[
EXCERPT] PART
IVD2C: THE PHEIDIAN PLANORBIDAE
A. TO ORDER ALL THINGS NOT ORDERED
A.1
THE PHEIDIAN PLANORBIDAE
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 centuriesroughly from
the time of Carl Linnaeus (17071778) 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 bestselling
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}
CHAPTEROPENING ILLUSTRATIONS
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, longextinct marine animals that
were so named because their coiled and chambered shells resembled
nothing so much as the horns of the ancient Egyptian ramgod, 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 speededup version of the
evolutionary advancement of the ammonite will be disappointed:
Ammonites floating, pulsating, slowswimming 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 smoothershelled 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 crosssection 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: ixx)
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 furtherAmmon, 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. Spira Solaris, Growth
factor k = Phi ^{2}
For more on
this spiral and the Capacious
Manitoba Ramshorn snail see Figure 13
{omitted here}
Thus
further Pheidian spirals with growth factors between 1.1739850
and 2.61803398874 : 1, five of which had already been
generatedtwo 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 allinclusive 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 investigationone that was to have a number of unexpected
results.
A.2. THE
PLANORBIDAE: FORMS
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 Centuriesespecially
with respect to snailsopened 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 ^{4/3}and 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 spiralslikely 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 employedthus 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
taskthe
basic mathematical elements have long been known, e.g., as
described
in detail by Sir D'Arcy Wentworth Thompson^{2 }(On
Growth and Form, 1917,1942, 1966 and 1992); by H.E. Huntley^{
3 }(The Divine Proportion, 1970my 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 asrequired basis.
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 additionsthe latter configuration identical to the former,
but without the crossreference 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
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 XARAX,
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 twodimensional spiral to drawings of
threedimensional 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 1d1d3 below).
Nevertheless, the investigationeven for the relatively
flat and largely symmetrical ammonitesbegan with no great
expectations, but happily with a wealth of available material. And, as
it turned out, Soun Vannithone's accurate planview 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
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^{ 6165}) 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
Jeanours and Rosemarie Filippi ^{67}.
Figure
1b2.
Ammonitesr^{ 61}
and the Dual Pheidian
configuration ( k = Phi
^{2})
^{
}
Figure
1c. Ammonites ^{6265}
and the Pheidian Planorbidae
I ( k = Phi,
Phi ^{4/3}, Phi ^{5/3} and Phi ^{2}
):
^{
}
Figure
1d.
Ammonites ^{66} and
the
Pheidian Planorbidae II (k = Phi)
Fig. 1d2. Ammonites ^{67} and the Pheidian Planorbidae
III ( k = Phi ^{4/3})
Fig. 1d3. Ammonites^{ 68} and the Pheidian Planorbidae
IV ( k = Phi ^{5/3})
Thus the Phedian
Planorbidae as applied to ammonites from
an initial surveyone small inroad into a complex subject with
accompanying dynamic, temporal and historical overtones that all
appeared to merit further examination.
Nextbased 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 laternot because of its simplicity, but the
exact oppositeits 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.
A.3. PHEIDIAN
PLANORBIDAE APPLIED TO SHELLS I
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 othersprimarily 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 yaxis with the corresponding equiangles of the
associated pheidian spirals along the xaxis 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)^{11}and 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:
 Figure 2A: Crosssection of Telescopium telescopium^{13
}(Linnaeus 1758; primary spiral: k = Phi
^{1/3}; growth factor: 1.1739850
).
 Figure 2B: Conus princeps f. lineolatus^{14}(Valenciennes
1832; primary spiral: k = Phi ^{2/3};
growth factor: 1.378240772 ).
 Figure 2C: Architectonica perspectiva^{15}
(Perspective Sundial; Linnaeus 1758; primary spiral: k
= Phi ^{3/3} thus the Pheidias
Spiral ; growth factor: 1.618033989 ).
 Figure 2D: Plan view of Harpa kajiyamai^{16}
(Rehder 1973; primary spiral: k = Phi ^{4/3};
growth factor: 1.899547627 ).
 Figure 2E: Pedinogyra hayii ^{17}(Hay's
Flatwhorled Snail; Griffith & Pidgeon 1833; primary spiral: k
= Phi ^{5/3}; growth
factor: 2.230040415
).
 Figure 2F: Planorbis corneus^{18}(Ramshorn
Snail; Linnaeus 1758; primary spiral: k = Phi
^{6/3} i.e., Phi ^{2}
thus Spira
Solaris; growth factor: 2.618034989 ).
