Geometry and self-righting of turtles

Abstract

Terrestrial animals with rigid shells face imminent danger when turned upside down. A rich variety of righting strategies of beetle and turtle species have been described, but the exact role of the shell's geometry in righting is so far unknown. These strategies are often based on active mechanisms, e.g. most beetles self-right via motion of their legs or wings; flat, aquatic turtles use their muscular neck to flip back. On the other hand, highly domed, terrestrial turtles with short limbs and necks have virtually no active control: here shape itself may serve as a fundamental tool. Based on field data gathered on a broad spectrum of aquatic and terrestrial turtle species we develop a geometric model of the shell. Inspired by recent mathematical results, we demonstrate that a simple mechanical classification of the model is closely linked to the animals' righting strategy. Specifically, we show that the exact geometry of highly domed terrestrial species is close to optimal for self-righting, and the shell's shape is the predominant factor of their ability to flip back. Our study illustrates how evolution solved a far-from-trivial geometrical problem and equipped some turtles with monostatic shells: beautiful forms, which rarely appear in nature otherwise.

Full text

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Draft of
Domokos, Gábor, and Péter L. Várkonyi. "Geometry and self-righting of turtles."
Proceedings of the Royal Society of London B: Biological Sciences 275.1630 (2008): 11-
17.
http://rspb.royalsocietypublishing.org/content/275/1630/11.short
Geometry and Self-Righting of Turtles
Gábor Domokos*†, Péter L. Várkonyi‡
*Budapest University of Technology and Economics, Department of Mechanics, Materials
and Structures, and Center for Applied Mathematics and Computational Physics,
‡Program in Applied and Computational Mathematics, Princeton University, Princeton,
Fine Hall, Princeton NJ-08544 USA
†corresponding author: H-1111 Budapest, Műegyetem rkp. 3, K242, Hungary. Phone: +36-
30-9502307, Fax: +36-1-463-1773, E-mail: domokos@iit.bme.hu
Keywords: shell morphology, self-righting, monostatic body, static equilibrium
Terrestrial animals with rigid shells face imminent danger when turned upside down. A rich variety of
righting strategies of beetle and turtle species have been described, but the exact role of the shell’s geometry
in righting has been unknown so far. These strategies are often based on active mechanisms, e.g. most beetles
self-right via motion of their legs or wings; flat, aquatic turtles use their muscular neck to flip back. On the
other hand, highly domed, terrestrial turtles with short limbs and necks have virtually no active control: here
shape itself may serve as a fundamental tool. Based on field data gathered on a broad spectrum of aquatic
and terrestrial turtle species we develop a geometric model of the shell. Inspired by recent mathematical
results, we demonstrate that a simple mechanical classification of the model is closely linked to the animals’
righting strategy. Specifically, we show that the exact geometry of highly domed terrestrial species is close to
optimal for self-righting and the shell’s shape is the predominant factor of their ability to flip back. Our study
illustrates how evolution solved a far-from-trivial geometrical problem and equipped some turtles with
monostatic shells: beautiful forms, which rarely appear in Nature otherwise.

2
Introduction
The ability of self-righting is crucial for animals with hard shells (Frantsevich &
Mokrushov 1980; Faisal & Matheson 2001; Frantsevich 2004; Uhrin et al. 2005), e.g.
beetles and turtles. It is often used as a measure of individual fitness (Freedberg et al. 2004;
Steyermark & Spotila 2001, Ashmore & Janzen 2003; Burger et al. 1998), although it is
also influenced by the environment e.g. temperature (Elnitsky & Claussen 2006). Both
righting behaviour (Rivera et al. 2004; Ashe 1970; Stancher et al. 2006; Wassersug &
Izumi-Kurotani 1993) and the evolution of shell morphology (Claude et al. 2003; Myers et
al. 2006; Rouault & Blanc 1978) of turtles have been studied recently. An example of their
interaction is the sexual dimorphism of species where males are often overturned during
combats (Mann et al. 2006; Bonnet et al. 2001; Willemsen & Hailey 2003), and their shell
has adapted to facilitate righting. Here we develop a geometric shell model based on field
data to uncover systematically the connections between righting strategies and turtle shell
morphology.
Righting is always performed via a transversal roll around the turtle’s longitudinal axis,
along the perimeter of the ‘main’ transversal cross section at the middle of the body. Thus,
the roll’s geometry is essentially planar and can be readily illustrated (Fig. 1), suggesting a
planar model: a convex, homogeneous disk rolling under gravity on a horizontal surface.

