Spira Solaris Archytas-Mirabilis Part IV

The present paper is a continuation of the investigation begun in previous sections concerning the spiral form in Nature, Time, and Place. Here, although more ancient roots undoubtedly exist the emphasis is now concentrated primarily on the past three centuries--roughly from the time of Carl Linnaeus (1707-1778) onward. But it is not an historical analysis per se, nor is it a commentary on the momentous changes that took place during this tortuous period. At least not directly, though a darker, negative side of the matter also surfaces as the investigation proceeds. This development itself is perhaps surprising since it is linked to a specialized yet apparently innocuous topic, namely the spiral formations evident in ammonites and shells. On the other hand, however, it is not quite so surprising, or entirely unexpected when the dynamics of the matter become apparent. 
    Nevertheless, on a more positive note the present study was precipitated somewhat fortuitously by the format adopted by Simon Winchester for his recent best-selling work:The Map that Changed the World: William Smith and the Birth of Modern Geology (2001). The format followed in this publication was explained by the latter as follows:1

Incorporated in eighteen of the nineteen chapter openings (including those of the prologue and the epilogue) will be found small line drawings of Jurassic ammonites, long-extinct marine animals that were so named because their coiled and chambered shells resembled nothing so much as the horns of the ancient Egyptian ram-god, Ammon. Soun Vannithone's drawings of these eighteen specimens are placed in the book in what I believe to be the ammonites' exact chronological sequence. This means that the book's first fossil, Psiloceras planorbis, which illustrates the prologue, is the oldest ammonite, and is to be found deepest down in any sequence of Jurassic sediments; by the same token the final fossil, Pavlovia pallasioides, comes from a much higher horizon, and is very much younger. Much like the epilogue it illustrates, it was fashioned last. It must be said, though, that anyone who flips rapidly from chapter to chapter in the hope of seeing a speeded-up version of the evolutionary advancement of the ammonite will be disappointed: Ammonites floating, pulsating, slow-swimming beasts that were hugely abundant in thc warm blue Jurassic seas do not display any conveniently obvious changes in their Idealures, they neither become progressively smaller with time, nor do they become larger; their shells do not become more complex, or less. True, some ammonites with very ridged shells do indeed evolve into smoother-shelled species over the ages, but these same creatures then become rougher and more ridged again as time wears on, managing thereby to confuse and fascinate all who study them. Only studies of ammonites from successive levels will reveal sure evidence of evolutionary change, and such study is too time consuming for the chance observer. Ammonites are, however, uniformly lovely; and they inspired William Smith: two reasons good enough, perhaps, for including them as symbols both of Smith's remarkable prescience and geological time's amazing bounty. However: eighteen ammonites and nineteen chapter openings? There is one additional illustration, of the microscopic cross-section of a typical oolitic limestone, which I have used to mark the heading for chapter 11. Since this chapter is very different in structure from all the others, and since much of its narrative takes place along the outcrop of those exquisitely lovely, honeycolored Jurassic rocks known in England as the Great Oolite and the Inferior Oolite, it seemed appropriate and reasonable to ask the legions of ammonites, on just this one occasion, to step or swim very slowly to one side.

: Psiloceras planorbis.... (Simon Winchester, The Map that Changed the World, Harper Collins, New York, 2001: ix-x)

A fascinating set of "Pheidian" spirals it would seem, and all carefully laid out in planview in addition. Then there was the line of development sketched out to feed curiosity even further--Ammon, Rams, Ramshorn Snails, Ammonites. Yes, of course! ...but planorbis? A strange name, but plentifully applied it would seem, and not only among ammonites either, but ramshorn snails and the like going back to at least the time of Linneaus (1758). And also thereafter into and beyond the early part of the Nineteenth Century, especially Say in the former period. As for the beautiful line drawings at the beginning of each chapter of The Map that Changed the World, they were that indeed, and although most were tighter spirals than Spira Solaris per se, they were nevertheless recognizable as equiangular spirals lying within the range already formulated and plotted in astronomical contexts, i.e., from the inverse velocity spiral Phi 1/3 to the planetary period spiral, Phi 2.

Figure 1. The Growth factor for Spira Solaris

Figure 1. Spira Solaris, Growth factor k = Phi 2
(For more on the Capacious Manitoba Ramshorn snail see Figure 13 below)

Thus further Pheidian spirals with growth factors between 1.1739850 and 2.61803398874 : 1, five of which had already been generated--two for the Periods, two for the Distances and one for the inverse Velocity. Up to this point, however, the emphasis had remained with the equiangular spiral based on Phi 2 in view of its all-inclusive nature on one hand and the confusion four additional spirals might have occasioned on the other. Now there was a practical reason for widening the range, though remnants of the earliest ammonite, Psiloceras planorbis provided insufficient definition to determine the fundamental spiral, at least with any degree of certainty. This said, however, it was still apparent that while the associated spiral in this instance was not Spira Solaris, it was nevertheless possibly related, for the Distance equivalent (i.e., the equiangular spiral k = Phi 4/3 with a growth factor of 1.899547627 : 1) did in fact provide a limited fit. Enough of an association, in fact, to give impetus to a more detailed investigation--one that was to have a number of unexpected results.

Although the assignment of the equiangular spiral k = Phi 4/3 to Psiloceras planorbis remains uncertain, the naming and classification of various "planorbidae" through the Eighteenth, Nineteenth and Twentieth Centuries--especially with respect to snails--opened up a fascinating and potentially useful line of inquiry. The first order of business here was obviously to conduct a survey to determine whether or not the "planorbidae" were indeed Pheidian in the sense stated in the previous section ("equiangular spirals based on the constant Phi raised to any power, whether integer, fractional part or any number whatsoever") and secondly, to establish whether or not the latter possessed the suspected relationship to Spira Solaris and associated spirals. For this purpose the range covered by the original five spirals mentioned earlier was extended to include further exponential "thirds" and "sixths." The former (in the simplest sense) being the natural continuation of the above mentioned range k = Phi 1/3 through k = Phi 6/3 with the insertion of the "missing" spiral k = Phi 5/3 between Phi 4/3and Phi 2; followed by the inclusion of k = Phi 7/3k = Phi 8/3 and finally, k = Phi 9/3 for a provisional upper limit. Next, the intermediate "sixths" were inserted to provide a test range that extended from k = Phi 1/6 to k = Phi 18/6 (growth factors 1.08450588 to 4.236067978) resulting in some 18 pheidian spirals--likely more than sufficient for ammonites, though clearly inadequate for all shells.

   Here it should be emphasized that this preliminary range was neither haphazardly nor arbitrarily determined. It was in fact specifically predetermined by what might best be called the test dynamics of the matter, with emphasis not only on the original five Spira Solarii, but also on the "thirds" on either side of Spira Solaris and possibly beyond. The rationale behind this selection will become apparent later; as it was, for spirals where the growth constant k was larger than unity standard computations involving six revolutions with 360 data points per revolution were employed--thus 6 complete cycles with a total of 2160 data points for each equiangular spiral. Where the growth constant k was less than unity additional cycles were added as the growth factors diminished. Nevertheless, the preparation of the test set was hardly a difficult task--the basic  mathematical elements have long been known, e.g., as described in detail by Sir D'Arcy Wentworth Thompson2 (On Growth and Form, 1917,1942, 1966 and 1992); by H.E. Huntley 3 (The Divine Proportion, 1970--my own introduction to the topic), by Jay Kappraff 4 (Connections, 1991) and for spreadsheet users, the PHB Practical Handbook of Spreadsheet Curves and Geometric Construction (1993) by Deane Arganbright.5 For the present analysis, however, each spiral was further expanded to include two intimately related forms locked in position and scale with respect to each other. As will be explained later, these double forms were rigidly determined from a fixed mathematical relationship. Thus, for example, the single equiangular spirals for the "Spiral of Pheidias" (Schooling 1914) and Spira Solaris were each joined by their respective additional pairs to form associated triple sets as shown below, thus raising the original test set to 54 pheidian spirals with more likely to be generated on an as-required basis.

Fig. 1a The Triple spiral configurations for Pheidas and Spira Solaris

Figure 1a. Triple spiral configurations for Pheidas (left) and Spira Solaris (right)

In the above, b represents the standard format, a and c the dual additions--the latter configuration identical to the former, but without the cross-reference lines. Both the origin and the technical details of these dual configurations will be supplied later, but in passing the extended forms may appear vaguely familiar to some readers, especially those acquainted with Sir Theodore Andrea Cook's The Curves of Life (1914:64, 278) 6 and Samuel Colman's Nature's Harmonic Unity (1911:115) 7. As for their application in the present survey, their occurrence was a continual series of surprises, for both dual formats seem to be apparent in certain classifications, i.e., form a appears to be a prominent feature among the more elongated Halitodae, while c is also evident among certain shells with smaller growth factors, the Spiral of Pheidias included. One other point of interest (though not pursued further here) is the change evident in c for the increase in growth factor between Phi and Phi 2. For the curious, the "natural" changes in form that accompany higher powers of Phi in this configuration also provide further room for thought, as shown below:

Figure 1b. Dual spiral configurations

Figure 1b. Dual Pheidian configurations:  k = Phi 2 to k = Phi 16 plus k = Phi 32

For the initial survey the pheidian thirds and sixths were applied in a standard manner, and apart from uniform scaling and rotation as required, the test spirals remained unmodified throughout. Lastly, the generated data were converted to standard graphical formats, rendered translucent to aid scaling and fitting, and then passed to a suitable platform for the testing phase. The software of choice here was XARA-X, which, as it turned out, was also capable of producing the output graphics and associated figures.
   Before describing the latter a few words concerning the initial testing phase and anticipated difficulties are perhaps in order. It was realized from the start that it is one thing to attempt to fit a two-dimensional spiral to drawings of three-dimensional objects, and yet another to attempt the same procedure with photographs, which may or may not have been influenced by perspective effects; optical systems, focal lengths, depths of field, and also quite possibly artistic licence. Some part, all, or none of which might also get carried over to line drawings. Then there were the many problems arising from natural growth itself to be taken into consideration, with no truly perfect spiral expected to be encountered and minor departures anticipated in certain cases. Fortunately, many ammonites possess relatively simple symmetrical forms, i.e., spiral growth largely confined to two dimensions without spires or appendages extending away from the primary spiral (e.g., Figures 1e, 1b2, 1c, and 1d-1d3 below)
   Nevertheless, the investigation--even for the relatively flat and largely symmetrical ammonites--began with no great expectations, but happily with a wealth of available material. And, as it turned out, Soun Vannithone's accurate plan-view line drawings provided both an ideal starting point and an excellent training ground; witness:

Ammonites and the Pheidian Planorbidae

Fig. 1e. Ammonites 69 and the Pheidian Planorbidae ( k = Phi  5/6 )

 I will not go further into the ammonite phase of the testing here except to say that overall (in spite of the complexity of the matter and the variations encountered) the initial ammonite survey provided sufficient information to yield positive answers to both the first and second questions posed. Namely, that the spiral configurations examined could indeed be considered in pheidian terms, and secondly, that the examples tested were also sufficiently relatable to the Pheidian sixths and thirds associated with Spira Solaris to merit the title Pheidian planorbidae, e.g., Figure 1b2,  k = 6/3 = Phi 2 (c: single, and b: dual Spira Solaris ),61 Figure 1C: k = Phi  to the 3/3,  4/3,  5/3,  6/3 powers respectively (five ammonites from: Ammonites et autres spirales by Hervé Châtelier 61-65) and Figure 1d: k = Phi to the 3/3 power (The Spiral of Pheidias), ammonite from Lower Jurassic Ammonites by Christopher M. Pamplin.66  Figure 1d2: k = Phi  4/3 and Figure 1d3: k = Phi  5/3 are from Jurassic ammonites and fossil brachiopoda 67  by Jean-ours and Rosemarie Filippi 67.