 Figure 2G: Nautilus pompilus in crosssection^{19}
(Chambered Nautilus; Linnaeus 1758; primary spiral: k = Phi
^{7/3}; growth factor: 3.073532624
).
 Figure 2H: Haliotis brazieri^{20}(Abalone;
Angas 1869; primary spiral: k = Phi ^{8/3};
growth factor: 3.608281187 ).
 Figure 2 I: Haliotis scalaris^{ 21}(Abalone;
Leach 1814, primary spiral: k = Phi ^{9/3}
; growth factor: 4.236067978 ).
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 welldefined open spiral forms) is also deferred until later
in view of the possible relationship between this type of shell and the
complexities inherent in phirelated "whirling rectangles" (esp.
Haliotis parva; k = Phi ^{4}
; see A4 below).
Fig. 2 The Pheidian
Planorbidae. Thirds: Growth Factors/Expansion Rates 1.174 to 4.236
[
COMMENTS AND ADDITIONAL GRAPHICS FOR FIGURES 2A AND 2B OMITTED ]
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 threefold number"Said
Aristotle, prince of philosophers and neverfailing 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.
The
underside and top of Architectonica perspectiva;
Single spiral (B and C); open double spiral (A); each k = Phi
^{1
}
[
COMMENTS AND INDIVIDUAL GRAPHICS FOR FIGURES 2D2I AND ADDITIONAL
SHELLS OMITTED ]
[ BIBLIOGRAPHY BELOW. REMAINING SECTIONS
OF SBB4D2, THE "DYNAMICS" OF THE MATTER, AND THE APPENDIX OMITTED ]
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1962:127128.
 Harris, John N.
"Projectiles, Parabolas,
and Velocity Expansions of the Laws of Planetary Motion, " JRASC,
Vol 83, No. 3, June 1989:216.
 Raup,
David. "Computer as aid in describing form in gastropod shells," Science
138, 1962:150152.
 Phillips, Tony and
Stony Brook, "The
Mathematical Study of Mollusk Shells" American Mathematical
Society; AMS.ORG.
 Thompson, Sir D'Arcy
Wentworth. On Growth and Form, Cambridge University Press,
Cambridge 1942; the complete unabridged reprint, Dover Books, Minneola
1992.
 Turritella duplicata,
Source: Canon Mosely, in Sir D'Arcy Wentworth Thompson, On
Growth and Form, the complete unabridged edition, 1992:772.
 Euhoplites truncatus
(Spath 1925).
Source: Jim Craig: Euhoplites
truncatus. Index: Fossils
of the Gault Clay
and Folkestone Beds of Kent, UK
 Dawkins, Richard. Climbing
Mount Improbable, W.W. Norton, New York 1996:198:223.
 _____________
Aruneus diademus Spider.Climbing
Mount Improbable, Norton, New York 1996:58.
 On Growth and Form,
1942:784.
 On Growth and Form,
1942:773.
 Lurie, E. (Ed.) Essay
On Classification, Belknap Press, Cambridge 1962:128.
 Church, Arthur Harry. On
The Relation
of Phyllotaxis to Mechanical Law, Williams and Norgate, London
1904; see also: http://www.sacredscience.com
(cat #154).
 Colman, Samuel. Nature's
Harmonic Unity, Benjamin Blom, New York 1971:3.
 Thatcheria mirabilis
(Angas 1877). Source: Mathew Ward, Photographer; in Peter S. Dance,
Shells, Stoddart, Toronto 1992.
 Hildoceras
bifrons, (Bruguière 1789). Figure 1b2.
Source: Hervé Châtelier, Ammonites et autres
spirales
 Hervé Châtelier.
 Dactylioceras
commune (Sowerby 1815). Figure
1Ca.
Source: Hervé Châtelier, Ammonites et autres
spirales.
 Porpoceras vortex
(Simpson 1855). Figure
1Cb.
Source: Hervé Châtelier, Ammonites et autres
spirales.
 Protetragonites
obliquestrangulatus
(Kilian 1888). Figure
1Cc.
Source: Hervé Châtelier, Ammonites et autres
spirales.
 Lytoceras cornucopia
(Young & Bird 1822). Figure
1Cd. Source: Hervé Châtelier, Ammonites
et
autres spirales.

Epophioceras
sp. (Spath, 1923). Figure
1D. Source: Christopher M.
Pamplin, (Defunct link: Lower
Jurassic Ammonites. http://ammonites.port5.com/epop.htm)


 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 April 2, 2009.
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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