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Gildner, Raymond F., 2003. A Fourier Method to Describe and Compare Suture Patterns. Palaeontologia Electronica 6(1):12pp, 4.1MB; http://palaeo-electronica.org/paleo/2003_1/suture/issue1_03.htm

A FOURIER METHOD TO DESCRIBE ANDCOMPARE SUTURE PATTERNS

Raymond F. Gildner

Paleontological Research Institution. Ithaca, NY 14853Department of Geosciences, Indiana University, Purdue University, Fort Wayne, Fort Wayne, IN 46805 gildnerr@ipfw.edu

ABSTRACT

Suture patterns in shelled cephalopods are periodic structures and can bedescribed using Fourier methods when points along the pattern are described by twoparametric equations. One equation describes the angular position along the circum-ference of the phragmacone, and another describes the height along the length of theshell. The angular position is amenable to Fourier description transformed to the differ-ence between the observed angle and the angle expected if the suture line werestraight. An accurate reconstruction of ammonitic suture patterns is accomplished withfew amplitudes. Applying the method to the digitized suture patterns provides a moreaccurate means of interpolation than linear interpolation, necessary for comparisonbetween suture patterns. Simple “nautilitic” and complex “ammonitic” suture patternsfrom the literature are used to demonstrate application of the method. Ontogeneticseries of suture patterns may develop first by increasing variability in their height, andonly later by increasing variability in the angular positions. The method invites newapproaches of analysis, including different approaches to nearest neighbor analysis todetermine patterns of similarity between suture patterns.

Copyright: Paleontological Society - September 2003Submission: 29 April 2002 - Acceptance: 6 August 2003KEY WORDS: cephalopods, ammonites, septa, suture patterns, Fourier analysis, modeling

INTRODUCTION

Among the externally shelled cephalopods,wall-like septa separate the shell into camerae andform sutures where each septum joins the externalshell. The shape of the suture has been used todistinguish cephalopod species and determine tax-onomic relationships, and often serves as a proxy

for examining the function of the septa, which hasbeen the focus of much attention in recent years.Although diagnostic and important, the numericaldescription of sutures has been elusive. This paperpresents a numerical method for describing suturepatterns and briefly investigates its application.

The terms “suture,” “suture line” and “suturepattern” have been used synonymously by various

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authors. In this paper, “suture” refers to the junc-ture of the septum and the shell wall, and is astructural part of the cephalopod shell. This paperdoes not consider the suture. “Suture line” refers tothe graphical representation of that juncture on theshell, and is typically drawn in twodimensions. Thispaper presents a method to reconstruct the sutureline using a mathematical description based onFourier series. A simulation would create suturelines from the physical processes which governthem; this paper does not present a method of sim-ulation. “Suture pattern” refers to the shape orform of the suture line. This paper presents somesuggestions for analyzing and comparing suturepatterns.

Suture patterns range from simple, straightlines (nautilitic patterns) to visually complex andintricate curves (ammonitic patterns). No explana-tion of sutural complexity is universally accepted.Proposed explanations focus upon a link betweensutural complexity and the shell’s resistance tobreaking due to hydrostatic or unidirectionalstresses (Hewitt and Westermann 1986, 1997,Westermann and Tsujita 1999), the participation ofthe septa in processes regulating buoyancy (Rey-ment 1958, Saunders 1995, Seilacher and LaBar-bera 1995), or other factors (e.g., viscous fingering,García-Ruíz et al. 1990; body-conch attachment,Lewy 2002). Quantitative modeling experiments ofthe effect of sutural complexity on shell strengthhave used artificially simple sinusoidal suture pat-terns with one or more frequencies (Daniel et al.1997, Hassan et al. 2002). The inability to mathe-matically describe more complex and realisticsuture patterns limits such analyses. (For moredetailed and complete reviews of this debate, seeJacobs 1996, Seilacher and LaBarbera 1995, Wes-termann 1996, Daniel et al. 1997, Olóriz et al.2002, Lewy 2002.)