Figure 1b2. Ammonites and the Dual Pheidian configuration

Figure 1b2. Ammonitesr 61 and the Dual Pheidian configuration ( k = Phi 2)

Figure 1c. Ammonites and Fig. 1c. Ammonites and the Pheidian Planorbiae I

Figure 1c. Ammonites 62-65 and the Pheidian Planorbidae I ( k = PhiPhi 4/3 Phi 5/3 and  Phi 2 ):

Figure 1d.

Figure 1d. Ammonites 66 and the Pheidian Planorbidae II (k = Phi)


  Fig. 1d2. Ammonites 67 and the Pheidian Planorbidae III ( k = Phi 4/3)

Ammonites and the Phedian Planorbidae IIv

Fig. 1d3. Ammonites 68 and the Pheidian Planorbidae IV ( k = Phi 5/3)

Thus the Phedian Planorbidae as applied to ammonites from an initial survey--one small inroad into a complex subject with accompanying dynamic, temporal and historical overtones that all appeared to merit further examination.
   Next--based on the positive indicators gained from the ammonite phase the testing moved on to "planorbid" snails, the treatment of which will also be deferred until later--not because of its simplicity, but the exact opposite--its undoubted complexity (see Figure 13 below).
   Finally, from these two bases the survey naturally turned to the more varied and extensive range of spirals found among seashells.

Part of the third phase of the testing is shown below in Figure 2. Although the selection includes some better known shells, it also omits others--primarily to emphasize certain points in each of the selected cases.To maximise relevant information the examples are also shown against a background plot of pheidian growth factors along the y-axis with the corresponding equiangles of the associated pheidian spirals along the x-axis for the successive exponential thirds from 1 through 9. For the study (following Mosely 1838) 8 the growth factor itself was taken to represent the "characteristic number" (n) of the associated "primary" spiral which was also the parameter k.
    Here the reader should be aware that little of what is presented below is new per se, nor is it presented as such here. Many of the assignments, although neglected at present, were obviously known in earlier times to one degree of accuracy or another, as the tables of related data for shells in Thompson's On Growth and Form (1917) clearly attest 9, e.g., the values determined by Nauman (1848, 1849),10 Muller (1850,1853)11and Macalister (1870)12.
    As for the primary spiral assignments for the nine shells shown in Figure 2, they proceed in due order from the lowest pheidian planorbidae (k = Phi 1/3) to the largest of this group, k = Phi 9/3.
Briefly, the assignments are as follows:
The above represent a small selection from the test survey. Although far from inclusive, the range for the shell phase of the survey extended from the tighest spiral (k = Phi 1/12; n = 1.040915886) out to Anadara brasiliana (Arc), k = Phi 10 (n = 122.991869381). Other shells tested included Terebra, k = Phi 1/6 (n = 1.083505882); Acropora, k = Phi 1/3 (n = 1.173984997);Turritella duplicata, (after Mosely, k = Phi 1/3; n = 1.173984997); Trochus (varied), k = Phi 1 (n = 1.618033989) and also one or two of the better known shells, e.g., Thatcheria Mirablis, k = Phi 7/6 (n = 1.753149344). There were additional assignments, but to "concatenate without abruption" (as Dr. Johnson was want to put it) would likely disrupt the general thrust of the paper, which is not the assignment of pheidian spiral forms to shells per se, but the dynamic, historical, and general implications. Moreover, in so much as a full description of the various assignments shown here should rightly follow after the dynamics of the matter are introduced the following descriptions are limited to a few notes concerning some of the major points of interest. Similarly, discussion of the Haliotidae (Abalone; excellent test subjects because of their generally flat shapes and well-defined open spiral forms) is also deferred until later in view of the possible relationship between this type of shell and the complexities inherent in phi-related "whirling rectangles" (esp. Haliotis parva; k = Phi 4 ; see A4 below).

Figure 2. Phedian Planobidae I

Fig. 2 The Pheidian Planorbidae. Thirds: Growth Factors/Expansion Rates 1.174 to 4.236

Figure 2A. Telescopium telescopium (Linnaeus 1758)
Figures 2A
and 2G provide examples of the kind of fit that can be obtained where high quality cross-sections are available, but this is not the only reason for their inclusion here. Figure 2A is also a miniture representation of the left vertical (Y) axis of the figure itself, which gives the successive values of the characteristic numbers for the nine sequential pheidian planorbidae. Expressed in thirds, the range extends from k = Phi 1/3 out to k = Phi 9/3. As shown in Figure 3 below, the superimposition of one half of the pheidian spiral k = Phi 1/3reinforces the applied scale while at the same suggesting an alternative approach to spiral assigments. Here matters become more complex, however, for there appears to be a three-dimensional pheidian relationship between the widths and heights of certain shells. So much so, in fact that it serves to aid the assignment process, but more on this useful variation later. 

Fig.3. Cross-section of Telescopium telescopium

Fig. 3. Cross-section of Telescopium telescopium (Linnaeus 1758)
with the Pheidian Spiral and vertical scale k = Phi

Figure 2B. Conus princeps f. lineolatus Linnaeus 175814 (above); Conus mercator (Linnaeus 1758)23 and Conus ammiralis f. hereditarius DA MOTTA, 198723 (below)
Were it not for the planview of this type of shell it would difficult to assign a spiral at all, yet as it happens--thanks to the superb collection and presentation of Conidae made available on the Internet by Giancarlo Paganelli (http://www.coneshell.net) sufficient examples exist to assign the primary spiral k = Phi 2/3 (growth factor: 1.378240772 ) with a fair degree of confidence. Although Conus tulipa 24 is an exception (at least from the test perspective; primary spiral: k = Phi 5/3) further examples show a number of intriguing variations, including possibly related scaling and multi-dimensional effects, varying from pronounced single line (e.g., Conus doreensis), double line (Conus mercator; Figure 4) and multiple spiral markings (Conus ammarilis; Figure 4b and Conus planorbis; Figure 4c). A more diffuse but similar example is Conus bandanus). Here the scale of the spiral applied in the vertical plane to the front of the shell is unchanged from that used for the planview. On the right (rear), still fixed with respect to the tip of the spire, the scale is increased until the outer whorl meets the same spiral markings that unwind on the other side of the shell. Similar test procedures involving 1, 2 and 3-whorl extensions are helpful in the case of more elongated Conidae, e.g, as applied in Figure 4b (a single whorl extension applied to Conus ammiralis). Though pertinent to the survey, the well defined "pheidian" markings on the three examples below are perhaps unusual; nevertheless the same basic technique remains applicable to other Conidae with more diffuse patterns, if not greater complexity.

Fig.4. Conus   Mercator (Linnaeus 1758)

Fig.4. Conus  Mercator Linnaeus 1758. Single spiral, k = Phi 2/3

Fig.4b. Conus  ammiralis (Linnaeus 1758)

Fig.4b. Conus ammiralis f. hereditarius DA MOTTA, 1987. Single spiral, 1-whorl extension; k = Phi 2/3

conus planorbis_vitu

Fig.4c. Conus planorbis f. vitulinus HWASS in BRUGUIÈRE, 1794. Single spiral; k = Phi 2/3

Figure 2C. Architectonica perspectiva (Linnaeus 1758) and Similar Shells
Figure 2C is shown in inverted planview for two major reasons. Firstly, a trio of like shells graced the dust cover of the 1942 edition of Sir D'Arcy Wentworth's On Growth and Form in this exact representation. The reason seems clear enough from the inverted "perspective" of the associated spiral (see Figure 5 below; perhaps the latter was also influenced by Aristotle, if not the three-fold number--"Said Aristotle, prince of philosophers and never-failing friend of truth: All things are three"). Here, one should also note that the "ratio of breadth of consecutive whorls" in Thompson's tabular data for shells of this general type, i.e., Solarium trochleare is given as 1.62 25, thus the Golden Section to two decimal places. The degree of accuracy is low, but hardly conclusive proof that a better value was not known. In fact there is very good reason (as explained in detail below) to believe that both Canon Mosely and Sir D'Arcy Wenthworth Thompson were intimately acquainted with the not only the Phedian planorbidae per se, but also the dynamics of the matter.
In the meantime, Figure 5 below emphasizes not only the distinct markings and underside perspective view, but also the fit in standard and open double form with respect to the top markings of the shell.

Fig.5 Architectonica perspectiva

Fig.5. The underside and top of Architectonica perspectiva;
Single spiral (B and C);
open double spiral (A); each k = Phi 1

Many shells with this assignment appear to be "golden" in more ways than one (see Figure 6 below), adding further merit to the observation attributed to Aristotle and Ovid30 in the last section, i.e.,

Said Aristotle, prince of philosophers and never-failing friend of truth : .........
All things are three; The three-fold number is present in all things whatsoever.
Nor did we ourselves discover this number, but rather natures teaches it to us...

Fig. 6.  Liguus virgineus (Linnaeus, 1758)

Fig. 6.  Liguus virgineus (Linnaeus, 1758)31
Fixed Inset scale, 1-whorl extension; k = Phi

Figure 2D. Harpa kajiyamai (Rehder 1973) above; Harpa goodwini (below)
The assignment of the primary spiral remains uncertain for Harpidae due to the small number tested to date. Nevertheless, the initial assignment (in spite the latter) would appear to be the pheidian spiral k = Phi 4/3 (growth factor: 1.899547627). The example of Harpa goodwini shown below (Defunct source: Guido T. Poppe,26 Conchology http://www.conchology.uunethost.be/ ) shows the planview fit (top and underside views) for the single spiral in question; as in the case of other intriguing examples (e.g., Haliotidae and Conidae) one could no doubt spend a long time working to decode the complex formations also evident here.

Fig. 7. Harpa goodwini (Top and underside)

Fig. 7.  Harpa goodwini (Top and underside) with the single Pheidian Spiral k = Phi 4/3

Figure 2H. Haliotis brazeri (Angus, 1869); unridged variant below
Unlike the double spiral
applied to the ridged form of Haliotis brazeri in Figure 2H, the single spiral applied to the smoother variant below 20a follows the central ridge, but in both cases the primary pheidian spiral nevertheless appears to be k = Phi 8/3. D. L. Beechey (source: http://seashellsofnsw.org.au/Haliotidae/Pages/haliotis_brazieri.htm) notes that: "There is continuing debate on whether the smooth form and the spirally ridged form are separate species, or forms of the one variable species. Intergrades between the two shell forms exist (although uncommon), supporting the view that there is only one variable species. Until anatomical studies are done the issue remains unresolved."