3
Although turtles are neither exactly convex nor exactly homogeneous, these seem to be
plausible first approximations. The difficulty of righting originates in the “potential hill”
(represented by the unstable equilibrium at the side); the turtle has to produce
biomechanical energy to overcome this obstacle. An obvious question is whether a contour
without this obstacle exists; such a hypothetical shape would have just one stable and one
unstable equilibrium point and, according to the planar model, turtles with this contour
could perform righting effortlessly. Nevertheless, one can prove (Domokos et al. 1994) that
each homogeneous, convex disk has at least two stable equilibria (like an ellipse). This
appears to be bad news for turtles. However, as we are about to show, the planar model
proves to be an over-simplified approach.
Although the geometry of the roll can be illustrated in two dimensions, the mechanics is
fundamentally three-dimensional (3D): the centre of gravity G is determined not just by the
main cross section but by the complete body. For example, a long, solid cylinder with ends
chopped of diagonally in opposite directions rolls on a circle embedded in its surface,
however G is not at the centre of the circle. V.I. Arnold suggested (Domokos 2006) that, in
contrast to planar disks, convex, homogenous 3D objects with just one stable and one
unstable equilibrium may exist. The conjecture turned out to be correct and led to the
classification of 3D bodies with respect to the number of stable and unstable equilibria
(Várkonyi & Domokos 2006a,b). Here we apply a simplified classification, referring only
to stable balance points (unstable ones are less relevant for turtles). Stability class Si
(i=1,2,3,…) contains all objects with i stable equilibria, when resting on a horizontal
surface. Of particular interest are objects in class S1 also called monostatic (Conway & Guy

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1969; Dawson & Finbow 1999). Turtles in S1 can self-right without any effort. Monostatic
bodies are rare in nature: systematic experiments with 2000 pebbles identified no such
object (Várkonyi & Domokos 2006a). Even rarer are monostatic bodies with two equilibria
altogether (satisfying Arnold’s above mentioned conjecture); nevertheless, the shape of
some highly domed terrestrial turtles is strongly reminiscent of them (Várkonyi &
Domokos 2006b).
The goal of this paper is to identify equilibrium classes Si of real turtles and the relation of
these equilibrium classes to righting behaviour. In particular, we are interested in whether
monostatic shell shapes exist or not. We develop a simple geometric model of the shell, and
fit model parameters to real measured shapes. The equilibrium class of the fitted model is
numerically determined and compared to known data about the righting behaviour of
turtles.
Methods: the geometric shell model
Here we summarize the construction of the shell model in three steps (transversal model,
longitudinal model, 3D model). We also describe the method of fitting the model
parameters to individual, measured turtle shapes, and the technique to determine the
equilibrium class of the 3D model.
In the shell model (Fig. 2), a planar curve represents the approximate transversal contour of
the shell (transversal model, see Fig. 2a). This curve has 3 parameters (Fig. 3): p,

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controlling the shape of the carapace, R, defining the height/width ratio of the contour (or
alternatively the relative positions of the carapace and the plastron) and k, determining the
transition between plastron and carapace.
The transversal model (Fig. 2a) is constructed in a polar coordinate system, with origin at
the middle of the contour, both horizontally and vertically. Height and width of the cross-
section are scaled to 2R and 2, respectively. The contour K of the cross-section is achieved
from the curves of the plastron (P) and the carapace (C). The plastron is approximated by a
straight line, given in our polar coordinate system by



otherwise) defined(not
2/2/ifcos/
),,( 0


p
kRP
(1)
where p0 is a scaling factor. The shape of the carapace is first approximated in an
orthogonal u-v coordinate system by the curve
2/12 )( pvvu 
(2)
(Fig. 2b). This curve is either an ellipse (p>0) or a hyperbola (p<0). Next, the curve is
expressed in the polar coordinate system of Fig. 2a by substituting
 