To date, quantitative measures of suture pat-terns fall into two general categories: statistics anddescriptions. Statistics are single values thatdescribe some aspect of the pattern that is usuallyconstrued as a measure of sutural complexity: anumber that increases with the suture’s visualcomplexity. Exactly what is meant by “complexity”is vague (McShea 1991); one reason such a statis-tic is pursued is to clarify its meaning. Descriptions,on the other hand, attempt to provide a mathemati-cal method by which the patterns can be recreatednumerically and do not address complexity. Thesemethods describe the shape of the suture pattern,and therefore must contain more values than a sin-gle statistic. Unlike a statistic, a mathematicaldescription of any suture pattern is unique to thatpattern.

Westermann proposed the length of a suturenormalized to the circumference of the phragma-cone as a measure of sutural complexity (suturalcomplexity index, Westermann 1971) Other work-ers adopted this statistic (e.g., the Index of SuturalComplexity, Ward 1980; suture-sinuosity index,Saunders 1995) or a variation of it (Suture Com-plexity Index, Saunders 1995). These statisticshave been used to study a variety of paleobiologi-cal problems (Ward 1980, Saunders 1995, Saun-ders and Work 1996, 1997). This statistic has thesame value for suture patterns of very differentshapes.

Fractal dimension is an alternative statistic tothe sutural complexity index. Fractal dimension(F.D., Boyajian and Lutz 1992; Df, Olóriz et al.1999) more directly measures recurving of a suturepattern. The first application of fractal dimension tosuture patterns used a space-filling method (Boya-jian and Lutz 1992, Lutz and Boyajian 1995), andmore recent applications use the more conven-tional step-line method (Olóriz et al. 1999). The val-ues of fractal dimension are different between thetwo methods. Fractal analysis has been applied toa variety of paleobiological problems (Olóriz andPalmqvist 1995, Olóriz et al. 1997), including thefunction of the septum (Olóriz et al. 2002). Like thesutural complexity index, different suture patternsshare the same value for fractal dimension.

Several attempts have been made to devise anumerical description of suture patterns. The gen-eral form of suture lines, a closed orbit wrappingaround the circumference of the phragmacone(Figure 1), has been recognized as periodic andtherefore potentially amenable to Fourier analysis.There is motivation for applying Fourier methods tosutures, as they have proven useful for morpho-

Figure 1. The suture line of Strenoceras in three dimen-sions. The venter, indicated by label and adorally ori-ented arrow, and dorsum, indicated by label and arrow,are indicated.

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metric analyses of ostracodes (Kaesler and Waters1972), bivalves (Gevirtz 1976), bryozoans (Ansteyand Delmet 1972, 1973, Anstey et al. 1976, Ansteyand Pachut 1980), trilobites (Foote 1990), humans(Palmqvist et al. 1996) and other taxa.

The complexity of suture patterns, particularlyof ammonitic patterns, precludes the direct applica-tion of Fourier methods due to non-unique points(Figure 2). It is not uncommon for a suture patternto recurve along its length, so that there is morethan one height for a single angular positionaround the circumference of the phragmacone.Less-detailed nautilitic suture patterns can be stud-ied using Fourier analysis because they lack non-unique points, but more elaborate ammonitic pat-terns cannot (Canfield and Anstey 1981). A previ-ous study attempted to circumvent the problem viaa transformation (Gildner and Ackerly 1985), butfails due to a dependence upon the scale of mea-surement.

The method described in this paper consistsof three parts: a parametric formulation for pointsalong the suture line, a transformation of the data,and a normalization of the series by trigonometricinterpolation. The parametric formulation removesthe problem of non-unique points. The transforma-tion solves the recognized problem confrontedwhen using Fourier methods on angular data: theremoval of the trend of increasing angle around thecircumference of the feature. This is analogous tothe perimeter method (Foote 1989), originallyapplied to describe trilobite cranidia as non-para-metric, planar curves in polar coordinates. Thisstudy extends Foote’s method into three dimen-sions. The trigonometric interpolation is not neces-sary for the description of a suture pattern but isneeded to standardize such descriptions for com-parisons between suture patterns. The index of thepoint along the suture line is used as the indepen-dent variable to calculate an initial pair of Fourierseries from digitized suture patterns, which arethen used to interpolate points for the calculation offinal, normalized Fourier series. The final seriesallows the numerical analysis of single suture pat-

terns, and the comparison of different suture pat-terns. This paper will briefly examine howontogenetic development and similarity betweensuture patterns may be approached using thismethod.