Fig. 7b. Haliotis Brazieri

Fig. 7b.  Haliotis brazeri with the single Pheidian Spiral k = Phi 8/3

Figure 2E. Hay's Flat-whorled Snail
The availability of accurate planviews of the top and the undersides of certain shells is obviously helpful in assigning the primary spiral, but there are still additional complications where pronounced curvature is present. For the examples selected, however, a useful modifying effect became apparent during the survey, namely what might be called "pheidian three-dimension growth" for lack of a better term. For example, the double spiral configuration applied to Hay's Flat-whorled snail is not only applicable to the underside, but to some extent it also follows the curvature of the shell along the top, as in Figure 2 and also below, where it not only fits the flatter interior around the low spire, but also generally corresponds to the linear markings of the shell between the inner and outer spirals further out:

Fig. 8. Hay's Flat-whorled Snail

Fig. 8. Hay's Flat-whorled Snail17 and the Double Spiral k = Phi 5/3 

Figure 2G. Nautilus pompilus (Linnaeus 1758)
Shown above in Figure 2G, Nautilus pompilus was assigned the primary spiral and characteristic number of k = Phi 7/3 and n = 3.073532624 respectively, which is hardly new. Nor is it a particularly difficult assignment. In fact Sir D'Arcy Wentworth Thompson, having provided the value 3.00 for the characteristic number of this shell in his tables preferred to devote his attention to more difficult problems, concluding that
"The numerical ratio in the case of the Nautilus happens to be one of unusual simplicity." 22 As indeed it is, at least in comparison to many other types of shells. Shown below is an enlarged version of the same cross-section of the Nautilus, but this time with the double primary spiral k = Phi 7/3. Here, as in other examples, the relationship between the two spirals remains fixed irrespective of scaling. Thus, although the outermost part of this variant shows some departure from the shell itself (i.e., at the bottom), the inner spiral (unaltered in position from its original form, scaled in situ and locked with respect to the outer) still follows the internal syphon tube, though departing slightly from it over the final three chambers.

Fig. 9. Cross-section of Nautilus pompilus (Linnaeus 1758)

Fig. 9. Cross-section of Nautilus pompilus (Linnaeus 1758); Double Spiral k = Phi 7/3

Figure 2F. Planorbis corneus (Linneaus 1758)
The Planorbidae were one of the main targets in the survey--not only because of a particular interest in their classification, but also for technical reasons amply demonstrated by the the standardized, technical photographs in Arthur H. Clarke's The Freshwater Molluscs of Canada (1981:175).27 The latter work included the "Superfamily Planorbacea, FAMILY PLANORBIDAE (Ramshorn Snails)" described in part as "Shells small to moderately large, dextral or sinistral, flatly coiled in most species and with a very low spire" 28 --the latter point clearly useful for testing. As for further general details of the Family Planorbidae, Martin Kohl (Defunct link; originally at Freshwater Molluscan Shells: Planorbidae; http://members.aol.com/mkohl2/Planorbidae.html ) provides the following:29

Generally referred to as "Wheel Snails", "Orb Snails", or "Ramshorn snails", the Planorbidae are the largest family of aquatic pulmonate gastropods, with species present on all continents and most islands. They are sinistral in their orientation, in spite of the fact that some may appear dextral due to the spire being sunken more than the umbilicus. Many of these species harbor the larvae of parasitic worms, particularly southern and Old-World taxa. Banarescu (1990) provides the following classification, here tabulated. Many subgeneric and subspecific names are in use, so for instance, the first specimens pictured are Planorbella (Pierosoma) trivolvis trivolvis (Say, 1817). [Martin Kohl; planorbidae.html ]

The example of Planorbis Corneus used in Figure 2 is from the above source; the parent photograph is shown below as Figure 10. Many members of this family appear to be pheidian in one form or another, and like  Architectonica, Trochus, etc., many seem to be "golden" in addition--a worthy subject in its own right, though it will not be pursued further in the present survey.

Fig. 10. Planorbis Corneus (Ramshorn snail; Linnaeus 1758)

Fig. 10. Planorbis Corneus (Ramshorn snail; Linnaeus 1758)18
and Spira Solaris
( k = Phi 2)

Returning to Planorbis Corneus, the partial fitting (as opposed to the whole outline) of the Pheidian spiral k = Phi 2 is shown here for a number of reasons. Firstly, although certain planorbidae appear to show little or no discernable changes in the primary spiral e.g., Promenetus exacuous megas, Dall 1905, primary spiral: k = Phi 5/3

Fig. 11. Broad Promenetus Ramshorn snails

Fig. 11. Broad Promenetus Ramshorn snails with single spiral: k = Phi 5/3

in others the visible inner regions differ from the general outline beyond that asssociated with curvature effects and the natural overlapping of the tube. Moreover (as seen in Figure 10 above), further differences are also evident depending on which side of the shell is under consideration. Finally, though another increase in complexity, in some cases the double spiral configuration is also apparent, i.e., with respect to ridge formations, as in the case of the Great Carinate Ramshorn and Capacious Manitoba Ramshorn snails:

Fig. 12. Great Carinate Ramshorn snail

Fig. 12. Great Carinate Ramshorn snail with double spiral: k = Phi 2

Although beyond the original intent of the survey, investigation of the apparent changes from the initial (inner) spiral k = Phi 2 for the Capacious Manitoba Ramshorn snail suggest that the relatively abrupt shifts may be interpreted as downward changes, i.e., from the initial spiral k = Phi 2 through  k = Phi 11/6 to k = Phi 5/3 (or expressed in sixths for "continuity", from k = Phi  12/6 through k = Phi 11/6 to k = Phi 10/6) as shown below:

Fig. 13. Capacious Manitoba Ramshorn snail I

Fig. 13. Capacious Manitoba Ramshorn snail with double and single spirals:
= Phi 12/6 k = Phi 11/6 and k = Phi 10/6.

In Figure 13 the double spirals (though still applicable) are omitted for clarity in all but the primary assignment. The other assignments remain provisional owing to the small sample tested. As for the whys and the wherefors of the matter, here it it necessary for a brief revision of the astronomical side of the Pheidian planorbidae before moving on to a description of the generation of the double spiral itself.


Up to the mid-point of Section III representations of the Solar System were largely logarithmic, two-dimensional in form, and also generally static in nature, despite discussions concerning periods of revolution, lap cycles, planetary orbits and varying velocities. Yet the complexity of the Solar System, its endless and varying motions, its waxings and wanings, its growth and decay, its anomalies and its regularities all suggest it is something far more than a mechanical clock or indeed anything that simplistic. But at least from the analyses presented in Section III there appears to be some justification for suggesting that an exponential component exists in the structure of the Solar System, and moreover, that remnants of it remain in the two log-linear zones and the three inverse-velocity relationships discussed in earlier sections. But where does this leave us? According to the methodology applied to the mean periods of revolution and the intervening synodic periods, the log-linearity in the Solar System largely translates into variants of the Phi-Series such that the mean periods (Sidereal and Synodic) increase sequentially by successive powers of Phi while the mean periods of the planets alone increase by Phi squared, i.e., relations 5a and 5b:

Relations 5a and 5b. The Fundamental Period Constants

Relations 5a and 5b. The Fundamental Period Constants

Correspondingly, because of the the third law of planetary motion and the relationship between the mean periods, mean distances and mean velocities, the factor Phi 4/3 (1.899547627) generates the mean planetary distances while the square root of the latter generates the mean distances throughout, i.e., including intermediate the synodic positions, thus:

Relations 6a and 6b. The Fundamental Distance Constants

Relations 6a and 6b. The Fundamental Distance Constants

Moreover, and in like manner, while the mean periods and mean distances increase with each successive revolution, the all-inclusive mean velocities and planetary mean velocities correspondingly decrease by Phi to the minus two-thirds and phi to the minus one-third respectively, i.e., the velocity constants are:

Relations 6c and 6d. The Fundamental Velocity Constants

Relations 6c and 6d. The Fundamental Velocity Constants

Lastly, such is the nature of the equiangular spirals under consideration that the inverse and normal spirals are virtually identical to each other (with minor reservations, i.e., the matters of phase and origin referred to in the previous section). Nevertheless, apart from the latter points, equiangular spirals based on relations 6e and 6d are (for present practical purposes) interchangeable with those based on the diminishing velocities of relations 6c and 6d.

Relations 6e and 6f. The Inverse Velocity Constants

Relations 6e and 6f. The Inverse Velocity Constants

By now the reader has no doubt already recognized that six of the equiangular spirals based on the constants in relations 5a, 5b, 6a, 6b, 6e and 6d are those applied above to Ammonites, Land, and Sea shells. These six spirals also represent the majority of the Pheidan planorbidae shown in Figure 2, and for the range in question, i.e., from the inverse velocity spiral Phi 1/3 out as far as Phi 3, only Phi 5/3 is missing below Spira Solaris, while the remainder extend sequentially beyond the latter, i.e.,  Phi 7/3Phi 8/3 and Phi 9/3.  But what do velocities have to do with the present discussion concerning the spiral formation evident in shells? Here it is perhaps helpful to give Sir D'Arcy Wentworth Thompson's description of this aspect in On Growth and Form: (1917) where it is included in the latter's detailed description of the equiangular spiral: 35

   Of the spiral forms which we have now mentioned, every one (with the single exception of the cordate outline of the leaf) is an example of the remarkable curve known as the equiangular or logarithmic spiral. But before we enter upon the mathematics of the equiangular  spiral, let us carefully observe that the whole of the organic  forms in which it is clearly and permanently exhibited, however different they may be from one another in outward appearance, in nature and in origin, nevertheless all belong, in a certain sense, to one particular class of conformations. In the great majority of cases, when we consider an organism in part or whole, when we look (for instance) at our own hand or foot, or contemplate an insect or a worm, we have no reason (or very little) to consider one part of the existing structure as older than another; through and through, the newer particles have been merged and cornmingled among the old; the outline, such as it is, is due to forces which for the most part are still at work to shape it, and which in shaping it have shaped it as a whole. But the horn, or the snail-shell is curiously different; for in these the presently existing structure is, so to speak, partly old and partly new. It has been conformed by successive and continuous increments; and each successive stage of growth, starting from the origin, remains as an integral and unchanging portion of the growing structure.
    We may go further, and see that horn and shell, though they belong to the living, are in no sense alive. They are by-products of the animal; they consist of " formed material," as it is sometimes called; their growth is not of their own doing, but comes of living cells beneath them or around. The many structures which display the logarithmic spiral increase, or accumulate, rather than grow. The shell of nautilus or snail, the chambered shell of a foraminifer, the elephant's tusk, the beaver's tooth, the cat's claws or the canary-bird's–all these shew the same simple and very beautiful spiral curve. And all alike consist of stuff secreted or deposited by living cells; all grow, as an edifice grows, by accretion of accumulated material; and in all alike consist of stuff secreted or deposited by living cells; all grow, as an edifice grows, by accretion of accumulated material; and in all alike the parts once formed remain in being and are thenceforward incapable of change.
    In a slightly different, but closely cognate way, the same is true of the spirally arranged florets of the sunflower.  For here again we are regarding serially arranged portions of a composite structure which portions, similar to one another in form, differ in age; and differ also in magnitude in the strict ratio of their age. Somehow or other, in the equiangular spiral the time-element always enters in; and to this important fact, full of curious biological as well as mathematical significance, we shall afterwards return.  
    In the elementary mathematics of a spiral, we speak of the point of origin as the pole (O); a straight line having its extremity in the pole, and revolving about it, is called the radius vector; and point (P), travelling along the radius vector under definite conditions of velocity, will then describe our spiral curve.
    Of several mathematical curves whose form and development may be so conceived, the two most important (and the only two, with which we need deal) are those which are known as (1) the equable spiral, or spiral of Archimedes, and (2) the equiangular or logarithmic spiral.
    The former may be roughly illustrated by the way a sailor coils a rope upon the deck; as the rope is of uniform thickness, so in the whole spiral coil is each whorl of the same breadth as that which precedes and as that which follows it. Using its ancient definition, we may define it by saying, that "If a straight line revolve uniformly about its extremity, a point which likewise travels uniformly along it will describe the equable spiral*." Or, putting the same thing into our more modern words, "If, while the radius vector revolve uniformly about the pole, a point (P) travel with uniform velocity along it, the curve described will be that called the equable spiral, or spiral of Archimedes." It is plain that the spiral of Archimedes may be compared, but again roughly, to a cylinder coiled up. It is plain also that a radius (r = OP), made up of the successive and equal whorls, will increase in arithmetical progression: and will equal a certain constant quantity (a) multiplied by the whole number of whorls, (or more strictly speaking) multiplied by the whole angle (θ) through which it has revolved: so that r = . And it is also plain that the radius meets the curve (or its tangent) at an angle which changes slowly but continuously, and which tends towards a right angle as the whorls increase in number and become more and more nearly circular.
    But, in contrast to this, in the equiangular spiral of the Nautilus or the snail-shell or Globigerina, the whorls continually increase in breadth, and do so in a steady and unchanging ratio. Our definition is as follows: "If, instead of travelling with a uniform velocity, our point move along the radius vector with a velocity increasing as its distance from the pole, then the path described is called an equiangular spiral."
Each whorl which the radius vector intersects will be broader than its predecessor in a definite ratio; the radius vector will increase in length in geometrical progression, as it sweeps through successive equal angles; and the equation to the spiral will be r = aø. (Sir D'Arcy Wentworth Thompson, On Growth and Form, 1917, 1942,1992:751-753; diagrams and footnotes omitted; the emphases are Thompson's alone)

Thus time and velocity, and both intimately associated.
Where next? Since time and motion are clearly involved, the matter of the "whirling rectangles" and the formation of the double spiral.

