Regularity in irregular echinoids

Digital Object Identifier (DOI):10.1007/s002850100126

J. Math. Biol. 44, 330–340 (2002) Mathematical Biology

M. Torres · J.L. Aragon · P. Domınguez · D. Gil

Regularity in irregular echinoids

Received: 19 June 2001 / Revised version: 2 October 2001 /Published online: 14 March 2002 – c© Springer-Verlag 2002

Abstract. Using the mathematical concept of eutactic star, we prove that it is possible todefine a morphospace for irregular echinoids by using a single parameter. In particular,we have found an extraordinary geometric property in the flower-like patterns of the fiveambulacral petals of these animals. This property is fulfilled with great accuracy for a largecollection of fossil specimens and provides new insights in the study of the viable skeletaldesigns of extinct and/or living organisms.

1. Introduction: general

Evolutionary forms in Nature are associated with symmetry and regularity. We havefound a strong regularity in flower-like patterns formed by the five ambulacral petalsof irregular echinoids [1]. We also show how they can be described mathematicallyby using a single parameter.

A two-dimensional regular vector-star, pointing to the vertices of a regularpolygon, is a clear example of radial symmetry as observed in regular echinoids.On the other hand, the restrictive condition of radial symmetry can be relaxed togenerate non-regular stars with bilateral symmetry, as it occurs in irregular echinoidswhere a bivium and a trivium of ambulacral petals appear [2]. As discussed in Ref.[3], bilateral symmetry in irregular echinoids is a structural adaptation facilitatingthe locomotion of these animals, which generally burrow just below the surface ofsediments on the sea bottom. If we try to classify the resulting structures of theseanimals, the direct symmetry analysis is quite limited because, once the pentagonalsymmetry of the point group C5 is lost, there remains only a mirror element whichis called Loven’s plane [2]. As we shall show in what follows, the concept of

M. Torres: Instituto de Fısica Aplicada, Consejo Superior de Investigaciones Cientıficas.Serrano, 144, 28006 Madrid Spain

J.L. Aragon: Instituto de Fısica, UNAM. Apartado Postal 1-1010, Queretaro, Qro. 76000,Mexico. Instituto de Fısica Aplicada, Consejo Superior de Investigaciones Cientıficas.Serrano, 144, 28006 Madrid Spain. e-mail: [email protected]

P. Domınguez: Departamento de Paleontologıa, Facultad de Ciencias Geologicas,Universidad Complutense. Ciudad Universitaria, 28040 Madrid, Spain. Department ofPalaeontology, The Natural History Museum, Cromwell Road, SW7 5BD London, England

D. Gil: Departamento de Paleontologıa, Facultad de Ciencias Geologicas,Universidad Complutense, Ciudad Universitaria, 28040 Madrid, Spain

Key words or phrases: Pentamerism – Irregular echinoids – Ambulacral petals –Morphospace – Eutactic stars

Regularity in irregular echinoids 331

the eutactic star [4] is a useful mathematical tool since allows us to describe andcharacterize bilateral stars.

2. Introduction: mathematical

N complex numbers z1, z2, ..., zN are eutactic if and only if [4]

z21 + z2

2 + · · · + z2N = 0 (1)

and not all zj are zero.If zj = aj + ibj , the eutacticity condition is equivalent to

a21 + · · · + a2

N = b21 + · · · + b2

N, (2a)

a1b1 + · · · + aNbN = 0, (2b)

that is, the N -dimensional vectors u1 =(a1, a2, . . . , aN) and u2 =(b1, b2, . . . , bN)

are orthogonal and are of equal norm. Eutactic stars were originally proposed in thestudy of regular polytopes [5,6] and extensively used in the field of quasicrystalswhere projections from higher-dimensional spaces are used to generate quasiperi-odic tilings [7] and to describe transformations from quasiperiodic to periodic tilings[8–11]. In these works, a plane eutactic star is the image of a general N -dimensionalorthonormal basis (N > 2) under the standard projection. Here, we consider theprojection from 5 to 2 dimensions.

Formally, the concept of eutactic star was introduced in the study of regularpolytopes [5]. A set of N vectors in Rn, {a1, a2, . . . , aN }, is said to form an eutacticstar if they can be regarded as a projection of N orthogonal vectors (N-cross) in aN-dimensional space (N > n). In simple terms, any eutactic star is the orthogonalprojection of an N-cross. If the N-cross is orthonormal then the projected star iscalled a normalized eutactic star. The condition under which a given star {a}i iseutactic are given by a theorem due to Hadwiger [5]:

A star {a1, a2, . . . , aN } in Rn is normalized eutactic if and only if for anyx ∈ Rn

x = T (x),

where T is defined by T = ∑Nj=1(x · aj )aj

Here we are interested in pentagonal eutactic stars in R2 so we shall adoptthe elegant approach to the eutacticity criterion recently introduced [4]. In thisapproach, the end points of a star vector are represented by complex numbers andthe eutacticity criterion is given by (2), for N = 5.