Suture patterns from the literature were trans-formed into digital images using a flat-bed scanner.The images were manually digitized directly on thecomputer, and the series were calculated from thedigitized data using a discrete Fourier transform.The suture patterns were reconstructed using theseries, and points equally spaced along the sutureline were trigonometrically interpolated using theseries. The final, normalized Fourier series werecalculated from the interpolated data. Digitizationof the images was done with a program written bythe author in Java (Frames); other processing wasdone with programs written by the author in REAL-basic, compiled for Mac OS X. The code and pro-grams are available for the three programs:Draw4096, Frames, and BothFourier., at the PEsite [http://palaeo-electronica.org/2003_1/suture/issue3_01.htm].

PARAMETRIC FORMULATION FOR POINTS ALONG THE SUTURE LINE

Sutures are three-dimensional structures (Fig-ure 3), but suture lines are usually illustrated astwo-dimensional, open curves in a plane. Whenviewed in three dimensions, a suture appears as aclosed orbital path around the circumference of thephragmacone. Therefore, the suture line is a three-dimensional, closed curve. It can be describedusing cylindrical coordinates. The position of pointsalong the suture line can be measured in terms ofan angle around the circumference of the phrag-macone (theta), the position along the shell's

FIGURE 2. The suture line of Strenoceras showing thenon-uniqueness of points. There are multiple values forheight h, along the blue lines, for several angular posi-tions around the suture line. This non-uniqueness makesa transformation of the angle necessary before Fouriermethods can be used.

FIGURE 3. The suture pattern of Strenoceras in cylin-drical coordinates illustrating the coordinates used inthis study; the angle around the circumference of thewhorl, theta, and the height h. The venter is indicatedby a white arrow.

GILDNER: FOURIER ANALYSIS OF AMMONOID SUTURES

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length (h) and the distance from the center of theshell whorl (r). Points along the suture line aredescribed by parametric equations, with n (theindex of the point) as the variable. If there are Npoints along the suture line, then the position S ofthe nth point (the point with index n of N) can bedescribed as

S(n) = ( r(n), theta(n), h(n) ) 1By using parametric equations, the problem of

non-uniqueness is eliminated, since each h(n) isunique with respect to n. This is in contrast to themore typical formulation where points are definedin linear terms (e.g., Canfield and Anstey 1981using Fourier series to describe nautilitic suturepatterns, and Daniel et al. 1997, Hassan et al.2002 using finite-element analyses).

A complete mathematical description ofsuture patterns would use the shape of the shellthat bounds the septum, rather than cylindricalcoordinates. Most cephalopod shells are logarith-mically spiraled cones (although the heteromorphammonites deviate radically from the logarithmicmodel). Unfortunately, the shape of the shell is sel-dom reported in detail in the literature, and thisstudy will use a cylindrical coordinate system. Thesuture patterns used in this study were from pub-lished suture patterns (principally Wiedmann1969), illustrated as two-dimensional curves. Infor-mation on h is incomplete in these sources, andinformation of the coiling parameters (e.g., W andD, Raup 1967), as well r, are missing. It is possible,however, to compensate for the lack of completeinformation on h by defining an origin on the sutureline (h(0) is defined as 0 where the suture crossesthe venter). Published suture lines commonly donot include information on r or the cross-section ofthe whorl at the position of the suture was drawn,and thus it is not possible to compensate for itsabsence so r is omitted from this analysis. (Itshould be noted that this is not always the case;see, for example, Olóriz et al. 1997.) The sutureline is scaled such that the length of the half-sutureis pi; this characterization has the effect of treatingthe cross-section of the whorl as a circle with aradius of 1. h is scaled such that proportionalitybetween h and the suture length is maintained.