In some respects the subject of "Whirling" rectangles represents a modern two-part puzzle--not so much the topic per se as the apparent stagnation and lack of understanding that (for whatever reason) currently attends it. This rectangle--"Golden" in the sense that the ratio between the length and the width is 1.61803398874 : 1 (i.e., Phi : 1 )--is more often than not shown in association with the side view of Nautilus pompilus, which is the first part of the puzzle, since the spiral assignment for Nautilus has long been known and the spiral in question has a growth factor (or expansion rate) more than twice that of the latter. The second part of the puzzle concerns why the matter is rarely taken further. It is surely a natural step when a spiral is shown in relation to a rectangle with attendant squares, etc., to investigate the details and if possible determine what lies behind the observed effect. By way of explanation, maintaining the same ratio between rectangle and square throughout, the original rectangle may be successively partitioned into firstly a square with both sides equal to the previous width, and secondly, into another similar golden rectangle, and so on, rotating 90 degrees with each successive partition. As it so happens, the combination of the quarter-perimeters inscribed in the resulting squares closely approximate an equiangular spiral, as is often demonstrated in discussions concerning this topic, though the spiral itself (actually k = Phi 4) is rarely identified (at least this was the case when I first mentioned it!). Nor is the representation a true spiral, as most commentators point out, though few tend to elaborate much further. An exception is Jay Hambidge,36 who also describes a similar (though not identical) treatment of "Whirling Rectangles" with respect to root-5 rectangles in Dynamic Symmetry, (1920).

Fig.14.  "Golden Rectangles," Squares, and the Equiangular Spiral
Fig.14.  "Golden Rectangles," Squares, and the Equiangular Spiral k = Phi 4

As for the "Golden Rectangle" and the observed spiralling effect, it is perhaps useful to remain with the astronomical side of the matter for a while and consider again what was stated in Section IV (Spira Solaris Archytas-Mirabilis), i.e.,

With respect to the present astronomical application and the exponential planetary framework it may be noted that all mean periods (planet-synodic-planet) increase by phi itself while all planetary periods per se increase by phi squared. Therefore the required period function should increase by the square root of phi per 90-degree segment and by phi squared per revolution. Thus for explanatory purposes, commencing with unity, the first 90-degree segment would have the value 1.27201965, the second (the half-cycle, or 180 degrees) 1.618033989 (phi itself), the third 2.058171027, and at the full cycle, phi squared = 2.618033989.

Not that this is new, though the above application is somewhat specialized. In fact Jay Kappraff 37shows quadrantal growth for the equiangular spiral in this exact manner in a schematic diagram of the logarithmic spiral, replete with attendant rotating and expanding rectangles (Figure 2.11,1991:46). Here it may be noted that in general terms the fixed increase per quadrant is the fourth root of the growth factor per revolution, as Sir D'Arcy Wentworth Thompson38 was obviously aware in citing the square root for the half-cycle and the square root again for the quarter (see also Figure 15c below). The fourth root in this context applies to all pheidian spirals and as such it is also inherent in the "Golden Rectangle," though this may not be immediately apparent for a number of reasons. Firstly, the associated spiral is in a sense incomplete with respect to the full rectangle and largest square, whereas it is always "complete" with respect to the smaller rectangles, etc.(see Figure 14 above). Secondly, quadrantal growth in this application is more naturally understood with respect to what may be termed the "outer" spiral as opposed to that enclosed by the rectangle itself. Thirdly, though identical in form, in addition to being external the outer spirals are also inclined at specific angles to both the rectangle and the inner spiral. This should become more apparent from Figure 15, which incorporates the above data for Spira Solaris and in addition shows the orientation of the double spirals to the parent rectangle:

Fig. 15.  The Rectangle with Inner and outer Spirals for Spira Solaris

Fig. 15.  Rectangle with Inner and outer Spirals for Spira Solaris, k = Phi 2

Thus while Figure 15 is a representation of quadrantal growth that results in an increase of Phi 2 per revolution, the same delineation is also traced out by both the inner and outer spirals associated with the rectangle. However, it is the outer spiral that shows the fourth root increases more clearly, and both are perhaps best demonstrated by animated graphics, firstly with respect to the original "Golden" rectangle and the spiral k = Phi 4 and secondly with respect to the associated quadrantal radii vectores for the same spiral; see Quarteranimation I (60kb) and Quarteranimation II  (117kb) respectively.
    As for the "Golden Rectangle" in this particular context, one of its main values would appear to be that it provides a natural lead-in to the above because of the close fit between the quarter-circumferences and the spiral k = Phi 4.

Golden Rectangle and associated spirals

Fig. 15b.  The Golden Rectangle with Inner and outer Spirals for k = Phi 4

After Figure 365, On Growht nad Form, 1917.

Fig. 15c.   The Golden Rectangle and the Outer Spiral by Sir D'Arcy Wentworth Thompson
(On Growth and Form
Chap. XI, The Equiangular Spiral, "Concerning Gnonoms," Fig.356; rotated, diagonals added)

But as the animations show, it is neither the rectangles nor the squares that are rotating, but the effects of spiral growth that the latter approximates so well in this particular example. Once the concept is understood, however, it may be extended to any and all pheidian spirals, which was in fact how the double spirals shown in Figures 1a and 1b were initially generated. It is at this juncture that the assignment of primary spirals to Haliotidae may be resumed.

Figure 2H
: Haliotis brazieri ( Angas 1869).Figure 2I Haliotis scalaris ( Leach 1814). Below:  Haliotis parva (Linnaeus 1758 ).
The two examples from this group shown in Figure 2 have growth factors considerably smaller than that discussed with respect to the Golden Rectangle (i.e., k = Phi 8/3 and k = Phi 9/3 respectively). Nevertheless, the spiral assignments for Haliotidae extend far higher and within this group it would appear that the primary spiral k = Phi 4 is applicable to Haliotis parva:

 Fig. 16.  The Golden Rectangle and Haliotis Parva

Fig. 16.  The "Golden Rectangle", Haliotis Parva 39 and single spiral: k = Phi 4

Here the flatter underside of the shell supplies additional reference markings. Neither the top, or the underside, or indeed the spires are truly flat, though the latter are relatively low in some cases and only slightly raised in others. Thus although well suited for the kind of testing carried out here there are nevertheless spatial complications to be taken into consideration. Then again, the Halitiotidae also seem to incorporate the double spiral form, especially in the higher assignments as seen in Figure 2 and the three examples of Haliotis parva 39 shown below in Figure 17a (topview) and 17b (underside).

Fig. 17a. Haliotis Parva (topview) with the Double Spiral

Fig. 17a. Haliotis Parva (topview) with the Double Spiral k = Phi 4

Fig. 17b. The Underside of Haliotis Parva with the Double Spiral

Fig. 17b. The Underside of Haliotis Parva with the Double Spiral k = Phi 4

Subject to revision and possible extensions, provisional assignments from the initial survey for some of the other Haliotidae are as follows:

For additional information on Haliotidae (and other shells) see the related sections in the following short list (in alphabetical order):
On a more general note, only a limited number of shell assignments from the initial survey are included here, partly because of the interim nature of the survey and partly because of "concatentation" concerns (perhaps already breached).  Additional assignments will likely be included in later papers following further refinements to the test procedures. For the time being, the last example from the survey is Thatcheria mirabilis (Angas 1877; k = Phi 7/6, n = 1.753149344) 60 shown below with an assignment based predominantly on the planview, but also reinforced by a full-height assignment of the same spiral in the vertical plane. This secondary, "three-dimensional" technique is applicable to other shells, but planviews are preferred for obvious reasons.

Fig. 18.  Thatcharia mirabilis

Fig. 18.  Thatcheria mirabilis with the spiral k = Phi 7/6

As for what may or may not lie at the root of such pronounced yet regular variations that give rise to the apparent fit of pheidian spirals to the shells discussed here, I do claim to have any concrete answers to this complex matter, but I can at least show in Part B what the Phedian Planorbidae may represent in fundamental dynamic terms.



The subject of angular momentum was addressed previously in both Section III and and Section IV--in hypothetical terms with respect to theoretical changes in the structure of the Solar System in the former, and in more practical terms in the latter concerning Period-Distance-Velocity relationships, i.e., T 2 = R 3 = Vr -6 (where T represents the mean period of revolution, R the mean heliocentric distance and Vr the mean orbital velocity) and: T 2 = R 3 = Vi 6 where Vi is the mean inverse orbital velocity. Here it may be observed that with respect to the Third (or Harmonic) Law of planetary motion the determination of the mean heliocentric distance (R) from the mean sidereal period (T) is often given by the exponential relationship: T 2 = R 3. However, both the heliocentric distance (R) and mean orbital velocity (Vr) may be obtained from the mean period (T) (and vice versa ) from a number of additional relationships;40 the first (a variant of the general relationship already given) being perhaps the best known:

Table 1. Pheidian Period, Distance and Velocity Relations

Table 1. Pheidian Period, Distance and Velocity Relations

Mean velocities may, of course, also be obtained from the fundamental relationship directly by using negative fractional exponents (relations 2 and 3) or more simply from the reciprocals of the inverse velocities (relations 4 and 5), which are, of course, simply cube roots and square roots. But the inclusion of the inverse velocity provides additional benefits, not least of which being that the Period-Distance-Inverse Velocity relationship T 2 = R 3 = Vi 6 allows the inverse velocity Vi to play a major role in the computation of Angular Momentum (L).  A brief and informative introduction to this subject is given below (source: Jeffrey K. Wagner, Introduction to the Solar System):41 