An interesting point arising in the study of eutactic stars is the possibility ofcontinuously transforming a star in such a way that the eutacticity property is main-tained in all the intermediate stages. This problem was solved for certain vectorstars related with quasiperiodic tilings in the study of the continuous transforma-tions between quasiperiodic and periodic structures [8–11]. In these works, thetransformations of eutactic stars are regarded as rotations in the N -dimensionalspace. In particular here we consider five vectors in R2 forming a star with 5-fold

332 M. Torres et al.

cyclic (rotational) symmetry C5 which is continuously transformed up to a star witha symmetry group B ⊂ C5. This is interesting because, in fossil specimens of rela-tively flat echinoid species, it is observed that the five ambulacral petals constitutea continuous series of pentagonal stars, within the morphospace defined by this setof transformations.

For study of the stars formed by the five petals of regular echinoids it is usefulto consider the following interpretation of the eutacticity criterion (Roger Thomas,personal communication). Consider a pattern in which ambulacra B and E areof equal length b and ambulacra C and D are of equal length c, with angles α

and β (see Figure 1). This defines a pattern with a unique ray (A, of length a)in the anterior/posterior axis and two pairs of bilaterally symmetrical rays. Thecoordinates of the five vectors defining the pattern are (0, a), (b cos α, b sin α),(c cos β, −c sin β), (−c cos β, −c sin β) and (−b cos α, b sin α), and we can con-struct the five-dimensional vectors

u1 = (0, b cos α, c cos β, −c cos β, −b cos α),

u2 = (a, b sin α, −c sin β, −c sin β, b sin α).

According to (2) this bilateral star is eutactic if u1 ·u2 = 0 and ‖u1‖ = ‖u2‖. Fromprevious equations we have that u1 · u2 = 0 for arbitrary a, b, c, α and β, that is,the bilaterality of the star accounts for the orthogonality of the five-dimensionalvectors. Now, the second condition, ‖u1‖ = ‖u2‖, requires that:

2b2(cos α)2 + 2c2(cos β)2 = a2 + 2c2(sin β)2 + 2b2(sin α)2,

x

y

AB

CD

E

ααββ

a bb

cc

Fig. 1. Diagram of the conventional representation of the ambulacra of irregular echinoidin plane geometry.


or

2b2 cos(2α) + 2c2 cos(2β) = a2, (3)

which imposes severe restrictions on the free parameters and limits the set of pos-sible forms more stringently than the requirement of bilateral symmetry alone. Theends of the ambulacra lie on a oval satisfying Equation (3) and, consequently, thepresent model of the eutactic star places new constraints on the set of possible formsbeyond that of Loven’s plane.

3. Results

We have measured the components (aj , bj ) of the petals (as depicted in Fig. 2, top),considered as the real and imaginary parts of the complex numbers z1, . . . , z5, for104 specimens of relatively flat fossil echinoid species in The Natural HistoryMuseum of London. Measurements were carried out on digital images using thedata acquisition software tpsDIG v.1.18 [12], and the data set was verified using thegeneralized rotational fit program GRF-ND [13]. The raw data1 are in TPS formatand, as an example, the entry corresponding to Mellita quinqueperforata reads:

LM=6185 240185 314247 252227 163136 167120 254

IMAGE=31128.tifSCALE=0.028571

The first row stands for the number of landmarks and the next ones are the sixcoordinates of the petals, including the origin (first coordinate), in pixels. The nextline after the coordinates refers the name of the TIFF image, in our records, ofthe corresponding specimen. The file name corresponds to the reference numberin the The Natural History Museum of London. Finally, the scale (in cm) is given.To simplify calculations, we work in arbitrary units so the scale is not relevant.The coordinates of the petals are (0, 74), (62, 12), (42, −77), (−49, −73) and(−65, 14). The five-dimensional vectors are u1 = (0, 62, 42, −49, −65) and u2 =(74, 12, −77, −73, 14). The eutacticity criterion requires that vectors u1 and u2be orthonormal. Since we are working with arbitrary units, let us consider the newvectors:

u′1 = u1

‖u1‖ = (0, 0.56054, 0.37972, −0.44301, −0.58766),

u′2 = u2

‖u1‖ = (0.66903, 0.10849, −0.69616, −0.65999, 0.12657),

1 Available upon request from the authors.


obtained using the standard scalar product in five dimensions. Notice that, by con-struction, ‖u′

1‖ = 1 so it remains to verify that ‖u′2‖ = 1 and that the angle

φ between u′1 and u′

2 is 900. From the above data we have ‖u′2‖ = 1.1814 and

φ = arccos(u′1 · u′

2/‖u′1‖‖u′

2‖) = 89.29830. The eutacticity criterion is thus accu-rately fulfilled.