This method requires the complete suture,around the entire circumference of the phragma-cone (from 0 to 2 pi). Suture patterns are often notpublished in their entirety, but omit the internal por-tion of the pattern that is covered by the succeed-ing volution. Such published patterns are notamenable to the method as described here,although it may be possible to extend the method.Some shelled cephalopods, most notably the het-

eromorph ammonites, are not coiled to the degreethat the shell formed by the more mature animaloverlaps the earlier formed shell; several of thesuture patterns used in this analysis are of hetero-morph taxa.

TRANSFORMATION OF ANGULAR POSITION AND FOURIER ANALYSIS

The h parameter of the curve is directly ame-nable to Fourier methods without transformation,but the non-uniqueness problem requires that thetabe transformed. Transforming the angular positionto the difference between the observed andexpected positions solves the problem of non-unique points along the suture (Foote 1989).

The transformation applied to theta is φ(n) = πn/N - θ(n) θ(n) = π n/N - φ(n)

and the inverse transformation is

where n is the index of the point, N is the total num-ber of points along the suture line, from the ventral(or external, n = 0) to the dorsal (or internal, n = N)points of symmetry.

Mirror-plane symmetry of suture lines pro-vides us with opportunities to simplify the analysis.This study uses the point where the suture linecrosses the venter as the origin (n = 0, h = 0, theta= 0). Because suture patterns are symmetric aboutthis origin, they can be fully described by cosineand sine series. Note that h is an even function(h(n) = h(-n)) and can be described using cosineseries, and that theta and phi are odd functions(theta(n) = -theta(-n)) and can be described usingsine series. Because points measured in one halfof the suture have counterparts in the other half,any number of points measured in the half suturedescribes an even number of points in the fullsuture.

The Fourier description of the coordinates ofpoint S(n) along the suture is

=n n( ) ( )n

N

πφ − θ 2A.

( ) ( )nn n

N

πθ = − φ 2B.

( ) h

N 1

i 0

h n in

A2 i

N( )cos

−

=

= π∑

( )N 1

i 0

nn i

N

nA 2 i

N( )sinφ

−

=

π θ = −

π∑

3A.

3B.

GILDNER: FOURIER ANALYSIS OF AMMONOID SUTURES

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where Ah(i), and A phi (i) are the Fourier amplitudesfor the frequency i.

Suture patterns of the same individual show agreat deal of variation throughout ontogeny.Sutures in an ontogenetic series are treated as dif-ferent patterns, as if they belonged to different spe-cies. Variation can also be found between the leftand right sides of the suture of the same individual.This variation? is typically not considered to beimportant, as shown by the convention of illustrat-ing only half-sutures. There is also variationbetween individuals of the same species, a com-mon situation in any morphometric study.

A discrete Fourier transform is used to calcu-late the Fourier series. There are two commonalgorithms for calculating Fourier series: the fastFourier transform (FFT) and the discrete Fouriertransform (DFT). For sine and cosine series, withno imaginary component to the amplitudes (phaseangles), FFT and DFT produce the same results.The FFT is a significantly faster computationalmethod for very large data sets, but is constrainedto data consisting of 2n points (where n is any inte-ger). The DFT is slower, but only requires an evennumber of points (2n). Since the use of half-suturesguarantees that there are an even number ofpoints in the full-suture, this requirement is auto-matically met. The FFT's requirement of 2n pointsis not automatically met and would require interpo-lating points along the suture before the Fourierseries could be calculated. Since the data forsuture lines is limited to a few thousand points, theconstraint on the number of points (2n versus 2n)

becomes more significant than any advantage inspeed the FFT would provide, and the DFT is pre-ferred. The algorithms for the sine and cosine DFTused in this study are modified from Pachner(1984).

