J. Moll. Stud. (1997), 63,547-554
© The Malacological Society of London 1996
SPIRAL GROWTH: THE 'MUSEUM OF ALL SHELLS'
REVISITED.
BERNARD TURSCH
Laboratoire de Bio-Ecologie, Faculty des Sciences, Universiti Libre de Bruxelles, 50 av. F.D. Roosevelt,
1050 Brussels, Belgium.
(Received 18 November 1996; accepted 11 March 1997)
ABSTRACT
Raup's model of shell growth, now in standard use,
is operational only for strictly conispiral shells.
The pertinence of evolutionary interpretations of
the distribution of existing shells in the morphospace
defined by Raup's parameters is questioned. A
simple, more general model accounts for the nonisometric growth of many shells. Some aspects of the
distribution of existing shells in the morphospace
derived from the new model are discussed.
INTRODUCTION
The standard tool for the geometrical analysis
of coiled shells is the model developed by Raup
in a series of papers culminating in his wellknown 1966 synthesis (Raup, 1966). In that
model (summarised below), coiled shells are
derived from an equiangular (logarithmic)
spiral and shell shape is determined by four
parameters: W (the rate of whorl expansion), T
(the rate of translation), S (the shape of the
generating curve) and D (the position of the
generating curve relative to the axis of coiling).
The model, coinciding with a boom in the
popularity of computers, caused a nourish
in theoretical morphology. An excellent
comparative review of the many later shell
models—and a very interesting analysis of their
evolution—is given by Stone (19%).
Within the frame of Raup's model, if one
assumes S to be constant (i.e. circular), the
shape of a helico-spiral shell is described by a
point in a tridimensional space with axes W, T
and D. This space is now commonly known as
the 'shell morphospace', the set of all possible
outcomes from a given geometrical/mathematical model. It can be perceived as the 'Museum
of All Shells', a most fitting name coined by
Dawkins (19%). Analysing the reasons why
existing shells are found only in a small, well-
defined region of the morphospace is now a
classic problem in evolutionary biology.
This paper questions the generality of Raup's
model, the pertinence of the interpretations
of his morphospace and advocates a simple,
alternative approach.
PROBLEMS WITH THE CLASSICAL
SHELL MODEL
If y is the axis of coiling, Ko the generating
curve and K] the generating curve after one
revolution (see Fig. 1), then Raup's parameters
(for a clear account, see Sheldon, 1993) can be
denned as:
= -r
D=
(a + b)
T=Cx
and must apply to every subsequent volution.
Figure 1. Construction for Raup's parameters (see
text).
548
BERNARD TURSCH
Figures 2. A. The distribution of shells in Raup's morphospace (modified from Raup 1965). B. The same
model, with calculated allowed distribution.
This model has been extremely useful but
has important drawbacks. It is not general, as it
was conceived to emulate only shells with a
regular, flat-sided spire (conispiral and planispiral shells). But a disquietingly large proportion of gastropod shells—most, according to
Vermeij (1993)—change their shape during
growth, resulting in concave or convex spires.
Such deviations from conicity were considered
to result from ontogenetic changes, that could
be emulated only by imposing large, ad hoc
variations to parameter T (Raup, 1966).
Another problem arises with bivalves: valves
produced by Raup's model cannot close tightly
(a gap remains open at the hinge). This was
explained by invoking a difference between a
'biological generating curve' and a 'geometrical
generating curve' (Raup, 1966). A similar explanation was needed to account for limpetshaped gastropods (Raup, 1966). Furthermore,
determining Raup's parameters from actual
shells is often not a straightforward operation
(for an example, see Kohn & Riggs, 1975).
There is also a serious problem with Raup's
morphospace (Fig. 2A) and its interpretation:
several authors—amongst others Schindel
(1990), Stone (1995)—have already pointed
out that Raup's parameters (the axes of the
morphospace) are not independent. The effects
of the interdependence of Raup's parameters
were experimentally evidenced by Schindel
(1990), who showed that 53 species of gastropods are more widely distributed in a morphospace defined by his own, independent
parameters (deriving from actual shell
measurements) than they are in Raup's morphospace.
I suggest a simpler way of showing that the
observed distribution of shell shapes in Raup's
system of interdependent coordinates could
reflect a mathematical constraint rather than a
biological phenomenon (favoured evolutionary
pathways). T is indeed linked to W and D by
the relation:
x
(a' + 672) - (a + 6/2)
bD
(T^D)
,
bDw
Jy^D)
and
where
b
=bW
As c and b being set at the start of the construction, Tis thus entirely determined by the values
of D and W. If one is concerned only with the
shape (and not the size) of shells, one should
compare models based upon the same generating curve (say with a diameter b = 1). Within
Raup's 'cube', one can then calculate all values
of Tfor values of c from 0 to 1.5 b units (a realistic range, encompassing practically all shells).