We followed the same procedure for all the specimens considered and resultsare shown in Table 1. A graphical representation of these results, which showsthe accuracy of the eutacticity property, is obtained if the unitary five-dimensionalvector u′

1 = u1/‖u1‖ is represented by the (1, 0) vector in two dimensions andthe vector u′

2 = u2/‖u1‖ is drawn at the respective angle φ and its correspondingnorm, as shown in Figure 2 (middle), where primes were eliminated to keep thenotation as simple as possible.

Table 1. Eutacticity of 104 flattened fossil specimens. In the second column we includethe corresponding reference number of the specimen in the The Natural History Museumof London. In columns three and four, the norm of the vector u2 and φ, the angle between5-dimensional vectors u1 and u2, are displayed.

Fossil specimens Reference ||u2|| φ (in degrees)

Arachnoides elongatus 75694 1.15 88.67Arachnoides elongatus E42457 1.15 89.11Arachnoides loveni E42444 1.06 90.29Arachnoides etheridgei E42458 1.11 91.34Arachnoides placenta E4355 1.11 89.93Dendraster E83633 1.01 89.51Dendraster E83634 0.95 93.39Dendrasteridae excentricus E17989 0.99 88.35Dendrasteridae excentricus E17990 1.01 88.60Dendrasteridae excentricus E18000 0.97 88.94Dendrasteridae excentricus E18001 1.22 91.18Dendraster gibbsi E41495 1.19 87.39Dendraster gibbsi E41499 1.09 87.27Dendraster gibbsi E41502 1.15 91.89Dendraster gibbsi E43746 1.01 87.87Dendraster gibbsi E75319 0.96 89.49Dendraster gibbsi E75321 0.99 88.07Dendraster gibbsi E42747 1.12 91.54Dendraster gibbsi E79582 1.17 93.77Dendraster vizcainoensis E42794 0.92 91.72Dendraster vizcainoensis E42798 0.97 89.56Dendraster vizcainoensis E42799 0.97 89.37Dendraster diagoansis E79338 1.10 92.66Dendraster diagoansis E83632 0.95 89.07Dendraster venturaensis E42796 1.01 92.08Dendraster calaster E43751 1.03 90.52Encope E41789 1.16 89.09Encope E83647 1.18 86.67

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Encope macropora 34363 0.93 87.01Encope macropora EE6085 1.14 90.98Encope macropora EE6086 1.17 88.36Encope macropora EE6088 1.07 89.94Encope macropora EE6089 1.25 90.53Encope macropora EE6090 1.15 89.81Encope micropora E42462 1.06 91.17Encope micropora E424623 0.98 93.85Encope micropora E79616 0.99 94.91Encope tenuis E17991 1.28 89.93Encope tenuis E17992 1.16 93.33Encope tianamiensis E75334 1.18 89.52Fellaster EE5320 1.08 91.87Laganum tumidum E11709 1.20 89.34Laganum tumidum E11710 1.19 89.27Laganum tumidum E11711 1.12 86.53Laganum tumidum E11713 1.19 87.93Laganum tumidum E11714 1.15 89.74Laganum tumidum E11715 1.17 91.49Laganum tumidum E11716 1.08 89.03Laganum tumidum E11717 1.18 86.48Laganum tumidum E11718 1.19 88.64Laganum tumidum E11719 1.24 91.34Laganum tumidum E11720 1.18 88.58Laganum tumidum E11721 1.24 89.27Laganum tumidum E11722 1.25 85.68Laganum tumidum E11723 1.17 87.30Laganum tumidum E11724 1.13 90.01Laganum tumidum E17567 1.11 92.59Laganum tumidum E17568 1.24 89.69Laganum tumidum E17569 1.23 90.03Laganum tumidum E17570 1.26 89.30Laganum depressum E19048 1.15 90.53Laganum depressum E19049 1.12 90.33Laganum depressum E19052 1.12 87.67Laganum depressum E19055 1.24 90.73Laganum depressum E19319 1.15 87.59Laganum depressum E19341 1.13 91.01Laganum depressum E19349 1.16 88.80Laganum depressum E19358 1.19 88.26Laganum depressum E19359 1.28 87.18Laganum depressum E19361 1.14 88.79Laganum depressum E19363 1.21 91.26Laganum ocalanum E75293 1.20 87.69Mellita quinqueperforata 31128 1.18 89.30Mellita quinqueperforata 31128-2 1.07 89.92Mellita quinqueperforata 31129 1.13 88.35Monostychia E1109 1.05 94.05