RCcv
Ccu




cos
sin
0
0
(3)
(4)
into (2) and solving it for C; c0 is again a scaling factor. The final contour K(,R,p,k) is
constructed as
 
k
kk PCK /1/1 
(5)
where the negative parameter 0>k>-1 determines the roundedness of the transition between
the plastron and the carapace. The factors c0 and p0 are determined numerically from the

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respective constraints that the lowest point of the final contour is at distance R and the
widest points are at distance 1 from the origin of the polar coordinate system. The highest
point of the carapace is automatically at distance R above the origin (since the point
(u,v)=(0,0) of the curve corresponds to (C,)=(R,), cf. (3),(4)). At some values of , the
curves P (for /2< ≤3/2) and C (in an interval around =0 if p<0) are not defined. Here,
(5) is replaced by KC and KP, respectively.
Contours (cf. Fig. 2a) of 30 turtles belonging to 17 species have been digitized and the three
parameters p, R and k of the transversal model were fitted to these contours by considering
n1000 equidistant angles i=2i/n (i=1,2,…,n) in our polar coordinate system and by
minimizing the mean square radial error
 

iii QK
n
e2
)()(
1

(6)
Here, K(i) and Q(i) denote points of the model and the measured contour, respectively.
The results of the parameter-fitting are summarized in Fig. 4a,b and in the Electronic
Supplementary Material (ESM), Table 1. ESM Fig. 7 compares measured contours to
optimally fitted model contours.
The longitudinal model, describing side- and top-view contours of the shell (Fig. 2c), has
been constructed by using averaged data from the above-mentioned 30 individuals.
Needless to say, the circular and straight contours and the parameter values of Fig. 2c do
not represent a precise fit to real animals. However, the mechanical behaviour is much less

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sensitive to these curves than to the shape of the transversal model: turtles always roll
transversally along the perimeter of the main cross-section. The only significant effect of
the longitudinal model is to modify the height of the centre of gravity. Slightly modified
longitudinal curves have been tested and showed identical qualitative behaviour.
Finally, the 3D model surface emerges as a series of horizontally and vertically scaled
versions of the main transversal section, fitting the longitudinal contours (Figure 2d, ESM
Fig. 8).
To determine the equilibrium class of the model at given values of p, R and k, the centre of
gravity G of the model was integrated numerically (due to the two planes of reflection
symmetry, only the y-coordinate of G needs to be computed). Equilibrium points of the 3D
model surface can be conveniently identified by considering the radius  pointing from the
centre of gravity G to the model surface, as a scalar function of two coordinates, e.g.  and
z (cf. Fig. 2a,c). As it was shown in Domokos et al. 1994, mechanical equilibria coincide
with stationary points of zi.e. points where the gradient vector [z is zero.
Specifically, stable equilibria occur at local minima of zunstable equilibria occur at
saddle points and local maxima. All these stationary points can be readily computed for the
given, 3D model surface zwith fitted p, R and k parameter values; the equilibrium
class Si is identified simply by counting the minima.

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Our 3D model leaves the rostral and caudal ends of the shell undefined; we assume that the
turtle has only unstable equilibrium points in these domains, in accordance with the facts
that the real shells are somewhat elongated and turtles do not tend to get stuck in head- or
tail-down positions.
Results: equilibrium classes, energy balance and righting strategies
The previously described procedure identifies the fitted shape parameters R,p,k
corresponding to any shell and thus the complete algorithm to determine the equilibrium
class of any individual turtle (as given in ESM Table 1). In order to analyse global trends,
we introduce simplified, lower dimensional (2D and 1D) models, trading accuracy of
individual predictions for visual and conceptual simplicity.
As a first step we eliminate parameter k. We consider the [R,k] projection of the [R,p,k]
space (Fig. 4a). Measured turtles are marked by squares; two dashed lines k+(R) and k–(R)
mark the approximate upper and lower envelopes of the data points. Fig 4b shows the [R,p]
projection of the [R,p,k] space. In addition to marking the measured turtles, we also
determined the equilibrium class of each [R,p] point by assuming for k the extreme values
k= k+(R) and k=k–(R). Dashed lines depict the boundaries between equilibrium classes S1,
S2 and S3 for both cases. As we can observe in Fig 4b, not even these extreme changes of k
have substantial influence on the borders. Few marks appear in the ambiguous grey zone
between the two sets of boundaries; the equilibrium class of the majority of individual

