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.
Prologue: 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
planetary period spiral, Phi 2.
1. Spira Solaris, Growth
factor k = Phi 2
(For more on the Capacious Manitoba Ramshorn snail see Figure 13 below)
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
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
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 7/3, k = 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.
Figure 1a. Triple spiral configurations for Pheidas (left) and Spira Solaris (right)
Figure 1b. Dual
k = Phi 2
to k =
Phi 16 plus k =
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
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,
it turned out, was also capable of producing the output graphics and
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:
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
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
relatable to the Pheidian sixths and thirds associated
with Spira Solaris
to merit the title Pheidian
= 6/3 = Phi 2 (c:
single, and b: dual
1C: k = Phi to the 3/3, 4/3,
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
Pheidias), ammonite from Lower
Christopher M. Pamplin.66 Figure
1d2: k = Phi 4/3
1d3: k = Phi 5/3 are from Jurassic
ammonites and fossil brachiopoda 67 by
Jean-ours and Rosemarie Filippi 67.
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.
Fig. 2 The Pheidian Planorbidae. Thirds: Growth Factors/Expansion Rates 1.174 to 4.236
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.
underside and top of Architectonica perspectiva;
Single spiral (B and C); open double spiral (A); each k = Phi 1
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,
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...
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) 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 brazeri with the single Pheidian Spiral k = Phi 8/3
Figure 2E. Hay's
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 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.
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
used in Figure 2 is from the above source; the parent
shown below as Figure 10. Many members of this family appear to be pheidian
in one form or another, and like Architectonica, Trochus,
many seem to be "golden" in addition--a worthy subject in its own
though it will not be pursued further in the present survey.
Capacious Manitoba Ramshorn snail with double and single
k = Phi 12/6, k = Phi 11/6 and k = Phi 10/6.
PHI-SERIES AND THE SOLAR SYSTEM REVISITED
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:
and 6b. The Fundamental Distance Constants
and 6d. The Fundamental 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
3, only Phi 5/3
below Spira Solaris, while the remainder extend sequentially
beyond the latter, i.e., Phi 7/3,
Phi 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
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 = aθ. 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.
WHIRLING RECTANGLES, SQUARES, AND EQUIANGULAR SPIRALS
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).
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
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
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
This should become more apparent from Figure 15, which incorporates the
above data for Spira Solaris and in addition shows the orientation of
spirals to the parent rectangle:
15. Rectangle with Inner and outer Spirals for Spira Solaris, k =
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
graphics, firstly with respect to the original "Golden" rectangle and
the spiral k = Phi 4 and
to the associated quadrantal radii vectores for the same spiral; see
(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.
15b. The Golden Rectangle with Inner and outer Spirals for k = Phi
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:
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 k = Phi 4
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:
B. TO GOVERN ALLTHINGS REVISITED
B.1. THE PHEIDIAN PLANORBIDAE AND ANGULAR MOMENTUM
ANGULAR MOMENTUM AND INVERSE VELOCITY
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:
Pheidian Period, Distance and Velocity Relations
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)
angular momentum is essentially the product of the mass (m),
the mean distance (R) and the mean
However, because of the Period-Distance-Velocity relationship
T 2 = R 3
= Vi -6 and the
velocity variant, the formula: L = Mav may be expressed
in the form Mass x Vi 2x
reduces further to Mass x Vi. Thus, in simpler and
more practical terms:
Angular Momentum L = mvi
Table 2. Percent
System L, The Phi-Series Exponential Framework
and the Modern Solar System (after Wagner 1991:426)
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):
- Vi Venus - Vi Mercury approximates the mean velocity of Uranus
- Vi Saturn - ViJupiter approximates the mean velocity of Mars. [Relation 4b]
- 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:
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
brings with it further complications in so much as it represents the
of the varying motion of three major superior
and Uranus--thus the effect of two adjacent difference cycles. Thus it
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:
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.
SPECULATION CONCERNING ORIGINS
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.
PHEIDIAN CONSTANT 2.61803398874 AND ITS INVERSE
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:
Table 5b. The
Mean Values, Jupiter, Saturn and the J-S Synodic Cycle.
Arithmetic Mean: ( SaturnVr + JupiterVr) = 0.381280708
* * * * *
THE MATTER OF LOST
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"
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.
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.
Fig. 23. Canon Mosely's Turritella Duplicata with the scale of the spiral k = Phi 1/3
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
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:
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:
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
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
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
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
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.
in 2002. Matters
have hardly improved since
this time ...
Copyright © 2002. John N. Harris,
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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