 Most of the angular momentum of the solar system is in the planets, not in the Sun. This is because the massive Sun rotates very slowly, whereas the planets, less massive but far away, move rapidly enough in their orbits that their angular momentum is greater. This is particularly true for the gas giants. (Angular momentum is a quantity for rotating or revolving objects that is somewhat analogous to momentum for objects moving in a straight line. The angular momentum, L, of an orbiting object is given by the equation  L  = mav,  where m is the mass of the object, a is its semi-major axis, and v is its average orbital velocity. For a rotating object, angular momentum is given by L = Cmr2w, where C is the object's moment of inertia coefficient, m is its mass, r is its radius, and w is its rotational velocity in radians per second. Table 29.1 lists the angular momentum of various solar system objects.) Other important characteristics of the solar system involve the physical properties of its various objects. The planets differ in composition, with the controlling factor being distance from the Sun. The inner, terrestrial planets are rocky and metallic, the outer gas giants are primarily hydrogen and helium, and outermost Pluto is icy. The planets and their satellites resemble miniature solar systems, and most satellites orbit regularly in the equatorial plane of their planet ... (Jeffrey K. wagner; Chap. 29.  Constraints on Solar System Formation in Introduction to the Solar System, Holt, Rinehart & Winston, Orlando 1991:426)

Consequently angular momentum is essentially the product of the mass (m), the mean distance (R) and the mean velocity (Vr). However, because of the Period-Distance-Velocity relationship T 2 = R 3 = Vi -6 and the inverse velocity variant, the formula: L = Mav may be expressed in the form  Mass x Vi 2x Vr, which reduces further to Mass x Vi.  Thus, in simpler and more practical terms:

Angular Momentum  L = mvi

     Angular momentum is often expressed in absolute terms, but for comparison and general purposes it may also be calculated with respect to unity with Earth providing the frame of reference for the mean periods, the mean distances, the mean velocities and the planetary masses (e.g., the mass of Jupiter from this viewpoint is 317.88 times that of Earth, etc.). Furthermore, while still remaining with mean values it is also possible to compute the angular momentum for the Phi-Series exponential planetary framework directly. Thus we obtain the following comparison with Jeffrey K. Wagner's estimates and the above relation:    

Table 2. Percent Solar System L, The Phi-Series Exponential Framework

Table 2. Percent Solar System L, The Phi-Series Exponential Framework
and the Modern Solar System
(after Wagner 1991:426)

As shown in Table 2, the estimated percentage of the total angular momentum of the Solar System possessed by each planet departs little from the modern estimates provided by Jeffrey K. Wagner (1991:426). Although minimal, the main differences are precisely those expected for the Phi-Series planetary framework, i.e., there is slightly less angular momentum for Uranus (the latter being theoretically closer to the Sun) and correspondingly more for Neptune, which is theoretically more distant.
    It remains now to present the actual angular momentum for the major planets computed for the Phi-series planetary framework with the relation L = Vi x Mass. Thus, starting from Mars (rather than Earth).
Working outwards it turns out that the resulting inverse velocities in Table 3 below are simply the Pheidian Planorbidae --and not only increasing by "Thirds" either, -- but also in due sequence, extending from the inverse velocity of Mars ( k = Phi 1/3) out to Neptune (k = Phi 11/3) with all intermediate synodic positions included.  Thus the Growth factor k is in fact the inverse velocity Vi --nothing more and nothing less, with the angular momentum in each case simply the product of the pheidian planorbidae and the planetary masses applied to modern data:

Table 3. Angular Momentum L, and the Phi-Series Exponential Framework

Table 3. Angular Momentum L, and the Phi-Series Exponential Framework

Lastly, for this phase of the investigation one further step is required, namely the insertion of the Pheidian "Sixths" which, (as explained in Section IV) is by no means as difficult as it might appear, as long as the associated procedures and techniques are understood. Historically, the following table is also of interest in so much as planet Earth, while representing the starting point with a "growth" factor of 1 : 1 and corresponding equiangle of ninety degrees, is in effect the "center" --a primary position that is nevertheless more theoretical than heliocentric per se
   The initial set of Pheidian Planorbidae as tested are given below in exponential sixths out to the twenty-fourth (corresponding growth factor/expansion rate: k = Phi 4):

Table 4. The Pheidian Planorbidae: Sixths; Growth Factors/Expansion Rates 1.084 to 6.854

Table 4. The Pheidian Planorbidae: Sixths; Growth Factors/Expansion Rates 1.084 to 6.854

In diagrammatic form with the thirds alone, the following graphical representation emphasizes the astronomical significance of the Pheidian planorbidae and their relationship to the four major planets. Here it now becomes apparent that Spira Solaris occupies a key position between the two most massive planets in the Solar System, Jupiter and Saturn--a situation further complicated by the elliptical nature of the respective orbits of the latter pair, their known resonances and further complexities concerning their inverse velocities with respect to the mean velocity of Mars as discussed in Sections II and III (see also below):
  Fig. 19.  The Phedian Planorbidae in Astronomical Context

Fig. 19.  The Phedian Planorbidae in Astronomical Context
In Figure 19 the numbers to the left of the planetary and intermediate synodic positions represent the corresponding exponents that generate the mean periods of the Phi-series exponential planetary framework. Thus the mean sidereal period of Jupiter is Phi raised to the fifth power ( Phi 5= 11.090169944 years), Saturn: Phi raised to the seventh power ( Phi 7= 29.034441854 years) and Uranus Phi raised to the ninth ( Phi 9 = 76.013155618 years). For the inverse velocities, a.k.a the Pheidian growth factors/expansion rates, these may be obtained by applying the cube root to the periods, or in a single step from the division of each period exponent by three. Thus for Jupiter k = Vi Phi 5/3; for Saturn k = Vi Phi 7/3; for Uranus k = Vi Phi 9/3, and for good measure (between the first pair) Spira Solaris: k = Vi = Phi 6/3 = Phi 2.
    In passing, it is not out of place to note here that the occurrence of the Pheidian spiral k = Phi 2/3 among Conidae may be back-tracked with respect to the double spiral and larger growth factors, i.e., k = Phi 2/3 raised to the fourth power is in turn k = Phi 8/3 now associated with the Saturn-Uranus synodic cycle, i.e., the eighth "third", with the other major synodic cycle (Spira Solaris) the sixth. Thus there is a continuous integer range from 5 though 9 with the odd-number planetary assignments predominating. For more on this aspect and its possible historical ramifications, see Historical Complexities.

As for the dynamics of the matter, this too is clearly a subject of great complexity--even restricted to mean values and the Phi-based exponential planetary framework. Although theoretical, the latter is at least complete, whereas the Solar System itself, with the apparent gap between Mars and Jupiter, regular variations arising from elliptical orbits, and inter-planetary resonances is vastly more complex. Nevertheless, the awareness of the relationship between inverse velocity Vi, planetary mass and angular momentum permits the re-examination of the Inverse Velocity relationships discussed in Section II and III with respect to the  superior planets (Uranus, Saturn and Jupiter) and the terrestrial planets from a more dynamic perspective. The three inverse-velocity relationships described in these earlier sections were as follows: 

  1. Vi Venus - Vi Mercury approximates the mean velocity of Uranus
  2. Vi Saturn - ViJupiter approximates the mean velocity of Mars. [Relation 4b]
  3. Vi Saturn/Uranus Synodic - ViJupiter/Saturn Synodic approximates the mean velocity of the Venus-Earth synodic cycle.[Relation 4s1]

Graphically (Figure 20 below), with the corresponding Phi-series inverse velocities for Jupiter, Saturn and the Jupiter-Saturn synodic (Spira Solaris) included for comparison, it will be noted that there are only minor differences between phi-series values and those of the modern Solar System, i.e., for Jupiter: 2.281082 compared to 2.2300404 ( Phi 5/3); Saturn: 3.088215 compared to 3.0735326 ( Phi 7/3) and in between, the difference cycle Spira Solaris: 2.709038 compared to 2.6180339 ( Phi 6/3) with the latter parameters also k, the Pheidian growth factor. The linkage between the superior and inferior planets is shown below:

Fig. 20.  The Inverse Velocity Relationships and the Solar System (Mean Values)

Fig. 20.  The Inverse Velocity Relationships and the Solar System (Mean Values)

Although the data for Figure 20 concerns mean values, for Spira Solaris in particular it is also possible to consider the true variations that results from the elliptical orbits of Jupiter and Saturn, as already treated in Section II with respect to the mean and varying orbital velocity of Mars. Here it was explained that subsequent testing provided the following results:

The inclusion of Earth in this context--synodic location notwithstanding--thus serves to augment the linkage between the Terrestrial planets of the lower log-linear zone and the three gas giants of the outer zone. One or two other inverse-velocity relationship also appear to exist that are almost sequential--a qualifier necessary here in so much as the latter appear to incorporate synodics and planetary inverse velocities. But there are also other considerations and complications to be addressed, for although mean values are applied in these relationships, in real time such functions vary according to the elliptical natures of the associated orbits. Nevertheless, in the case of the Mars-Jupiter-Saturn relationship, with frames of reference provided by the mean orbital velocity of Earth of 29.7859 kilometers per second and 24.1309 kilometers per second for that of Mars, real-time maxima and minima for Relation [4b] range between 19.66 and 28.3 kilometers per second, well exceeding the extremal velocities of Mars itself. However, utilizing the methods of Bretagon and Simon 44 adapted to generate sequential data for 5-day intervals from 1700 to 2000 A. D., the mean value nevertheless still turns out to be 24.0938 kilometers per second. Similarly, the data for the real-time function based on Relation [4s1] reveals that although there is an even wider swing in extremal values, the mean value is also comparable to that obtained from Relation [4s1] directly. All of which is further complicated by the proximity of the Mars-Jupiter synodic to the Earth-Mars synodic and various resonances known to exist in the Solar System. (Spira Solaris, Part II. The Alternatives)

The third inverse velocity relationship brings with it further complications in so much as it represents the combination of the varying motion of three major superior planets--Jupiter, Saturn and Uranus--thus the effect of two adjacent difference cycles. Thus it also involves the three major adjacent planets that together possess over 90 percent of the entire Solar System's angular momentum, with the Jupiter-Saturn difference cycle (i.e., Spira Solaris) at the 42 percent position on the following chart:

Fig. 20.  The Distribution of Angular Momentum among the Major Planets

Fig. 21.  The Distribution of Angular Momentum among the Major Planets

Given their massive sizes, their prominence in terms of angular momentum and the cyclic, resonant effects of their regular motions--all, if not synchronized with the phi-series exponential framework, then periodically sweeping across it, it is perhaps not that surprisingly if smaller objects such as Earth and the other inferior planets should be influenced in one way or another by the combined effects of the angular motion of larger bodies such as Jupiter, Saturn, Uranus and Neptune. To which may also be added further resonances among the inferior planets themselves, augmenting perhaps the natural, local, diurnal, monthly and annual cycles in turn.
    How such events might or might not filter down to Earth itself is a question that perhaps rightly belongs to the first of things. And as such, it is perhaps largely unknowable, but if in our times it is possible to a pursue a "Theory of Everything" with a straight face and an apparently unerring aim, then we might at least take up a few threads already woven into this highly complex tapestry, starting with the lead provided by Louis Agassiz.