9
turtles is predicted correctly by the simplified, 2D [R,p] model, yielding the following
observations:
- although class S1 is represented by a rather small domain (small ranges both for R
and p), nevertheless, tall turtles are remarkably close to S1, i.e. they tend to be
monostatic.
- flat turtles fall into S2, the majority of medium turtles falls into S3.
Next we further simplify our model to better understand global trends: strong linear
correlation between the parameters (corr<R,p>=0.73; corr<R,k>=0.64) suggests that a
one-parameter (R) model family is sufficient to approximate the geometry. From the
biological point of view, this implies that a single, visually significant parameter (the
height/with ratio R) basically determines the equilibrium class, and, as we will see soon,
the righting strategy. Figure 4c illustrates the angular location of equilibria as R is
varied in this one-parameter model and reveals a classical pitchfork bifurcation. This
simple 1D model illustrates the transition between three different types of turtles:
- individuals below R=0.57 tend to have two stable equilibria, one on the plastron
and one opposite, on the carapax,
- as R increases, two additional stable equilibria emerge on the back and their
distance is growing monotonically,
- the additional stable equilibria vanish at R=0.92, here monostatic turtles appear.
A closer look at an actual shell (Fig. 2d) explains why monostatic bodies may exist in
3D in contrast to 2D: the front and the back part of the shell are lower than the main

10
cross-section, thus the centre of gravity G of a complete turtle body is closer to the
plastron than the centre of its ‘main’ cross-section. The lower position of G makes self-
righting easier than predicted by the planar model of the Introduction. While this simple
qualitative observation is intuitively helpful, it certainly does not account for the
quantitative agreement between model parameters of highly domed turtle shells and
monostatic bodies (cf. Fig. 4b, ESM Table 1).
The energy balance of righting on a horizontal surface reveals a close relationship between
the equilibrium class and the righting strategy (for the latter, see Ashe 1970, Rivera et al.
2004). Non-monostatic turtles have to overcome a primary potential barrier (primary
energy deficit Dp, Fig. 5a) due to the unstable equilibrium at the turtle’s side. Additional,
secondary deficit (Ds) results from shell imperfections (Fig. 5a). Turtles with high energy
barriers use primarily their necks for righting (Rivera et al. 2004), thus the excess neck
length N (Fig. 5b) is the dominant factor of primary available biomechanical energy (Ap).
Additional, secondary energy (As) is generated by limb- and head-bobbing, ventral
orientation of the head or feet (to move centre of gravity), and nearly horizontal pushing
with the legs (using friction) (Ashe 1970). The latter often results in rotation around a
vertical axis during righting efforts, helping the feet find support.
Here we perform a qualitative, theoretical analysis of the energy balance based on two
simple assumptions: Dp(R) is monotonically decreasing and vanishes for monostatic (S1)
turtles (Fig 5c); Ap(N) is monotonically increasing if N>0, and Ap=0 for N≤0 (Fig. 5d).

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Energy balance curves, i.e. solutions of Ap(+As)= Dp(+Ds) can be plotted in the plane of
geometric parameters R and N (Figure 5e). Turtles represented by points to the right of the
curves stand a good chance for righting themselves. Turtles to the left of the curves are
expected to have serious difficulty. Our assumptions lead to the following qualitative
conclusions:
 For flat turtles (R under ~0.6, Fig 6a,) inside S2, the curves form a narrow band,
indicating that the primary parameters R and N dominate righting fitness. The
associated righting strategy is based on primary energies: righting is accomplished
via a strong vertical push with the neck, lifting the turtle sufficiently to overcome
the primary energy barrier. Most aquatic and semi-aquatic turtles, e.g. side-necked
turtles (Pleurodira), snapping turtles (Chelydridae), mud turtles (Kinosternidae)
follow this strategy.
 Tall turtles (R above ~0.8, Fig 6b,) inside or close to the monostatic class S1 usually
have shorter necks than their carapace heights, i.e. N<0. Thus their R, N values are
in the wide grey zone between the curves of Fig. 5e, indicating the dominance of
secondary effects in righting fitness. The associated righting strategy is based on
secondary energies: Righting either starts spontaneously (Ashe 1970) or it is
accomplished by dynamic motion of the limbs, to overcome small shell
imperfections. Subsequently, when the plastron is already close to vertical and the
legs reach the surface, horizontal pushing with the legs (using friction) produces
additional moments to overcome secondary energy barriers. This is the primary