One finds that T has values in the range 0 to 4
(the range considered by Raup) only in a restricted volume, bounded by a surface (shaded
in Fig. 2B). This 'mathematically allowed' volume is largely filled by the observed distribution
of gastropods and coiled cephalopods. It also
accommodates the thin volume occupied by
brachiopods. No shell strictly obeying Raup's
conditions should be found below the boundary
surface, for purely mathematical reasons (the
SPIRAL GROWTH OF SHELLS
location of bivalves is unexpected and probably
results from the adjustments mentioned above).
So, it is likely that the observed general distribution simply reflects a starting condition: most
real shells begin their growth with values of c in
the range 0 to 1.5ftunits.
The general problem of the classical model
—having to make constants vary in order to
approximate reality—stems from the assumption that all shells can be 'shoehorned' into a
conispiral paradigm.
AN OPERATIONAL MODEL
Most of the difficulties encountered with Raup's
model can be solved by using a conventional,
elementary description of the helico-spiral in
which all parameters are independently defined. The model described here is intended as
a probe for biological studies rather than for
realistic simulation of specialized structures, so
it can be kept very simple.
The generating curve Ko is, as usual, taken to
549
be an ellipse because the aperture of most
shells can be approximated by (or inscribed in)
this shape. The size and proportions of the
ellipse Ko are determined by its smallest diameter vv0 (here always equal to 1) and its ellipticity
e (see Fig. 3A). As in previous works (e.g.
Illert, 1989; Cortie, 1989), axes Xj, yd and z*
parallel to x, y, z are traced from the centre of
the ellipse. Another system of orthogonal axes
x,, ys (the axes of the ellipse) and z, (perpendicular to the plane of the ellipse) is constructed
on the centre of Ko. The angles a, p and 8
determine the spatial orientation of Ko. The
characteristics of the generating curve Ko are
summarised in a complex parameter S, here
reported as S = (e\a\p\8).
The position of the centre of Ko (the generating curve) as well as that of K, (the generating
curve after one revolution) relative to the axis
of coiling y are set at the start (see Fig. 3B). r0 is
the radial distance of the centre of Ko and <4
is the distance separating the centres of Ko and
K, along axis y. One then sets the distance d,
separating the centre of K, from the centre of K2
Figure 3. A. Determination of the initial parameters of the generating curve. B. Determination of the growth
parameters $^ £ and •H'.
BERNARD TURSCH
550
(the generating curve after two full revolutions).
One can now define 3 growth parameters:
<£= r,/r0 = rjr^
L
<£is the rate of Radial expansion, L the rate
of Longitudinal expansion and 'W the rate of
Whorl expansion (the same as Raup's parameter W). The three parameters 4^ L, <W are
entirely independent. One will notice that <£
and iVare determined after one revolution of
the generating curve, while £ is determined
only after a second revolution.
The internal definition of L (the only originality, if there is any, in this otherwise obvious
model) allows one to avoid the problem of
having to select a point of origin for the helicospiral. The position of this point in relation to
Ko determines much of the shape of the resulting surface of revolution, a difficulty that has
plagued previous models.
dn-i (the distance, along axis y, of the centre
of the generating curve Ko to the centre of the
curve K^, after n revolutions) is given by the
geometrical progression
where (see Fig. 3 A) v0 is the projection of Ko on
axis y and ho is the projection of Ko on axis x.
If a = 0 (a very common case) then p =
ewo/2'
if P = 0 then q =
In this simple model, the outcomes of the
construction are largely predictable by comparing the values of the parameters (more detailed
examples will be given in a separate, forthcoming paper). For example, ifflf,= L then, in
sagittal view, the revolution of the centre of
the generating curve (but not necessarily the
outline of the shell) takes place on a conical
surface. If <S^ = L = iVthen growth will be isometric, leading to shells with true conical
spires. If ^.= VJ+ L then growth will be nonisometric, the shape of the shell varying during
growth (see Figs. 4A, 4B, 4C), hence the
importance of specifying n, the number of volutions. In this case, \i <R^< L the spire will be
convex (see Fig. 4D); if ^, > L the spire will be
concave (see Fig. 4C).