The DFT returns a description of the suturepattern as two series of amplitudes, each withmany elements. The suture pattern can be consid-ered as comprised of two different signals, h andphi. The number of frequencies needed for thereconstruction of suture patterns depends upon thecomplexity of the suture and the detail needed(Figure 4). Simple patterns, such as that of Agonia-tites, require as few as 64 amplitudes (32 frequen-cies for both phi and h). More complex suturepatterns, such as that of Scaphites, require manymore data. The reconstruction of the suture patternof Scaphites requires 512 frequencies. More com-plex suture patterns require even more data.

Much of the visual complexity of a suture pat-tern appears to be the result of the phi-frequencies,at least subjectively (Figure 5). Uniformly reducingall h-frequencies by the same factor while leavingthe phi frequencies unchanged produces suturepatterns that are vertically shortened, yet still retaintheir saddles and lobes in recognizable shapes.Uniformly reducing all the phi-frequencies whileleaving the h-frequencies unchanged produces adifferent result. The number and magnitude of non-unique points along the suture are reduced. Thenew pattern is no longer recognizable as being thesame as the unaltered suture pattern. An ammo-

FIGURE 4. Reconstructed suture lines of Agoniatites (below) and Scaphites (above). The reconstruction of Scaphitesuses 1024 amplitudes; that of Agoniatites uses only 64 amplitudes. Black curve in background is the suture line asdigitized from published image. The white lines in the foreground are the reconstructions, calculated from Fourieramplitudes. Figures modified from Wiedmann (1969).

GILDNER: FOURIER ANALYSIS OF AMMONOID SUTURES

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nitic suture pattern plotted with reduced values forphi appears to be a less complex structure.

APPLICATION OF FOURIER DESCRIPTION OF SUTURE PATTERNS

It is useful for analysis to reduce these manyterms in a meaningful way. In time-series analysisand signal processing, the sum of the squares ofthe amplitudes of a signal is the signal’s power.That is

The combined power of a suture pattern is:P(all) = Power(h) + Power(theta) 4C

Each of these definitions of Power describesdifferent properties of the suture pattern. Power(h)describes variability in the height of the suture, thevertical distance between the top and bottom of thesuture. A high Power(h) indicates that the suture istall, along the longitudinal direction of the whorl. Inthe same way, the Power(phi) is a description ofvariability along the radial direction of the suturepattern. A suture pattern with non-singular pointscan be expected to have a high Power(phi).Power(all), Power(h) and Power(phi) of a straightsuture line are all 0.

For this study, ontogenetic series of Strenoc-eras and of Scaphites were analyzed using thismethod, and Power(h) and Power(phi) were plottedfor each (Figure 6). The ontogenetic series showthat h and phi do not develop synchronously. Theamplitudes of the h-frequencies increase morequickly than those of the phi-frequencies in theearly stage of development, but in the middle stageof development this relationship changes. In thelast stage of development of Scaphites, both h andphi increase at an intermediate rate. Only two onto-genetic series are shown, and other ontogeneticseries may display significant deviation from thesimple pattern illustrated. More data are needed to

( ) ( )2h

N 1

i 0

P h A i

−

=

= ∑

( ) ( )2N 1

i 0

P A iφ

−

=

φ = ∑

4A.

4B.

( )2h

N 1

i 0 i 0

A i ( )2A iφ

N 1 −( )P all = +∑ ∑

= =

−4D.

FIGURE 5. The suture pattern of Scaphites drawn with the amplitudes of h and phi scaled. phi amplitudes have beenscaled by the same amount in each column: from left to right, by 0.25, 0.5, 0.75, and 1.0 (no scaling). h amplitudeshave been scaled by the same amount in each row: from left to right, by 0.25, 0.5, 0.75, and 1.0 (no scaling). Theshape of the suture pattern is recognizable when both h and phi are scaled equally, even to 0.5 times their originalvalues.

GILDNER: FOURIER ANALYSIS OF AMMONOID SUTURES

7

determine the general applicability of the trend ofsuture development through ontogeny.