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strategy of highly domed terrestrial tortoises with short necks and rounded cross-
sections, such as Geochelone, Stigmochelys, Astrochelys, and some Terrapene and
Testudo species.
 For medium turtles (~0.6 <R<~0.8, Fig. 6c,) in or close to S3, the energy diagram
shows regions of both types: the three curves initially form a narrow band and
subsequently diverge. The associated righting strategy is composed as a mixture of
the previous two: if placed on the middle of the back, the turtle starts rolling
spontaneously, assisted by dynamic limb and head motion to overcome shell
irregularities (similar to class S1) until it reaches stable equilibrium. From there,
the successful righting strategy is based on a vertical push with the neck (similar to
class S2), accompanied by pushing with the legs. Many tortoises (e.g.
Psammobates, many Testudo, and Terrapene species) belong to this group.
Due to strong correlation between model parameters, by measuring the R=height/width
ratio one can often correctly predict the righting strategy (Fig.4b,d). Quantitative analysis
of the individual’s energy balance would require more detail about the neck, the shell
imperfections and other ingredients.
Discussion
Many factors have been identified to affect the shape of turtles. Flat shells with sharp and
smooth edges are advantageous, respectively, for swimming (Claude et al. 2003), and for
digging (Willemsen & Hailey 2003). On the other hand, increased shell height was found
to offer better protection against the snapping jaws of predators (Pritchard 1979); it also

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protects from desiccation and improves thermoregulation (Carr 1952). While these factors
indicate qualitatively that higher shells might be of advantage, they can neither be applied
for quantitative prediction, nor do they determine the optimal height/width ratio. Our study
is much more specific: it not only shows that there exists a narrow optimal (monostatic)
range of the height/width ratio for self-righting in a terrestrial environment, it also predicts
the exact optimal geometry.
The height/width ratio of highly domed species (G. Elegans, G. radiata) is near the
minimum for monostatic shapes (R0.9), indicating an optimal tradeoff between self-
righting ability and other factors penalizing increased height (e.g. decreased stability). The
shape parameter p is also in the optimal range for these species (0.8<p<1.1). Thus, the
advantage of being close to monostatic not only determines the height/width ratio, but the
exact shape (e.g. roundedness).
It would be worth exploring how other morphological differences between aquatic and
terrestrial turtles (Mann et al. 2006; Acuna-Mesen 1994) affect their righting fitness.
Another observation of particular interest is that bad nourishment often produces shell
imperfections (Wiesner & Iben 2003; Highfield 1989), decreasing the chances of successful
righting, according to the presented theory.
Acknowledgements

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This research was supported by OTKA grant T72146 and the Zoltán Magyari Postdoctoral
Fellowship. We thank Gyula Hack, Doma Csabay, Mrs. József Bíró, Diana Cseke, Judit
Vörös, Zoltán Korsós and Timea Szabó for providing access to turtles. Comments from
John Iverson, Géza Meszéna, István Scheuring, Patrick Simen and Andy Ruina as well as
from anonymous referees are appreciated.
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Figures & captions:
Figure 1: Rigid disk (representing the main transversal section of the turtle shell)
rolling on a horizontal line. Thick dashed line denotes the orbit of the centre of gravity
(G) during rolling. Peaks/valleys of the orbit correspond to local extrema of the potential
energy, i.e. unstable/stable equilibria of the rigid body. For convex, homogenous 2D disks,
there are always at least two stable points (Domokos et al. 1994).

18
Figure 2: Turtle shell model. a: Frontal view of shell (Stigmochelys pardalis) and three-
parameter transversal model K,Rp,k) of main cross section. Plastron is approximated
by straight line P (eq. (1)). Carapace is approximated by curve C (eqs. (2)-(4)); smooth
transition between plastron and carapace achieved by merging function (eq. (5)). b:
Carapace shape at various values of parameter p. c: Longitudinal model: schematic side-
and top-view contours are circular arcs obtained as averages from measurements. Sizes are
normalized. d: Visual comparison of digitized shell image and model surface.

