APPENDIX TO THE PHEIDIAN PLANORBIDAE
THE MATTER OF LOST
LIGHT
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 Raup^{48 }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
138:150152)
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 barebones 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 duplicata^{ 51}.
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 crosssection 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 twodimensional 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
fourdecimal place pheidian growth factorsspecifically Phi
^{1/3}, Phi ^{7/6}and 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. 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}):
Fig. 22a. The single spirals k = Phi ^{7/3}, k = Phi ^{5/3
}and k = Phi ^{2/3
}
plus bivalve 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 multidimensional 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 ^{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 constructionlogical
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 admirationas D'Arcy
Wentworth Thompson duly noted in On Growth and Form"Canon
Mosely
was a man of great and versatile ability"^{ 55}and 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. 3,1989:207218) 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.
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: T^{2} = R^{3} (
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 matterinformation 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 incorrectthe actual result1.1785being 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 counterparts1.618 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 Phiseries 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 timehonoured 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 footnote^{57}discussed
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) ^{59}allowed
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, allinclusive 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
wellserved 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 ...
END
OF PART
IVD2c
REFERENCES
 Winchester, Simon. The
Map that Changed the World, Harper Collins, New York 2001.
 Thomson, Sir D'Arcy
Wentworth. On
Growth and Form, Cambridge University Press, Cambridge 1942;
Dover Books, Minneola 1992.
 Huntley, H. E. The
Divine Proportion: A Study of Mathematical Beauty, Dover,
New York 1970.
 Kappraff, Jay. Connections:The
Geometric Bridge Between Art and Science, McGrawHill, New York
1991.
 Arganbright, Deane.
PHB Practical Handbook of Spreadsheet Curves and Geometric Construction,
CRC
Press, Boca Raton 1993.
 Cook, Sir Theodore
Andrea. The
Curves of Life, Dover, New York 1978; republication
of the London (1914) edition.
 Colman, Samuel. Nature's
Harmonic Unity, Benjamin Blom, New York 1971.
 Mosely, Rev. H. "On
the geometrical forms of turbinated and discoid shells," Phil. trans.
Pt. 1. 1838:351370.
 Thomson, Sir D'Arcy
Wentworth. On Growth and Form, Cambridge University Press,
Cambridge 1942; Dover Books, Minneola 1992.
 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, 1849:171195; "Spirale von Nautilus u. Ammonites galeatus, Ber.
k. sachs. Ges. II, 1848:26; Spirale von Amm. Ramsaueri, ibid.
XVI, 1864:21.
 Muller, J. "Beitrag
zur Konchyliometrie," Poggend. Ann. LXXXVI, 1850:533; ibid.
XC 1853:323.
 Macalister, A.
"Observations on the mode of growth of discoid and turbinated shells," Proc.
R.S. XVIII, 1870:529532.
 Telescopium
telescopium Linneaus
1758. Source: S. Peter Dance, Shells and Shell Collecting,
Hamlyn Publishing Group, London 1972:32.
 Conus princeps f.
lineolatus Valenciennes1832. Source: G. Paganelli, Conus
princeps f. lineolatus 1197 Index: www.coneshell.net
 Architectonica
perspectiva Linneaus 1758. Source: S. Peter Dance, Shells and
Shell Collecting, Hamlyn Publishing Group, London 1972:5253.
 Harpa kajiyamai Rehder
1973.Source: Machiko Yamada, http://www.bigai.ne.jp/pic_book/data20/r001332.html
Index: Micro shells Homepage
 Pedinogyra hayii
Griffith & Pidgeon 1833 (Hay's Flatwhorled Snail). Source: Machiko
Yamada, http://www.bigai.ne.jp/pic_book/data10/r000918.html
Index: Micro shells Homepage
 Planorbis corneus,
Linnaeus 1758;
Source: Martin Kohl, Freshwater
Molluscan Shells: Pulmonata; Index: Freshwater
Molluscan Shells
 Nautilus pompilus,
Linnaeus
1758. Source: SEASHELLS. World of Nature Series, W.H.
Smith, New York.