The Fourier description of suture patterns alsoprovides a basis by which suture patterns may becompared. Previous quantitative comparisons ofsuture patterns have not been based on methodsthat are capable of reconstructing suture patterns,but have been based on suture statistics. Thesutural complexity index (Westermann 1971) hasbeen used in several studies with limited success.Its usefulness is limited because many differentsuture patterns have the same value. However, thesutural complexity index can be measured can becalculated directly from the Fourier series. Thelength of the suture line is the line integral of theparametric Fourier series, and the normalizationcan be accomplished by dividing the length by pi.

As a practical matter, it is more efficient to cal-culate the index of sutural complexity by recon-structing the suture pattern and calculating andsumming the distance between calculated points.

The two Fourier series describe the shape ofthe suture pattern in detail, as can be demon-strated by recreating the suture pattern (Figure 4).The series also record the shape in a more generalfashion. Suture patterns reconstructed using onlythe first few frequencies display the same generalshape. For Strenoceras and Scaphites, reconstruc-tions of stages in an ontogenetic series using the

first 32 frequencies generate patterns that are simi-lar (Figure 7) to other members of the same series.The basic pattern of major and minor lobes can beseen in each reconstruction, from the earliest to themature suture pattern.

The frequencies of both h and phi can betreated as orthogonal axes defining a multidimen-sional space (the number of dimensions equaltwice the number of frequencies). The amplitude ofeach frequency of a suture pattern describes apoint in this space. Another way of thinking of thisconcept is to consider the suture pattern a vector inthe multidimensional space, starting at the originand ending at the point defined by the amplitudesof each frequency. The magnitude of the vector isthe square root of Power(all); the direction is deter-mined by the relative amplitudes of each fre-quency. Each suture pattern defines a uniquepoint. Visualizing the pattern in this way is a usefulway to think about the analysis and comparison ofsuture patterns. Suture patterns can be comparedby analyzing the relationships of points and vec-tors.

Suture patterns whose points are coincidentare the same pattern; they have the same shape.Those whose points are near each other in spacehave similar shapes. Because the Euclidean dis-tance between the points representing suture pat-terns is a reflection of their similarity, comparison ofthese distances is a comparison of suture patterns.This difference can be demonstrated by varyingthe amplitudes of fossil suture patterns (Scaphites,Figure 4) to generate suture patterns that can then

( ) ( )1

0

1 dh x d xSCI

dx dxdx+

θ=

π ∫ 5.

FIGURE 6. Plot of Power(phi) and Power(h) for suture patterns in the ontogenetic series of Strenoceras (6a) andScaphites (6b). The trend of power through the ontogeny is indicated by the thick, bent arrow, with the arrow head atthe mature suture pattern. Colored arrows connect the point plotted to the suture pattern. Early in ontogeny, h-ampli-tudes increase relatively faster than phi, then enter a stage in which the relationship is reversed.

GILDNER: FOURIER ANALYSIS OF AMMONOID SUTURES

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be compared to the original: the greater the differ-ence in power, the greater the difference in shape.The Euclidean distance between points is

where C is the distance between the points, andA(i) and B(i) are the amplitudes of the ith frequen-cies for the two suture patterns.

A dendrogram of nearest neighbors can becreated from the distances (Figure 8). The taxaused in this study are not meant to be exhaustive,but were chosen to capture a range of sutureshapes. Five relatively simple nautilitic suture pat-terns were chosen: Agoniatites, Anetoceras, Bac-trites, Cyrtobactrites, and Mimagoniatites. Seven(or eight if Scaphites and Strenoceras are countedseparately?) more complex suture patterns werechosen: Chelinoceras, Diabologoceras, Douville-iceras, Hamites, Leptoceras, Paraspiticeras,Scaphites, and Strenoceras. The ontogeneticseries of Scaphites and Strenoceras used in theanalysis above were included, with each stage ofthe series treated as a separate suture pattern.Among these taxa, Agoniatites, Anetoceras, andMimagoniatites are agoniatitid genera, Bactritesand Cyrtobactrites are bactritid genera, and Douvil-leiceras, Cheloniceras, and Paraspiticeras aredouvilleiceratid genera; other families are repre-sented by single taxa. All genera are currentlyassigned to the ammonoids (Order AmmonoideaZittel, 1884).