19
Figure 3: Main cross section at various parameter values. Parameter p determines the
shape of the carapace, R determines the relative positions of the carapace and plastron
(height/width ratio), k determines the roundedness at the carapace-plastron transition.

20
Figure 4: Measured turtles and equilibrium classes. a,b: squares show the fitted model
parameters corresponding to individual turtles in the [R,k] and [R,p] parameter planes.
Linear regression (continuous lines) reveals
01.190.0)(  RRk
,
51.159.2)(  RRp
. In Fig4a,
dashed lines
 
2.0)(,0max)( 
RkRk
mark approximate upper and lower envelope of
the data points. In Fig4b, the equilibrium class of the model was determined for each point
of the parameter space [R,p] under the assumptions of k=k+(R) and k=k–(R); boundaries of
classes S1, S2, and S3 are shown by dashed lines in both cases. The grey shading between

21
the two sets of boundaries shows regions of the [R,p] space where the equilibrium class of
an individual may depend on k. Domain „X” corresponds to parameter values not
compatible with the model. Fig4c: angular positions  of equilibria on the main cross-
section of the turtle shell as function of R under assumptions of
)(Rkk 
and
)(Rpp 
.
Equilibria in all classes marked on main transversal cross section by black (stable) and grey
(saddle point) circles. Unstable equilibria off the main cross-section (near head and tail) are
not indicated. d: fitted R value and contour of some measured individuals.

22
Figure 5: Energy balance of righting. a: Illustration of primary (Dp) and secondary (Ds)
energy deficit of rolling turtle due to potential energy barrier between stable equilibria.
Continuous/dashed line denotes potential energy U of perfect/imperfect shell normalized by
body size. b: Schematic frontal view of a righting turtle. The neck’s excess length is
defined as N=(Hn- Hmin)/Hmin, i.e. scaled difference between the neck’s length (Hn) and the
distance (Hmin) from the neck’s base to the top of the carapace. c: Primary (Dp) and
secondary (Ds) energy deficit as functions of R. Dp(R) is decreasing, and vanishes for
monostatic (R>0.92) turtles. d: Available biomechanical energy as function of the excess
neck length N: primary (Ap) due to excess neck length N (Ap=0 for N<0; Ap(N) increasing
for N>0), secondary (As) due to dynamical effects, head and limb bobbing, horizontal push
by leg, etc. e: Energy balance curves in the [R,N] plane. In the grey region secondary
components (Ds, As) determine righting success. To the right of the grey region, righting is

23
successful even in the presence of secondary deficits (shell irregularities). To the left,
righting is unsuccessful even if secondary available energy (from head-bobbing, etc.) is
used. Observe characteristic behavior in the three principal equilibrium classes: narrow
grey band in S2 , wide region in S1 and mixed in S3.

24
Figure 6: Righting strategies. Each strategy is characterized by the typical shape of the
rolling main cross-section (grey contours) as well as the orbit (dashed line) of the centre of
gravity G. Arrows denote key elements of righting, dashed arrows apply in presence of
secondary energy barriers. a: flat turtles (R<~0.6 inside Class S2, photo: Hydromedusa
tectifera): high primary energy barrier between stable and unstable equilibria is overcome

25
by primary biomechanical energy resulting from vertical push with neck. Righting fitness is
determined by primary geometric parameters (height/width ratio R, excess neck length N).
b: tall turtles (R>~0.8, inside or close to monostatic Class S1, photo: Geochelone elegans):
small, secondary energy barriers (resulting mainly from shell imperfections) are overcome
by secondary sources of biomechanical energy: head- and foot-bobbing, push by feet. c:
medium turtles (~0.6<R<0.8, inside or close to Class S3, photo: Terrapene carolina): in the
first phase of roll, secondary barriers are overcome by dynamic (secondary) energy, in the
second phase, the primary energy barrier is overcome by primary energy (push with neck,
feet).

26
Electronic Supplementary Material (ESM)
Figure 7: Model fit to real contours. From left to right: Geochelone elegans (number 1 in
ESM Table 1), Stigmochelys pardalis (number 11), Chelonoidis nigra (number 20),
Cuora amboinensis (number 26), Chelonia mydas (number 30). The model fits well to
terrestrial species with highly domed cross-sections (three contours on the left side), but
less so to flatter, semi-aquatic and aquatic turtles (two contours on the right) where sharp
edges appear, improving swimming performance.