Just prior to launching into a detailed exposition of the application of phyllotaxis to the Solar System first proposed by Benjamin Peirce in 184945 Louis Agassiz wrote in his famous Essay on Classification (1857):46

    It must occur to every reflecting mind, that the mutual relation and respective parallelism of so many structural, embryonic, geological, and geographical characteristics of the animal kingdom are the most conclusive proof that they were ordained by a reflective mind, while they present at the same time the side of nature most accessible to our intelligence, when seeking to penetrate the relations between finite beings and the cause of their existence.
    The phenomena of the inorganic world are all simple, when compared to those of the organic world. There is not one of the great physical agents, electricity, magnetism, heat, light, or chemical affinity, which exhibits in its sphere as complicated phenomena as the simplest organized beings; and we need not look for the highest among the latter to find them presenting the same physical phenomena as are manifested in the material world, besides those which are exclusively peculiar to them. When then organized beings include everything the material world contains and a great deal more that is peculiarly their own, how could they be produced by physical causes, and how can the physicists, acquainted with the laws of the material world and who acknowledge that these laws must have been established at the beginning, overlook that à fortiori the more complicated laws which regulate the organic world, of the existence of which there is no trace for a long period upon the surface of the earth, must have been established later and successively at the time of the creation of the successive types of animals and plants?
(Louis Agassiz,Essay on Classification,  Ed. E. Lurie, Belknap Press, Cambridge 1962:127-128; emphases supplied)

Thus two points to consider, firstly the physical environment, and secondly the inception and the later flourishing of life as we understand it on Earth. As for possible physical causes that may pertain to both, there cannot be one without the other, and perhaps their origins are also intertwined. To this end and with this in mind, consider the following short discourse occasioned by Galileo's treatment of planetary motion described somewhat circuitously in The Two New Sciences (1609).  Published in the Journal of the Royal Astronomical Society of Canada in 1989 the parent paper was not merely a historical commentary, it was also the source of the velocity expansions of the laws of planetary motion applied throughout the present series of essays, angular momentum included. Given below is the short discussion that occurs at the end of the paper concerning a possible percussive origin for the planets attributed to Galileo, or at least the general line of inquiry that he appears to have been pursuing:47

   Could Galileo have extended his treatment of terrestrial projectile paths to embrace satellite orbits and also have expanded the idea one step further to include the planets as satellites of the Sun? While acknowledging that there are dangers in attributing to Galileo modern or Newtonian concepts, it is necessary to recall that the initial discussion of the parabola concerned the path traced by a projectile with uniform horizontal velocity applied down the horizontal axis, and "naturally accelerated" velocity applied down the vertical axis. Visually, a projectile launched almost horizontally will obviously gain very little height before falling back to ground when the initial velocity is relatively low. As the initial velocity increases, however, some height will be gained because of the curvature of the Earth, and although the projectile may still fall to ground, with sufficient velocity, a projectile will finally "fall" into orbit around Earth itself. Thus in general, by reversing matters, all objects in specific orbits may be treated in terms of a "percussive origins theory" with the parent body the initial source. The hypothesis may therefore be applied to the planets and the Solar System with the Sun as the single percussive point of origin.
    Could Galileo have taken this final step? If he did, then undoubtedly criteria provided by Galileo in his historical aside becomes more significant than ever, i.e., if planetary origins are considered in terms of projectiles originating from the Sun, the planets would indeed "start with zero velocity" and "move through successive speeds" until their initial "rectilinear motion" changed into "circular motion" (or orbital motion) as they "fell" into their respective orbitals positions. And once established, the planets would then "revolve without either receding from or approaching" their common point of origin, or deviating from their "final" positions in the Solar System.
   Although no causal mechanism is associated with this "percussive origins" (or "Small Bang") theory, the hypothesis might possibly assume that the Sun was essentially formed at this stage, and for whatever reason, the planetary material was ejected from the Sun in one enormous explosion.
   In this sense the hypothesis is a variation of catastrophe theory, with the exception that the source of the catastrophe is internal rather than external. The latter, involving collisions or near misses with double or triple stars, etc., are not generally well supported today, but the percussive elements of the basic hypothesis may perhaps have some affinity with the massive explosion of the solar core (i.e., the "T Tauri winds") thought by some accretion theorists to be a possible explanation for the expulsion of unaccreted dust and gas from the Solar System. (John N. Harris, "Projectiles, Parabolas, and Velocity Expansions of the Laws of Planetary Motion, " JRASC, Vol 83, No. 3, June 1989:207-218)

And the point here?  The matter of origins and the possible inception of ordering and marshalling forces from the dynamic, resonant motions of the major superior planets, not perhaps so much initially as later, though a violent birth may also present an option.
Lastly, a real-time illustration of the significance of the motions of
the two most masssive, fibonacci-resonant planets in the Solar System, their relationship to Spira Solaris, and also perhaps, to natural growth.

From Part III (The Exponential Order) the planetary period constant for the Phi-series exponential framework was determined to be Phi 2  = 2.61803398874, whereas the inverse (Phi -2 = 0.381966011) is known to be closely related to the "ideal" convergence angle in phyllotaxic contexts. However, the reader will recall that "Spira Solaris" (the all-inclusive equiangular "period" spiral) was also based on the fundamental constant Phi 2 = 2.618033989. This was the final result of an investigation that had included mean planetary periods, synodic motion and orbital velocities in a search for logarithmic order in the structure of the Solar System - a search that was ultimately reduced to the solution of the quadratic formula that defines the Golden Section ( X 2 2- X - 1 = 0 ).  The initial result was therefore Phi itself = 1.618033989 for an exponential planetary framework incorporating the intervening synodic (lap) cycles between adjacent planets. For planet-to-planet increments the final period constant was therefore Phi 2 = 2.618033989.  
  As it so happens,
the reciprocal  of this last value, Phi -2 = 0.381966011 is not only intimately related to "ideal angles" and natural growth, it is also a repeat parameter in the Phi-Series planetary framework, i.e., it occurs twice for Mercury (once for the heliocentric distance, and again for the Mercury-Venus synodic period. Thirdly, it is also the "velocity" (Vr) of the synodic difference cycle between Jupiter and Saturn:

Fig. 21b. Relations 14a-14e and the Ideal Growth Angle

Fig. 21b. Relations 14a-14e and the Ideal Growth Angle

These multiple occurrences arise from the three-fold nature of the Phi-Series Planetary Framework, which necessarily incorporates identical values for periods, distances and velocities according to their exponential position in the planetary framework, including the Inverse Velocities, which (as we have already seen above) play an important role in the computation of angular momentum.

With this basis revisited, the well-known 60-year, 2 : 3 : 5  fibonacci resonance between the two most massive planets in the Solar system (Jupiter and Saturn) takes on further significance as Table 5 and Figure 21c show, for the arithmetic mean of the actual Jupiter and Saturn mean velocities (Vr) is not only 0.381280708; the daily average function for this pair of planets computed for the 400-year interval from 1600 - 2000 CE [Julian Day 2305447.5 through Julian Day 2451544.5 ] is in turn found to be 0.381071579

Planets and



Velocity Vr

Velocity Vi




(Ref. Unity)

k (Growth)
 JUPITER 11.868991 5.203264 0.438391444 2.28106642
Synodic 19.925328 7.349712
0.368862787 2.71103519
 SATURN 29.354971 9.516000 0.324169972 3.08480145

Table 5b. The Solar System Mean Values, Jupiter, Saturn and the J-S Synodic Cycle.
Arithmetic Mean:
( SaturnVr + JupiterVr) = 0.381280708

Fig. 21b.  The Jupiter-Saturn Cycle and the Fundamental Constant

Fig. 21c. The Jupiter-Saturn Average Velocity Function (JS-Avg.Vr) and the Phyllotaxic Constant k = Phi -2

The above 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) spiralling towards the constellation of Hercules -- at which point readers may like to examine -- either from the present perspective, or indeed in its own right -- the insights and complexities in Gabi Mueller's Qualitative Torkado Model (includes English translation links); see also the latter's Golden Mean iterations: http://www.torkado.de/phi.htm.

   As for the development of life on Earth, or indeed anywhere, this no doubt remains one of the great mysteries, but it may be that the three-fold number and pythagorean tenets lie at the root of it to some extent at least, and that there is both verity and wisdom in what has been handed down to us concerning this matter, and the forces that:
"Govern all things, and Order all things not Ordered."

* * * * *


As Sir Theodore Andrea Cook pointed out long ago (1914:414), there are any number of equiangular spirals that lie between the limits set by a straight line and a circle. Which may or may not provide a partial explanation for the present lack of progress in coming to terms with the many spiral configurations so clearly evident in shells.
This state of affairs is especially surprising when the universal availability of the modern computer is taken into consideration, but perhaps this is also part of the problem, i.e., a general lack of focus allied to the use of the computer as a toy rather than scientific tool. In fact forty years have passed since David Raup48 first introduced computer simulation of shells, but little in the way of qualification or quantification appears to have followed thereafter. Instead, the subject appears to have been spread so wide of late that it is in grave danger of dissipating entirely rather than being consolidated and refined further. Which again is puzzling given the start obtained in 1962, as Tony Phillips recounts 49 in "The Mathematical Study of Mollusk Shells"

The paleontologist David Raup, then at Cornell, published a paper in 1962 (Science 138:150-152) entitled "Computer as aid in describing form in gastropod shells." He showed how a computer could be programmed to make images of the equiangular spiral model with several parameters. To put this achievement in perspective: the term "computer graphics" was coined around 1960, and Spacewar, the first video game ever, was designed in 1961. What is even more remarkable is that a video sampling of Raup's models (which were made by tailoring waveforms on an oscilloscope screen) is still on display in the Mollusk galleries of the American Museum of Natural History in New York. My chancing on that exhibit was the impetus for this column.
    The video is part of an installation called "Spirals and Shell Variation" which also includes wire models and a variety of specimens. In the video, entitled "The Geometry of the Coiled Shell," Raup gives a bare-bones presentation of the potential of his method. Only three parameters are illustrated: rate of aperture expansion, rate of departure from axis, and rate of descent along axis; the parameter values are described ("0," "small," "large") qualitatively.  Seven different natural morphologies are simulated (Nautilus, Spirula, Valvata, Goniobasis, Vermicularia, Anadonta (a bivalve) and Bulla) with in most cases a specimen for comparison.

Nor can it be said that considerable strides had not already been made years ago by the likes of Canon Mosely (1838) and his contemporaries, or that the continuance of the latter's treatment by Sir D'Arcy Wentworth Thompson was not widely available (at least in the complete and unabridged edition of On Grow and Form published in 1917, 1942 and 1992)50. Take, for example, the Nautilus, the first shell mentioned in the above quotation, most likely the same Nautilus that Sir D'Arcy Wentworth Thompson declined to discus in detail because he preferred instead (along with Mosely) to deal with the more complex turbinated shell Turritella duplicata51.
   For my own part I must admit that prior to April of this year (2002) that I had not addressed the spiral formation in shells at all, and moreover, I came across Sir D'Arcy Wentworth's seminal On Growth and Form and the contributions of Canon Mosely rather late in my inquiry. Partly, no doubt because of my less than perfect methodology, but also partly because although the latter pair provide by far the best starting point for spiral forms in shells, no clear signposts to this effect were available to point the way. Not only this, but two quite different versions of the latter work exist, one with a sizeable amount of material pertaining to shells and the entire chapter on phyllotaxis expunged in an abridged edition first published a year before the Raup paper in 1961 and reprinted thereafter in 1969, 1971, 1975, 1977, 1981, 1983, 1984, 1988, 1990, 1992, 1994, 1995, 1997and 2000. Here even the cross-section of the Nautilus shown on the front page is degraded, though ultimately it is still identifiable as k = Phi 7/3.
   But in any case, my own analyses proceeded from ammonites (by way of Simon Winchester's introduction to the subject as noted above) to ramshorn snails, seashells and the associated works of Sir Theodore Andrea Cook (The Curves of Life, 1914), Samuel Colman (Nature's Harmonic Unity, 1911) and then finally to the details in Thompson's On Growth and Form (1917,1942). Because of this circuitous route I found that by initially concentrating on two-dimensional growth factors that I had perhaps naturally emulated Mosely's "characteristic" numbers. In the interim I had also been working my way through various shells (some easy, some difficult and some still unassigned) before finally coming across the latter's analyses laid out in great detail by Sir D'Arcy Wentworth Thompson. Having followed my own route, however, the "characteristic" numbers discussed by the latter were by this time hardly new, in fact for the most part they were immediately recognizable as two or four-decimal place pheidian growth factors--specifically  Phi 1/3, Phi 7/6and Phi 1/4--the latter pair being rarer in my own limited experience, with k =  Phi 7/6 (growth factor: 1.753149344) given by Mosely to two decimal places as 1.75.
   All this, mind you, by 1838, while the Raup approach still apparently provides little or no integration or semblance of order. Indeed, during the ammonite phase of testing it so happened that one of the ammonites examined was Euhopites truncatus Spath 1925 from the collection of the late Jim Craig,52 which coincidentally was also discussed in the above paper. The results of the respective assignments for this example are shown below with Spira Solaris positioned firstly with respect to the inner visible spiral, secondly with respect to the outline [ I ], and next [ II ] superimposed on the somewhat angular spiral generated by the Raup approach, the latter thus also essentially Spira Solaris.