Euclidean distance between the points repre-senting suture patterns will not necessarily be a

very sensitive indicator of similarity between sutureshape. As seen earlier, the shape of a suture pat-tern is conserved if all amplitudes are scaledequally. A greater distance than one might expectfrom a visual examination separates the points forthese similar suture patterns. The general shape ofthe suture pattern is preserved in the relativeamplitudes between frequencies, not in their abso-lute magnitude. That is, the general shape of thesuture pattern is recorded in the orientation of thevector, not its magnitude.

The distance-based dendrogram reflects thisinsensitivity. Although there is an apparent visualpattern in the tree, the bactritids, douvilleiceratidsand ontogentic series are split between differentbranches, separated by the suture patterns of moredistantly related taxa. Only the suture patterns ofthe agoniatitids are grouped into the same branchof the dendrogram. This moderate success is less-ened by the placement of the agoniatitids amongthe other taxa.

The orientations of the vectors can be moresensitive indicators of the similarity between suturepatterns. The orientations can be compared byexamining the angle between the vectors repre-senting them. The Cosine Law resolves the anglebetween two suture patterns (A and B) and originas:

where delta is the angle between the two suturepatterns, A and B are the magnitudes of the vec-tors (equal to the square root of Power(All) foreach), and C is the distance between the endpoints

( ) ( )( ) ( ) ( )( )N 1

22

h h

i 0

C A i B i A i B i

−

φ φ=

= − + −∑ 6.

2 2 21 A B C

ABcos

−

+ −∆ =

7.

FIGURE 7. Suture patterns for stages in an ontogenetic series of Strenoceras (7a) and Scaphites (7b) reconstructedwith the first 16 frequencies (left), and with all frequencies (right). The similarity between each of the stages showsthat the fundamental shape of the suture pattern is conserved and can be extracted using Fourier series.

GILDNER: FOURIER ANALYSIS OF AMMONOID SUTURES

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of the vectors, and is the same distance that isused to compare suture patterns by Euclidean dis-tance.

As with Euclidean distances, a dendrogram ofnearest angular neighbors can be created (Figure9). The dendrogram based on angles is similar in

some ways to the distance-based dendrogram, butin other ways is significantly different. The agoni-atitids remain collected into the same branch, andthe bactritids, douvilleiceratids and ontogeneticseries of Scaphites remain separated. However,

FIGURE 8. Dendrogram of similarity between suturepatterns, based on Euclidean distance between pointsrepresenting the sutures in multidimensional space.Related taxa are coded by colored boxes: douvilleicer-atids - white with black border, bactritids - yellow, agoni-atitids - green, Scaphites - red, and Strenoceras - blue.Suture patterns included are Agoniatites (Agon), Aneto-ceras (Anet), Bactrites (Bact), Cyrtobactrites (Cyrt) andMimagoniatites (Mima), Chelinoceras (Chel), Douville-iceras (Douv), Hamites (Hami), Leptoceras (Lept),Paraspiticeras (Para), Scaphites (Scap) and Strenoc-eras (Stren). The numbers after Scaphites and Strenoc-eras refer to the position in the ontogenetic series with 1being earliest.

FIGURE 9. Dendrogram of similarity between suturepatterns, based on angular distance between points rep-resenting the sutures in multidimensional space.Related taxa are coded by colored boxes: douvilleicer-atids - white with black border, bactritids - yellow, agoni-atitids - green, Scaphites - red, and Strenoceras - blue.Suture patterns included are Agoniatites (Agon), Aneto-ceras (Anet), Bactrites (Bact), Cyrtobactrites (Cyrt) andMimagoniatites (Mima), Chelinoceras (Chel), Douville-iceras (Douv), Hamites (Hami), Leptoceras (Lept),Paraspiticeras (Para), Scaphites (Scap) and Strenoc-eras (Stren). The numbers after Scaphites and Strenoc-eras refer to the position in the ontogenetic series with 1being earliest.