The angular orientation of the generating
curve has a determining influence on the final
rn, the radial distance of the centre of K,, after n shape. For instance, when t+'has the very large
values required to emulate the shape of limpets
revolutions, is rn = ro<R?
It follows that the final shape will be largely and bivalves, increasing values of p (all other
determined by the initial lengths r0 and do. So it parameters being equal) yield shapes varying
is useful to define the two non-dimensional from that of a typical Bivalve to that of a
typical Limpet (see Figs. 5A, 5B, 5C, 5D).
parameters:
When n > 0.5 and p = 0, the construction of
bivalves encounters a fabricational problem.
The two valves cannot fit tight (see Fig. 5E)
ho/2
vo
R = 1 3 L = 1 05
W^ 1 3
S = 1 5VCMK)
q=1
p»15
R=15
L*1.8
W=15
N»5
S = 1 5W\0V0
1 p=06
Figure 4. Some properties of growth parameters ^ C and ftf All shell models in sagittal view. A to C. If ^.= W*
* C, the shape of the shell varies during growth. If <R_> L the spire is concave. D. If <Sj= W>=fr C and 1{_< C the
spire is convex.
SPIRAL GROWTH OF SHELLS
551
because their protruding umbos would have to parameters yield impossible or unworkable
overlap [the old 'problems of bivalveness' shapes, stressing the importance of fabrica(Raup, 1966)]. The same steric hindrance could tional constraints.
prevent a limpet to stick to a flat surface. This
The shapes of some shells, for instance
hitherto vexing problem simply ceases to exist Eulimidae with a curved coiling axis, convexif the generating curve has a small initial angle sided limpets, shells in the family Clausiliidae
p (see Fig. 5F).
or in the genus Cerion (with diameters diminThis very elementary model has many advan- ishing near the end of growth), Vermetidae and
tages. It can, of course, produce all the shapes other shells with irregular coiling, still remain
derived from the model of Raup. In addition, it unexplained if one does not invoke ontogenetic
explains variations of shape during growth and changes. But the number of such cases is now
thus accounts for the very common occurrence very much reduced.
of shells with concave or convex spires, without
Amongst the many previous shell models
any need to postulate 'ontogenetic changes' (see Stone, 1996), three do share with the pre(Raup, 1966) or 'inadequacies of the logarith- sent approach the characteristics of being based
mic model' (Vermeij, 1993). It accounts for upon algebraically independent parameters,
bivalves and limpet-shaped gastropods without using an external coordinate system (coiling
having to invoke differences between 'geomet- axis) and accounting for non-isometric growth.
ric' and 'biological' generating curves. It also These are the models of Cortie (1989), Schindel
answers another 'unexplained' problem
(1990) and Stone (1995). In all of these, anisomrecently brought forwards (Vermeij, 1993): etry can be simulated only by varying the values
why the apex of high-conic limpets is more cen- of shell parameters during growth. In the
trally located than that of low-conic ones (see present model, the mode of growth is preFigs. 5A, 5B, 5C, 5D). Many combinations of
determined by the relative values of the initial
R-W L"0
W-10 4 N-025
S - VOPC
q• 1 p•0
= 10
R-W L"0
W-100 N-064
S- 1WN3
P = 60
P = 30
R-W L-0
W-100 N-064
Figure 5. Some properties of angular parameter p. All shell models seen in apical view. A-D. Effect of variations of angle p (all other parameters constant) for W> = 104. A matches a typical bivalve, D a typical limpet.
E, F. A solution to the 'problem of bivalveness'. When p = 0 the umbos overlap (E, black arrow). When p =
1Q (F) the fabricational problem disappears. G, H. For multispiral shells, prefered values for parameter p are in
a range (shadowed), which decreases sharply with increasing values of •W.
552
BERNARD TURSCH
shell parameters. The model of Schindel (1990)
differs very much because the aperture is not
defined by a generating curve and the growth
parameters are defined by measurements made
at specific points of the surface of the shell.
The model proposed here is so simple that
rough simulations of sagittal shell sections can
be made without any calculation, by using a
small computer equipped with one of the many
drawing programs allowing rotation and proportional scaling of objects. All one has to do
(see Fig. 3B) is to apply successive scalings by
factor 1\> to the generating surface Ko (with its
centre marked, for convenience) to obtain Kj,
K2, K3. . . The surface K, is then positioned by
constructing the rectangle [d0, r,] as indicated
in the description of the model. K2, K3 . . . are
then positioned by constructing the rectangles
[di, r2], [D2, r}],. . . , all obtained by successive
scalings of rectangle [do, rx] (by factor c in the
dimension d^ and by factor <Rjn the dimension
TV This simple procedure will be detailed stepby-step in a separate, forthcoming paper.