Fig. 21. Ammonite Euhoplites truncatus Spath 1925

Fig. 22. Ammonite Euhoplites truncatus Spath 1925 and the single spiral k = Phi 2

The difference, being of course, that there are no trial and error operations attending Spiral Solaris, or indeed any of the Pheidian planorbidae. Moreover, there is a great deal that can be accomplished before the introduction of the third dimension, which has perhaps been part of the problem in recent times, namely a general lack of focus coupled with a dearth of choices. Too inviting a diversion, and also too distracting, it would seem. Other matches from the above paper include a side view fit for Nautilus pompilus (as before, k = Phi 7/3); the ammonite Astroceras obtusum ( k = Phi 5/3), Bellerophina minuta ( k = Phi 2/3):

Methodologies and Single Spirals

Fig. 22a. The single spirals k = Phi 7/3, k = Phi 5/3  and k = Phi 2/3

plus bi-valve Mya arenia (a rare side view: k = Phi 10). It should be noted here that mathematical details are certainly presented in abundance the above paper, but perhaps in too much multi-dimensional detail to provide order and connectivity, at least in pheidian terms.

  This appears to be a prevailing problem at present, for computer generated profusion and dispersion also occurs in science popularist Richard Dawkins'  "Museum of All Shells" (Climbing Mount Improbable, 1996).53 Then again, it is difficult to know what to make of a book that although dealing with natural growth does not even mention the Fibonacci Series in the Index. Nor do some of the  more sophisticated methods involving the use of computers in this book appear to have been well maximized either, e.g., Fig. 2.10.d (
"Computer tracing of a particular spider's positions as it spins a web. MoveWatch program written by Sam Zschokke.") records an auxiliary spiral constructed by the spider Aruneus diademus,54 which though somewhat irregular nevertheless provides a ready fit for the pheidian spiral k = Phi 2/3.

Fig. 22b. Web of Aruneus diademus and the single spiral k = Phi to the two-thirds power

Fig. 22b. Web of Aruneus diademus and the single spiral k = Phi 2/3

But there is more to the engineering prowess of the spider in any case, not least of all its outer commencement point and inward motion during web construction--logical enough and necessary perhaps, but the implications remain profound.

   And so, sadly, are the implications of the apparent decline in understanding that appears to have followed successful investigations into the spiral formation in nature carried out over the last three centuries. Particularly in the case of shells, it would seem, but it was not the general treatment of this subject in Sir D'Arcy Wentworth Thompson's voluminous On Growth and Form that served to emphasize the decline, but Canon Mosely's convoluted treatment of the shell Turbo duplicata in 1838. Indeed, one glance at the data and the means of presentation was enough to elicit both surprise and great admiration--as D'Arcy Wentworth Thompson duly noted in On Growth and Form--"Canon Mosely was a man of great and versatile ability" 55and here was at least partly why.

   But for myself it was something else in addition. In pursuing the line of inquiry followed by Galileo in a 1989 paper entitled: " Projectiles, Parabolas, and Velocity Expansions of the Laws of Planetary Motion" (Journal of the Royal Astronomical Society of Canada; RASC, Vol 83, No. 3,1989:207-218) I had long been puzzled by the obvious fact that if I could deduce what Galileo had laid out in the New Sciences (he had, after all, left sufficient clues) then others should surely have been able to do the same. Now it would appear that some indeed had, and well before my time in addition, including both Canon Moseley (1838) and Sir D'Arcy Wentworth Thompson (1917). Nor does there seem to be much doubt about this either; it is quite clear what Mosely delivered with his analysis of Turbo duplicata, and equally clear that Thompson understood it when he in turn passed along the essence of the matter.
  As for the relevant details, first of all, how well did Canon Mosely fare with Turbo duplicata ? Well enough, even in general terms, successfully arriving at the characteristic number (k) of 1.1806 compared to that of the pheidian spiral k = Phi 1/3 (1.1739)--as the latter scale applied to Mosely's illustration of the shell in question shows:

Fig. 22.  Canon Mosely's Turritella Duplicata

Fig. 23.  Canon Mosely's Turritella Duplicata with the scale of the spiral k = Phi 1/3

But Mosely's treatment was far from simple, and to explain it in detail it is necessary to ask the reader to recall some of the steps taken so far in the present inquiry. Firstly, with respect to the astronomical side of the Pheidian planorbidae, it is necessary to remember that the growth factor k (Mosely's "Characteristic number") is the mean orbital inverse velocity (Vi); secondly, remember also the relationships between the mean planetary periods (T), the mean heliocentric distances (R) and the mean inverse velocities (Vi), especially relations 4 and 5 from Table 1: Mean Inverse Velocity (Vi) = T 1/3 and also Mean Inverse Velocity (Vi) =  R 1/2. Finally, note that the published "Velocity Expansions of the Laws of Planetary Motion" attributed to Galileo in the above mentioned paper were those given in the first line of the abstract, i.e.:

Kepler's Third Law of planetary motion: T2 = R3 ( T = period in years, R = mean distance in astronomical units ) may be extended to include the inverse of the mean speed Vi ( in units of the inverse of the Earth's mean orbital speed ) such that:  R = Vi 2 and T 2= R 3 = Vi 6

Additional relationships were also introduced, but the above represents the deducable essence of the matter--information that is quite sufficient for present purposes in so much as it leads readily enough to relations 4 and 5.  With this in mind we may now turn to Canon Mosley's unusual treatment of the spiral formations of Turbo duplicata recounted by Sir D'Arcy Wentworth Thompson (1917:773) 56

From the apex of a large Turritella (Turbo) duplicata a line was drawn across its whorls, and their widths were measured upon it in succession, beginning with the last but one. The measure ments were, as before, made with a fine pair of compasses and a diagonal scale. The sight was assisted by a magnifying glass. In a parallel column to the following admeasurements are the terms of a geometric progression, whose first term is the width of the widest whorl measured, and whose common ratio is 1.1804. [tables and data omitted ]
The close coincidence between the observed and the calculated figures is very remarkable, and is amply sufficient to justify the conclusion that we are here dealing with a true logarithmic spiral. Nevertheless, in order to verify his conclusion still further, and to get partially rid of the inaccuracies due to successive small measurements, Moseley proceeded to investigate the, same shell, measuring not single whorls but groups of whorls taken several at a time: making use of the following property of a geometrical progression, that "if u  represent the ratio of the sum of every even number (m ) of its terms to the sum of half that number of terms, then the common ratio (r ) of the series is represented by the formula:  r = (u - 1) 2/m .

So far, all of this is fascinating in its detail, exactitude and the amount of measurement involved, but it is next part that contains the hidden pearl. Given below in graphical form to match that presented by Sir D'Arcy Wentworth Thompson, the analysis proceeds as follows:

Canon Mosely and Turbo Dupicata

Notwithstanding the methodology, nor being being overly critical, it still seems an unnecessarily convoluted determination, and moreover, in spite of Mosely's confident statement that "It is scarcely possible to imagine a more accurate verification than is deduced from these larger admeasurements," the last relation "r = (1.389)1/2 = 1.1806" is in fact incorrect--the actual result--1.1785--being on the other side of  Phi 1/3 (1.17398) . But what certainly is correct is the following restatement of the last two relations using accurate pheidian values, firstly to four decimal places (after Mosely), and secondly to ten:

Canon Mosely II

And yes, the key values given by Mosely (1.645 and 1.389) are indeed "larger admeasurements" compared to their pheidian counterparts--1.618 and 1.378 respectively. It is true that in the above r is applied in all cases, but it is still astronomically correct in the lower instance. Moreover, Mosely provided in thinly disguised form not only the Golden Ratio, but also by demonstrating the application of the cube and square roots the methodology leading to the mean sidered period (1.618033989), the mean heliocentric distance (1.378240772) and the corresponding mean inverse orbital velocity (1.173984997), i.e., the essence and the root of the Phi-series planetary framework with the last value also the characteristic number, or growth factor k for the shell in question. And having achieved this considerable goal, Mosley did indeed "with safety" not only annex the species Turbo duplicatus, he also passed on his insights in time-honoured tradition, ably assisted by Sir D'Arcy Wentworth Thompson, who in turn passed it on into the next century.

     Unfortunately, this part is missing in the abridged edition of On Growth and Form, as is the entire chapter on Phyllotaxis with its copious notes and related references.

   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 Thompson's voluminous On Growth and Form. 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 Life and Theology of Orpheus) speaks of social decline, loss of knowledge in ancient times and the efforts to preserve it by those who, "though they lived in a base age" nevertheless "happily fathomed the depth of their great master's works, luminously and copiously developed their recondite meaning, and benevolently communicated it in their writings for the general good," that sadly, such times are still upon us. Thus, just as Sir Theodore Andrea Cook, who in the Curves of Life (1914) was unable to define the "well known logarithmic spiral" equated in 1881 with the chemical elements (see the previous section), neither Mosely nor Thompson were able write openly about the either the Golden Ratio or the Pheidian planorbidae. Nor unto the present day, it seems have others, for if not a forbidden subject per se, it long seems to have been a poor career choice, so to speak. Moreover, even after Louis Agassiz introduced Benjamin Pierce's phyllotaxic approach to structure of the Solar System in his Essay on Classification (1857) the matter was swiftly dispatched and rarely referred to again. A possibly momentous shift in awareness, shunted aside with greatest of ease, as the editor of Essay on Classification, (E. Lurie) explained in the short loaded footnote57discussed in the previous section. Nor it would seem, were the works of Arthur Harry Church (On the Relation of Phyllotaxis to Mechanical Law, 1904)58 or Samuel Colman (Nature's Harmonic Unity, 1911) 59allowed to take root. Nor again were the lines of inquiry laid out in Jay Hambidge's (Dynamic Symmetry 1920) permitted to have much on effect on the status quo either, not to mention Sir Theordore Andrea Cook's Curves of Life (1914) and the general the thrust of the many papers published during the previous century.

   Where does this obfuscation and stagnation leaves us now? Wondering perhaps where we might be today if the implications of the phyllotaxic side of the matter introduced in 1849 by Benjamin Pierce had at least been allowed to filter into the mainstream of knowledge with its wider, all-inclusive perspective concerning "life" as we currently understand it. The realization, perhaps, that we may indeed belong to something larger than ourselves, and that as an integral, living part of the Solar System rather than an isolated destructive apex, that we should conduct ourselves with more care and consideration towards all forms of life. Nor can we be in the least encouraged by the fact that since that time there have been almost continual outbreaks of local and international violence on Earth, including two major global conflicts with the imminent threat of another looming on the darkening horizon.*

   And here we might also ask whether humankind was truly well-served over the past century or so by the continued preservation of the status quo and the agendas of special interest groups, and because of this, whether we will even survive the madness of our times, let alone come of age.