GILDNER: FOURIER ANALYSIS OF AMMONOID SUTURES

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the ontogenetic series of Strenoceras is nowgrouped together in the same branch. In addition,the agoniatitids are now separated from the othertaxa at a much higher level, indicating that the dif-ference in their shape compared to the shape ofthe other taxa is more significant.

The ontogenetic series of Strenoceras waseasily grouped into one cluster using the angularmeasure, but not using Euclidean distance. Eventhough the first, youngest suture in the series isvery “simple,” clustering by angle tightly groups itwith the rest of the taxon. The ontogenetic series ofScaphites is not grouped using either Euclideandistance nor angular measures of similarity,although clustering is more complete using theangular measure. In both instances, the douville-iceratids Douvilleiceras and Chelinoceras aregrouped closely, but separated from the other dou-villiceratid Paraspiticeras.

In neither case does the nearest-neighbordendrogram accurately reflect taxonomy. This lackof correlation is not surprising, because it has beenshown that determining taxonomic relationshipsrequires examination of both the patterns and theirontogenetic development (House 1980, Wiedmannand Kullman 1980). The effort reinforces commonknowledge: that similarity of suture shape alone,without regard to the shapes ontogeny, is not suffi-cient for unraveling the phylogeny of theammonoids.

SUMMARY

The analyses presented in this paper are pre-liminary efforts. The suture patterns used for analy-sis were chosen to test the reconstruction method,and not to resolve taxonomic or evolutionary ques-tions. More complex analytical techniques wouldundoubtedly prove useful and informative.

Combined with a model of the septum (Ham-mer 1999, Daniel et al. 1997, Hassan et al. 2002)and the outer shell (Raup 1967, Ackerly 1989,Okamoto 1988), the model of suture patterns pre-sented here can be used to improve the accuracyof computational mechanical models. The modelpresented here does not include the expansionand curvature of the phragmacone, but treats theshell as a cylinder, as do many of the currentmechanical models. However, it is possible to inte-grate it into a more realistic model of the ammoniteshell and septa. This extension would allow studyof the shell-suture-septa complex in a systems-approach.

A thorough test of the usefulness of Fourieranalysis to the study of suture patterns must use

more accurate data and data from more taxa. Datafor this study were manually digitized, scannedimages of photocopies of published suture pat-terns. These images have gone through a numberof reproductions before being digitized for the rawdata used for the Fourier analysis, and an unknownamount of error has been introduced at each step.The accuracy and precision of published images ofsuture patterns is variable. Use of original suturetracings, rather than mass-produced images,would improve the reliability of the results of analy-ses such as this study. More accurate yet would bedata collected directly from the specimen, usingprecision systems such as three-dimensionalpoint-digitizing arms (Lyons et al. 2000; Wilhite2002).

Along with applications in the study of suturepatterns, a quantitative description of suture pat-terns has benefit for digital records. The ampli-tudes of a Fourier series can be stored in asurprisingly small amount of data. For moderatelycomplex suture patterns such as that of Strenoc-eras, the amount is only 1 or 2 kilobytes (depend-ing on whether single or double precision numbersare used). This size issue is an important consider-ation for the electronic communication of suturepatterns and in the memory demands for comput-erized databases, such as Ammon (Liang andSmith 1997). The low data requirements could alsoprove useful for applications where the rate of datatransfer is limited, such as the Internet.

ACKNOWLEDGMENTS

I especially want to thank two anonymousreviewers of the submitted manuscript for theircomments, some of which prevented me frommaking some simple and embarrassing errors. R.Kaesler, M. Foote, W. B. Saunders and Ø. Hammerprovided information and feedback that helpeddevelop the concept. E. G. Allen generously sharedher knowledge and insight, above and beyond thepale. S. C. Ackerly originally directed my attentionto the problem of quantifying suture patterns.Finally, R. Linsley’s simple question (“But what is itgood for?”) greatly changed the direction of thisresearch.

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