 Haliotis brazieri,
Angas 1869. Source: D. L. Beechey, Haliotis
brazieri; Index: Shells of New South Wales. 20a. Haliotis brazieri (smooth form variant)
 Haliotis
scalaris, Leach 1814, Source: Machiko Yamada, http://www.bigai.ne.jp/pic_book/data20/r001967.html
Index: Micro shells Homepage
 Thompson, Sir D'Arcy
Wentworth. On Growth and Form, 1992:
 Conus mercator, Linnaeus 1758 and Conus ammiralis f. hereditarius DA MOTTA, 1987. Source: G. Paganelli,
Index: www.coneshell.net
 Conus tulipa, Linnaeus
1758. Source: G. Paganelli, Conus tulipa 710
Index: www.coneshell.net
 Thompson, Sir
D'Arcy Wentworth. On Growth and Form, 1992:816.
 Harpa goodwini.
Source: Guido T. Poppe, Conchology http://www.conchology.uunethost.be/
 Clarke, Arthur H.The
Freshwater Molluscs of Canada, National Museum of Natural Sciences,
Ottawa 1981.
 ibid., p.175.
 Kohl, Martin. Freshwater
Molluscan Shells: Planorbidae ; Index: Freshwater
Molluscan Shells
 Ovid, as quoted by
Nicole Oresme in Du Ciel et du monde, Book II, Chapter
25, fols. 144a144b, p.537.
 Liguus virgineus
Linnaeus, 1758. Source: Harry Lee, jaxshells.org: http://www.jaxshells.org/ligver.htm
Index: http://www.jaxshells.org/
 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.
 Helisoma
(pierosoma) corpulentum corpulentum (Capacious Manitoba Ramshorn
Snail^{ }, Say 1824). Source: Arthur H. Clarke. The
Freshwater
Molluscs of Canada, Ottawa 1981:206.
 Promenetus
exacuous megas (
Broad Promenetus Dall, 1905. Source: Arthur H. Clarke.The Freshwater
Molluscs of Canada, National Museum of Natural Sciences, Ottawa
1981:189.
 Thompson, Sir
D'Arcy Wentworth, On Growth and Form, 1992:751753.
 Hambidge, Jay. Dynamic
Symmetry,
Yale University
Press, New Haven 1920:1618.
 Kappraff, Jay. Connections:The
Geometric Bridge Between Art and Science, McGrawHill, New York
1991:46.
 Thompson, Sir
D'Arcy Wentworth, On Growth and Form, 1992:791.
 Haliotis parva,
Linnaeus 1758. Source: Molluscs.net: Haliotis
parva; Index: http://www.molluscs.net/
 Harris, John N.
"Projectiles, Parabolas, and Velocity Expansions of the Laws of
Planetary Motion," JRASC, Vol 83, No. 3, June 1989:207218.
 Wagner, Jeffrey K. Introduction
to the Solar System, Holt, Rinehart & Winston, Orlando 1991:426.
 Marine decorated
rhyton from Zakros
(Crete). Wondrous Realms of the Aegean, selected by the
editors, Lost Civilizations Series, TimeLife Books, Virginia 1993:110.
 Embossed, carved
12inch rhyton from Zakros (Crete). Wondrous Realms of the Aegean,
selected by the editors, Lost Civilizations Series, TimeLife Books,
Virginia 1993:99.
 Bretagnon, Pierre and
JeanLouis Simon, Planetary Programs and Tables from 4000 to +2800,
WillmanBell, Inc. Richmond, 1986.
 Pierce, Benjamin. "Mathematical Investigations of
the Fractions Which Occur in Phyllotaxis," Proceedings, AAAS, II
1850:444447.
 Agassiz, Louis. Essay
On Classification, Ed. E. Lurie, Belknap Press, Cambridge
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.



 Ethioceras raricostatum (Figure 1e). Line drawing by Soun Vannithone, in Winchester, Simon. The
Map that Changed the World, Harper Collins, New York 2001:1
Copyright © 2002. John N. Harris,
M.A.(CMNS). Last
updated on July 20, 2004.
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.
RETURN TO THE MAIN PAGE
PREVIOUS SECTION
NEXT SECTION