* Written in 2002.  Matters have hardly improved since this time ...



  1. Winchester, Simon. The Map that Changed the World, Harper Collins, New York 2001.
  2. Thomson, Sir D'Arcy Wentworth. On Growth and Form, Cambridge University Press, Cambridge 1942; Dover Books, Minneola 1992.
  3. Huntley, H. E. The Divine Proportion: A Study of Mathematical Beauty, Dover, New York 1970.
  4. Kappraff, Jay. Connections:The Geometric Bridge Between Art and Science, McGraw-Hill, New York 1991.
  5. Arganbright, Deane. PHB Practical Handbook of Spreadsheet Curves and Geometric Construction, CRC Press, Boca Raton 1993.
  6. Cook, Sir Theodore Andrea. The Curves of Life, Dover, New York 1978; republication of the London (1914) edition.
  7. Colman, Samuel. Nature's Harmonic Unity, Benjamin Blom, New York 1971.
  8. Mosely, Rev. H. "On the geometrical forms of turbinated and discoid shells," Phil. trans. Pt. 1. 1838:351-370.
  9. Thomson, Sir D'Arcy Wentworth. On Growth and Form, Cambridge University Press, Cambridge 1942; Dover Books, Minneola 1992. 
  10. Nauman, C.F. "Ueber die Spiralen von Conchylieu," Abh. k. sachs. Ges. 1846; "Ueber die cyclocentrische Conchospirale u. uber das Windungsgetz von Planorbis corneus," ibid. I, 1849:171-195; "Spirale von Nautilus u. Ammonites galeatus, Ber. k. sachs. Ges. II, 1848:26; Spirale von Amm. Ramsaueri, ibid. XVI, 1864:21.
  11. Muller, J. "Beitrag zur Konchyliometrie," Poggend. Ann. LXXXVI, 1850:533; ibid. XC 1853:323.
  12. Macalister, A. "Observations on the mode of growth of discoid and turbinated shells," Proc. R.S. XVIII, 1870:529-532.
  13. Telescopium telescopium Linneaus 1758. Source: S. Peter Dance, Shells and Shell Collecting, Hamlyn Publishing Group, London 1972:32.
  14. Conus princeps f. lineolatus Valenciennes1832. Source: G. Paganelli,  Conus princeps f. lineolatus 1197. coneshell.net
  15. Architectonica perspectiva Linneaus 1758. Source: S. Peter Dance, Shells and Shell Collecting, Hamlyn Publishing Group, London 1972:52-53.
  16. Harpa kajiyamai Rehder 1973.Source: Machiko Yamada, (Defunct link: http://www.bigai.ne.jp/pic_book/data20/r001967.html)
  17. Pedinogyra hayii Griffith & Pidgeon 1833 (Hay's Flat-whorled Snail). Source: Machiko Yamada  (Defunct link: http://www.bigai.ne.jp/pic_book/data20/r001967.html)
  18. Planorbis corneus, Linnaeus 1758; Source: Martin Kohl, (Defunct link: http://members.aol.com/Mkohl1/Pulmonata.html)
  19. Nautilus pompilus, Linnaeus 1758. Source: SEASHELLS. World of Nature Series, W.H. Smith, New York.
  20. Haliotis brazieri, Angas 1869. Source: D. L. Beechey, Haliotis brazieri; Index: Shells of New South Wales 20a. Haliotis brazieri (smooth form variant)
  21. Haliotis scalaris, Leach 1814, Source: Machiko Yamada, (Defunct link: http://www.bigai.ne.jp/pic_book/data20/r001967.html)
  22. Thompson, Sir D'Arcy Wentworth. On Growth and Form, 1992:
  23. Conus mercator, Linnaeus 1758 and Conus ammiralis f. hereditarius DA MOTTA, 1987. Source: G. Paganelli, coneshell.net
  24. Conus tulipa, Linnaeus 1758. Source: G. Paganelli, Conus tulipa 710, coneshell.net
  25. Thompson, Sir D'Arcy Wentworth. On Growth and Form, 1992:816.
  26. Harpa goodwini. Source: Guido T. Poppe, Conchology (Defunct link: http://www.conchology.uunethost.be/ )
  27. Clarke, Arthur H.The Freshwater Molluscs of Canada, National Museum of Natural Sciences, Ottawa 1981. 
  28. ibid., p.175.
  29. Kohl, Martin. Freshwater Molluscan Shells: Planorbidae (Defunct link: http://members.aol.com/mkohl2/Planorbidae.html)
  30. Ovid, as quoted by Nicole Oresme in Du Ciel et du monde, Book II, Chapter 25, fols. 144a-144b, p.537.
  31. Liguus virgineus Linnaeus, 1758. Source: Harry Lee, jaxshells.org: http://www.jaxshells.org/ligver.htm  Index: http://www.jaxshells.org/
  32. Helisoma pilsbryi infracarinatum (Great Carinate Ramshorn Snail, Baker 1932). Source: Arthur H. Clarke.The Freshwater Molluscs of Canada, National Museum of Natural Sciences, Ottawa 1981:210.
  33. Helisoma (pierosoma) corpulentum corpulentum (Capacious Manitoba Ramshorn Snail , Say 1824). Source: Arthur H. Clarke. The Freshwater Molluscs of Canada, Ottawa 1981:206.
  34. Promenetus exacuous megas ( Broad Promenetus Dall, 1905. Source: Arthur H. Clarke.The Freshwater Molluscs of Canada, National Museum of Natural Sciences, Ottawa 1981:189.
  35. Thompson, Sir D'Arcy Wentworth, On Growth and Form, 1992:751-753.
  36. Hambidge, Jay. Dynamic Symmetry, Yale University Press, New Haven 1920:16-18.
  37. Kappraff, Jay. Connections:The Geometric Bridge Between Art and Science, McGraw-Hill, New York 1991:46. 
  38. Thompson, Sir D'Arcy Wentworth, On Growth and Form, 1992:791. 
  39. Haliotis parva,  Linnaeus 1758. Source: Molluscs.net: Haliotis parva; Index: http://www.molluscs.net/
  40. Harris, John N. "Projectiles, Parabolas, and Velocity Expansions of the Laws of Planetary Motion," JRASC, Vol 83, No. 3, June 1989:207-218.
  41. Wagner, Jeffrey K. Introduction to the Solar System, Holt, Rinehart & Winston, Orlando 1991:426.
  42. Marine decorated rhyton from Zakros (Crete). Wondrous Realms of the Aegean, selected by the editors, Lost Civilizations Series, Time-Life Books, Virginia 1993:110.
  43. Embossed, carved 12-inch rhyton from Zakros (Crete). Wondrous Realms of the Aegean, selected by the editors, Lost Civilizations Series, Time-Life Books, Virginia 1993:99.
  44. Bretagnon, Pierre and Jean-Louis Simon, Planetary Programs and Tables from -4000 to +2800, Willman-Bell, Inc. Richmond, 1986.
  45. Pierce, Benjamin. "Mathematical Investigations of the Fractions Which Occur in Phyllotaxis," Proceedings, AAAS, II 1850:444-447.
  46. Agassiz, Louis. Essay On Classification,  Ed. E. Lurie, Belknap Press, Cambridge 1962:127-128.
  47. Harris, John N. "Projectiles, Parabolas, and Velocity Expansions of the Laws of Planetary Motion, " JRASC, Vol 83, No. 3, June 1989:216.
  48. Raup, David. "Computer as aid in describing form in gastropod shells," Science 138, 1962:150-152. 
  49. Phillips, Tony and Stony Brook, "The Mathematical Study of Mollusk Shells" American Mathematical Society; AMS.ORG
  50. Thompson, Sir D'Arcy Wentworth. On Growth and Form, Cambridge University Press, Cambridge 1942; the complete unabridged reprint, Dover Books, Minneola 1992.
  51. Turritella duplicata, Source: Canon Mosely, in  Sir D'Arcy Wentworth Thompson, On Growth and Form, the complete unabridged edition, 1992:772.
  52. Euhoplites truncatus (Spath 1925). Source: Jim Craig: Euhoplites truncatus. Index: Fossils of the Gault Clay and Folkestone Beds of Kent, UK
  53. Dawkins, Richard. Climbing Mount Improbable, W.W. Norton, New York 1996:198:223. 
  54. _____________  Aruneus diademus Spider.Climbing Mount Improbable, Norton, New York 1996:58. 
  55. On Growth and Form, 1942:784. 
  56. On Growth and Form, 1942:773. 
  57. Lurie, E. (Ed.) Essay On Classification,  Belknap Press, Cambridge 1962:128.
  58. Church, Arthur Harry. On The Relation of Phyllotaxis to Mechanical Law, Williams and Norgate, London 1904; see also: http://www.sacredscience.com (cat #154).
  59. Colman, Samuel. Nature's Harmonic Unity, Benjamin Blom, New York 1971:3.
  60. Thatcheria mirabilis (Angas 1877). Source: Mathew Ward, Photographer; in Peter S. Dance, Shells, Stoddart, Toronto 1992.
  61.  Hildoceras bifrons, (Bruguière 1789).  Figure 1b2. Source: Hervé Châtelier, Ammonites et autres spirales - Hervé Châtelier.
  62. Dactylioceras commune (Sowerby 1815).  Figure 1Ca. Source: Hervé Châtelier, Ammonites et autres spirales.
  63. Porpoceras vortex (Simpson 1855).  Figure 1Cb.  Source: Hervé Châtelier, Ammonites et autres spirales.
  64. Protetragonites oblique-strangulatus (Kilian 1888).  Figure 1Cc. Source: Hervé Châtelier, Ammonites et autres spirales.
  65. Lytoceras cornucopia (Young & Bird 1822).  Figure 1Cd.  Source: Hervé Châtelier, Ammonites et autres spirales.
  66. Epophioceras sp. (Spath, 1923). Figure 1D. Source: Christopher M. Pamplin, (Defunct link: Lower Jurassic Ammonites. http://ammonites.port5.com/epop.htm)
  67. Acanthopleuroceras valdani (D'Orbigny). Figure1d2. Source: Jean-ours and Rosemarie Filippi: Jurassic ammonites and fossil brachiopoda.
  68. Aegoceras (Aegoceras) capricornus (Schlotheim). Figure 1d3:  Source: Jean-ours and Rosemarie Filippi: Jurassic ammonites and fossil brachiopoda.
  69. Ethioceras raricostatum (Figure 1e). Line drawing by Soun Vannithone, in Winchester, Simon. The Map that Changed the World, Harper Collins, New York 2001:1

For U.K. Ammonites, see:  FOSSILS OF THE GAULT CLAY AND FOLKESTONE BED OF KENT, UK  by the late Jim Craig, and  FOSSILS OF THE LONDON CLAY by Fred Clouter.

Copyright © 2002. John N. Harris, M.A.(CMNS). Last updated on March 31, 2009. Cosmetic update October 9, 2010.
Ammonite graphics (Figures 1b2, 1c and 1d) added on April 29, 2003; Figure 21c on June 4 2003; Figure. 7b added 11 May, 2004; Figures 22a, 22b, and 1d3 added 17 July, 2004. Figures1 and 1e on 18 July, 2004.

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