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Monostychia australis 48803 1.29 87.76Monostychia australis E39196 1.13 88.06Monostychia australis EE1218 1.14 84.81Monostychia australis EE599 1.09 87.88Monostychia australis EE601 1.13 87.71Monostychia australis EE602 1.09 89.29Monostychia australis EE603 1.07 89.67Monostychia australis EE604 1.14 88.46Monostychia australis EE605 1.07 90.90Monostychia australis EE606 1.12 92.01Monostychia australis EE607 1.16 90.02Peronella platymodes E74600 1.24 91.96Peronella decagonale E3770 1.16 89.48Scutella 38600 1.06 88.95Scutella 40076 1.03 86.42Scutella E2495 0.97 88.23Scutella E40168 1.01 88.86Scutella E4778 1.07 91.90Scutella E73266 1.09 88.25Scutella E76167 1.04 90.24Scutella E860 1.08 92.22Scutella E9204 1.05 90.83Scutella E9209 1.14 88.25Scutella E9406 1.06 88.35Scutella pileus 34360 1.09 90.59Scutella pileus 34361 1.09 87.96Scutella striatula E2353 1.01 90.33Scutella striatula E3775 1.10 88.69

The taxa studied were selected because their skeleton are quite flat, so theirambulacra lie close to a plane. These species all happen to be roughly circular inconsequence of the adaptation of the echinoid body plan to their particular modeof life. The calculated values of the angle φ between u1 and u2 are all within50 of 900. More precisely, the average values of ‖u2‖ and φ for each genus are,respectively, Arachnoides: 1.11, 89.860; Dendraster: 1.03, 90.100; Encope: 1.12,90.360; Fellaster: 1.08, 91.870; Laganum: 1.14, 89.090; Mellita: 1.12, 89.190;Monostychia: 1.12, 89.210; Peronella: 1.20, 90.720; Scutella: 1.05, 89.330. Fromthese values we obtain that the mean of ‖u2‖ is 1.10, with a population standarddeviation, σ9 = 0.04, and the mean of φ is 89.970, with σ9 = 0.86. The modestvariance observed here has three origins: asymmetry in individual echinoids (so-called “fluctuating asymmetry”), variation among individuals within species, andmeasurement errors (Roger Thomas, personal communication).

As discussed in Section 2, a pentagonal vector star in the plane is eutactic if it canbe regarded as the projection (shadow) of an orthonormal basis in five dimensions.If we rotate the five-dimensional orthonormal basis according to a specific rotation


matrix, described by a single parameter θ , the projections of the star will change intheir geometry but the property of eutacticity is preserved.

Starting from mathematical results we have previously developed for hyper-spatial rotations in five dimensions [9,10], we are able to describe the continuoustransformation of the star {z1, z2, . . . , z5}, as a function of a single parameter θ ,which preserves the eutacticity and describe the arrangement of petaloid ambulacrafor each specimen. This parameter θ turns out to be the same one that we used todefine the rotation matrix of the orthonormal basis in the five-dimensional space,corresponding to a rotation angle in the hyperspace, as detailed in Ref. [10].

The variable star is given by

z1 = {0, 2 cos2 θ/N2}, (4a)

z2 = {(cos θ + sin θ)/N1, cos 2θ/N2}, (4b)

z3 = {cos θ/N1, −(sin2 θ + sin 2θ)/N2}, (4c)

z4 = {− cos θ/N1, −(sin2 θ + sin 2θ)/N2}, (4d)

z5 = {−(cos θ + sin θ)/N1, cos 2θN2}, (4e)

where N1 = N1(θ) and N2 = N2(θ) are normalization functions of the five-dimensional vectors u1 and u2, given by

N1 =√

3 + cos(2θ) + 2 sin(2θ),

N2 =√

17

4cos(2θ) + 3

4cos(4θ) + 2 sin(2θ) − sin(4θ).