27
Figure 8: Model shape at various parameter values. From top to bottom: R=0.4, 0.65,
0.9. In all cases the remaining two parameters p and k are obtained from the regression lines
p=2.59R-1.51, k=0.90R-1.01.

28
Table 1: Measured data and fitted model parameters. Measured individuals are in
decreasing order based on their height/width ratio. In the last column, a second class is
shown in parenthesis for individuals with parameter values very close to the boundary of
two equilibrium classes. We also measured turtles where the shell’s shape has adapted to
special habits (e.g. Geochelone sulcata, digging itself into the sand, Malacochersus tornieri,
squeezing the shell between rocks). Although the model could still be fitted to the shell’s
contour with reasonable error (e<0.08), these data have been excluded from the statistics in
the [R,p] and [R,k] parameter planes.
SERI
AL #
INPUT DATA FROM MEASUREMENT
FITTED MODEL
PARAMETERS
ERROR
CLASS
species
contour of
main cross
section
R=height/
width
ratio
k
p
R
e
cf. eq. (6)
1
Geochelone
elegans
0.99
-0.19
0.65
1.00
0.0006
S3 (S1)
2
Geochelone
elegans
0.90
-0.00
1.04
0.88
0.0013
S1
3
Stigmochelys
pardalis
0.87
-0.11
0.83
0.89
0.0009
S3 (S1)
4
Stigmochelys
pardalis
0.85
-0.27
0.63
0.85
0.0008
S3
5
Stigmochelys
pardalis
0.85
-0.14
0.63
0.85
0.0025
S3
6
Stigmochelys
pardalis
0.85
-0.32
0.49
0.85
0.0015
S3

29
7
Geochelone
elegans
0.85
-0.06
1.07
0.84
0.0012
S2 (S1)
8
Astrochelys
radiata
0.84
-0.00
0.84
0.85
0.0012
S3 (S1)
9
Stigmochelys
pardalis
0.83
-0.39
0.48
0.83
0.0012
S3
10
Stigmochelys
pardalis
0.82
-0.31
0.93
0.81
0.0007
S2
11
Stigmochelys
pardalis
0.81
-0.38
0.55
0.80
0.0014
S3
12
Stigmochelys
pardalis
0.79
-0.39
0.51
0.79
0.0007
S3
13
Stigmochelys
pardalis
0.79
-0.27
0.49
0.81
0.0010
S3
14
Psammobates
tentorius
0.79
-0.42
0.34
0.79
0.0013
S3
15
Stigmochelys
pardalis
0.79
-0.42
0.34
0.79
0.0013
S3
16
Terrapene
carolina
triungis
0.78
-0.47
0.21
0.78
0.0028
S3
17
Stigmochelys
pardalis
0.78
-0.38
0.23
0.78
0.0009
S3
18
Chelonoidis
carbonaria
0.78
-0.29
0.50
0.79
0.0022
S3
19
Testudo graeca
anamurensis
0.72
-0.33
0.84
0.73
0.0013
S2

30
20
Chelonoidis
nigra
0.72
-0.19
0.95
0.72
0.0013
S2
21
Chelonoidis
carbonaria
0.68
-0.12
0.96
0.70
0.0023
S2
22
Eurotestudo
hermanni
0.67
-0.31
0.59
0.66
0.0009
S3 (S2)
23
Eurotestudo
hermanni
0.65
-0.34
0.34
0.65
0.0011
S3 (S2)
24
Terrapene
carolina bauri
0.64
-0.57
-0.06
0.65
0.0013
S3 (S2)
25
Trachemys
scripta elegans
0.63
-0.73
-0.06
0.67
0.0032
S3 (S2)
26
Cuora
amboinensis
0.58
-0.58
-0.29
0.63
0.0071
S3 (S2)
27
Rhinoclemmys
pulcherrima
manni
0.56
-0.55
-0.36
0.58
0.0020
S3 (S2)
28
Phrynops hilarii
0.45
-0.73
0.13
0.47
0.0028
S2
29
Carettochelys
insculpta
0.43
-0.61
-0.18
0.46
0.0051
S2
30
Chelonia mydas
0.41
-0.33
-1.02
0.43
0.0050
S3 (S2)