Uncoiled
GASTROPODS
Coiled
CEPHALOPODS
60 / 4 0 /20 / 0
-20\ V40 -60
Most of Coiled
GASTROPODS
Figure 6. A view of the shell morphospace. Axes vl, p and q. All shell models seen in apical view, at the same
scale and derive from the same generating curve.
553
SPIRAL GROWTH OF SHELLS
AN ALTERNATIVE SHELL
and are built on other parameters (p = 0, <fi
MORPHOSPACE
» W).
This constraint is lifted above the bottleneck,
What happens now to the 'Museum of All in the region of shells with very large values of
Shells' in the morphospace derived from the
"MA These shells are necessarily limited to a
present model? One deals now with a hyper- fraction of a whorl (or else they would reach
space with 10 dimensions: 'W, <R, £, p, q, e, a, p, gigantic sizes). Other limitations on p then
8 and n. As such, it is of little heuristic value, as come into play. For the uncoiled limpets and
it can be approached only by multidimensional bivalves, even small negative values of p would
analysis. Bi- or tridimensional representations dramatically reduce the space left for the aniyield more direct information and do, of mal—the old 'problem of univalveness' (Raup,
course, reflect actual distributions in the hyper- 1966)—and values beyond the constant tangent
space. Many combinations of axes can be angle of the logarithmic spiral (a small angle
considered. For example, Fig. 6 illustrates the for large 'W) are impossible. In contrast, large,
distribution of the shells of living molluscs in positive values of p increase (to a limit) the
the system of axes W, q and p (an angle so far volume available to the animal or limpets
much overlooked, although easily estimated on (see Figs. 5A to 5D). Large, positive values of
actual shells by examining the aperture or the p are not expected for bivalves because they
growth lines in apical view).
would yield shells in which the very divergent
Simulations matching shells of existing mol- umbos are located near the centre of each
valve.
luscs (for realistic values of n, the number of
volutions) were positioned on the graph and
As much of the previous discussions about
bounded by roughly approximated surfaces. shell morphospace focussed on the problem of
Only a small portion of the morphospace (now selection vs. genetic availability, one should
based upon independent parameters) contains keep in mind that all the axes do not have the
existing organisms. As fully expected, bivalves same significance. The values of q, p, e, a, p and
overlap in part with uncoiled gastropods, and 8 are fixed initial conditions, and it is tempting
coiled cephalopods entirely overlap with gastro- to speculate that they reflect an embryonic
pods. Scaphopods (the model does not accu- repertoire. Parameter p only sets ihe pitch of
rately depict their growth, because their early the first volution. Parameter q was defined here
shell is decollated) appear isolated and centred from an initial distance r0 for model construcaround the plane p = 0.
tion and analysis convenience. It seems unlikely
The shape of the space occupied by gas- that a larva with a large value of q could calcutropods and coiled cephalopods is somewhat late its distance from a remote, immaterial axis
unusual, but could largely have been predicted. of coiling. The biological instruction correMost shells (save scaphopods) can be simu- sponding to q is probably given in terms of curlated with q = 1, so their distribution volume vature. In contrast, 'W, <K.and L are expansion
should be quite flat around the plane q = 1. rates. They just selectively amplify the starting
Why the many, seemingly possible shapes built parameters during growth, as long as n (the
upon other values of q are not encountered in number of volutions) has not reached its maxiliving shells is an example of a question open to mum value.
discussion in biological terms.
The distribution observed in this novel view
In the plane (p, rH/), the presence of a
of the morphospace stresses that the shapes of
narrow bottleneck in the distribution is also existing shells should not be assumed (as it is
fully expected because the likely values of p now usual) to derive mostly from growth rates.
are limited by fabricational problems (as for The distribution is also largely determined by
other parameters). If one observes actual, starting conditions, hitherto much neglected,
multispiral shells in apical view, nearly all have and by the fabricational constraints resulting
values of p within a range roughly limited by from their amplification.
the angle of the two tangents drawn from the
lip to the previous volution (see Fig. 5G). This
range decreases sharply with increasing values
of 0V(see Figs. 5G, 5H). Beyond that range, the
whorls of multispiral shells become 'unwound'
ACKNOWLEDGEMENTS
(do not touch each other anymore), with a
predictable loss in mechanical strength. Such
I am grateful to the F.R.F.C. and to Biotec, S.A. for
shells are the exception in living molluscs financial support.
554
BERNARD TURSCH
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