As said above, and detailed in Ref. [10], the transformation of the star vectorcan be viewed as a rotation in the 5-dimensional space. Notice that starting fromthe orthonormal vectors u1 and u2 a basis for the 5-dimensional space can begenerated by means of the Gram-Schmidt procedure. By assembling this basisu1, u2, . . . , u5 ∈ R5 as the rows of an 5 × 5 matrix, the rotation that describes thetransformation in the hyperspace is defined.

We can translate this parametrization to describe the transformation of the stardefined by the five ambulacra as follows. From (4) and Figure 1, we have that

a = 2 cos2 θ

N2, (5a)

b2 = (cos θ + sin θ)2

N 21

+ (cos 2θ)2

N 22

, (5b)

c2 = (cos θ)2

N 21

+ (sin2 θ + sin 2θ)2

N 22

, (5c)

tan α = N1 cos 2θ

N2(cos θ + sin θ), (5d)

tan β = N1(sin2 θ + sin 2θ)

N2 cos θ, (5e)


Fig. 2. Mellita quinqueperforata (top left) and Dendraster venturaensis (top right) withtheir respective stars of petaloid ambulacra. Graph showing how the eutacticity property of113 fossil specimens belonging to nine genera of irregular echinoids is fulfilled (middle).Evolution of the eutactic star model for different values of the angular parameter θ (bottom).The regular pentagonal star corresponds to θ = θ0 = 31.71740 and, as an example, starsat θ = 380 and θ = 180 reproduce patterns observed in Mellita quinqueperforata andDendraster venturaensis, respectively. The star for θ = 450 represents the extreme patternof Brissus.

which satisfy the constraints of (3). Notice that with this parametrization the set offive variables a, b, c, α and β is reduced to a single one: the angle θ .

In Fig. 2 (bottom) we show four variants or regions within the morphospacedefined by the model of the eutactic star, generated by using (4) or (5). The regular


Fig. 3. Brissus lasti (Ref. E17907) represents the extreme of the transformation, at θ = 45o,which is a limit of the trvium-bivium ambulacral structure.

pentagonal star is obtained for θ0 = 12 arctan(2) = 31.7174o. Stars for θ = 38o and

θ = 18o are displayed at the left and right. They reproduce the patterns observedin Mellita quinqueperforata and Dendraster venturaensis.

Both possibilities, θ > θ0 and θ < θ0, are represented in the collection studied.Moreover, among the examples not included here, since they can not be consid-ered as flat structures, let us mention Brissus (Fig. 3), which exhibits a limit of thetrivium-bivium pattern at the extreme of the transformation for θ = 450 (Fig. 2,left bottom). This structure is a limit of the trivium-bivium ambulacral structuresince no pattern is observed with four ambulacra in the same half plane, i.e., a“quadrivium” has been never observed. The spacing of ambulacra on the test issurely determined primarily by the need to distribute the food grooves efficientlyacross the available space; this functional consideration mitigates against the occur-rence of a quadrivium of ambulacra on the posterior half of the test (Roger Thomas,personal communication). According this interpretation, the constraint on ambu-lacral patterns that we have documented has a functional, rather than geometrical,origin.

From our results we conjecture that all possible values of θ are occupied inthe morphospace here described. This claim is however hard to prove since theparameter θ (ranging from 00 to 450) is a continuous variable and the fossil registeris essentially discrete. In any case, irregular echinoids are a remarkable example of


the fact that any constructively viable skeletal design has been completely exploitedby extinct and/or living organisms as proposed in Ref. [14].

4. Conclusions

The concept of the eutactic star is applied to account for an extraordinary regular-ity observed in the flower-like patterns of the five ambulacral petals of irregularechinoids. It is shown that the five petals of these animals are living examples ofeutactic stars. Using a model of the corresponding eutactic star, parameterized bya single angle, it is observed that the morphospace of the petals is fully occupiedby extinct and living irregular echinoids as predicted, in a more general context, byR.D.K. Thomas et al. [14].

Acknowledgements. Two of us (M.T. and J.L.A.) want to thank Profs. P.K. Maini and R.A.Barrio for useful discussions and hospitality at the Center for Mathematical Biology inOxford. This work was supported by DGICYT, Spain, through Project No. PB97-1232, andnetwork RAIMA-VIIIH, of CYTED, is also gratefully acknowledged. One of us (J.L.A)acknowledges financial support from the Ministerio de Educacion y Cultura of Spain (grantSAB1999-0197) and DGAPA-UNAM (grant IN-108199).

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Regularity in irregular echinoids

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