font color="#000000"><small><font face="Helvetica, Arial, sans-serif"><big><big><big>RELATED PAPERS AND TOOLS</big></big></big></font></small></font></font>
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<p style="text-align: justify"><font color="#000000" face="Arial,Helvetica"><a href="../../sbb7a.html">Velocity Expansions of the Laws of Planetary Motion</a>. <br> </font><font face="Helvetica, Arial, sans-serif">http://www.spirasolaris.ca/sbb7a.html</font></p>
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<p style="text-align: justify"><font color="#000000" face="Arial,Helvetica"><span style="font-weight: bold">Abstract</span><br> Kepler's Third Law of planetary motion: T<sup>2</sup> = R<sup>3</sup> (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 = Vi<sup>2</sup> and T<sup>2</sup> = R<sup>3</sup> = Vi<sup> 6</sup>. The first relation - found in Galileo's last major work, the <em>Dialogues Concerning Two New Sciences</em> (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.</font></p>
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<blockquote>
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<p><font color="#000000" face="Arial,Helvetica">Note: </font><font color="#000000" face="Arial,Helvetica">This paper (which deals with the resurrection of the<em> Fourth Law of Planetary Motion</em>, i.e., the velocity component) was written north of the 70th parallel during the Summer of 1988.<br> It was subsequently published in the<em> Journal of the</em> <em>Royal Astronomical Society of Canada</em> (JRASC) the following year. It is reproduced here with the permission of the Editor of the Journal.<br> </font></p>
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</blockquote>
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<p style="text-align: justify"><font color="#000000" face="Arial,Helvetica"><a href="../../time1.html">Times Series Analysis.</a></font><font face="Helvetica, Arial, sans-serif"><br> http://www.spirasolaris.ca/time1.html</font><font color="#000000" face="Arial,Helvetica"> <br> 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.</font></p>
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<p style="text-align: justify"> <font color="#000000" face="Arial,Helvetica"><a href="../../times2.html">Time Series Graphics</a></font><font face="Helvetica, Arial, sans-serif"><br> http://www.spirasolaris.ca/times2.html<br> </font><font color="#000000" face="Arial,Helvetica">Examples of chaotic and resonant planetary relationships in the Solar System and a possible link with Solar Activity.</font></p>
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<p style="text-align: justify"><font face="Arial">Copyright © 1997. John N. Harris, M.A. (CMNS). This section last updated on July 28, 2004.</font></p>
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