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tocGeometric morphometrics in ammonoids based on virtual modelling

Daniel A. Morón-Alfonso, René Hoffmann, and Marcela Cichowolski

Article number: 24.3.a29
Copyright Paleontological Society, September 2021

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Submission: 24 March 2021. Acceptance: 6 September 2021.


Linear morphometrics is the most widely applied technique to study the variation of the conch morphology in ammonoids and other ectocochleate cephalopods. However, because this method frequently relies upon a few linear measurements, it lacks the explanatory power to accurately characterize the shape of the whorl cross-section, which is instead discussed solely in descriptive terms, e.g., elliptical, triangular, or subquadrate. Here, we introduce a landmark-based geometric morphometric approach to study ammonoid whorl cross-sections, derived from the regularly used morphometric parameters in cephalopods. This new technique uses virtual modelling to generate semilandmark configurations and virtual models of whorl cross-sections. We applied it to study 50 ammonoid specimens belonging to 48 genera exhibiting a wide range of morphologies and ages. Results indicate that this new method is appropriate to describe the shape of ammonoid whorl cross-sections, allowing us to construct a morphospace showing several biological patterns (e.g., clustering and homeomorphy), and complex morphological transformations that, in some cases, correlate with evolutionary tendencies described by previous authors. Further, this technique can be used to generate the basic segment required for the elaboration of the virtual models employed in hydrostatic and hydrodynamic studies.

Daniel A. Morón-Alfonso. Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Departamento de Ciencias Geológicas, Área de Paleontología, Ciudad Universitaria, Pab. 2, C1428EGA, Buenos Aires, Argentina. CONICET-Universidad de Buenos Aires, Instituto de Estudios Andinos “Don Pablo Groeber” (IDEAN), Buenos Aires, Argentina.
René Hoffmann. Institute of Geology, Mineralogy, & Geophysics, Ruhr-Universität Bochum, 44801 Bochum, Germany.
Marcela Cichowolski. Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Departamento de Ciencias Geológicas, Área de Paleontología, Ciudad Universitaria, Pab. 2, C1428EGA, Buenos Aires, Argentina. CONICET-Universidad de Buenos Aires, Instituto de Estudios Andinos “Don Pablo Groeber” (IDEAN), Buenos Aires, Argentina.

Keywords: Cephalopoda; Ammonoidea; geometric morphometric; conch; whorl

Final citation: Morón-Alfonso, Daniel A., Hoffmann, René, and Cichowolski, Marcela. 2021. Geometric morphometrics in ammonoids based on virtual modelling. Palaeontologia Electronica, 24(3):a29.

Copyright: September 2021 Paleontological Society.
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s figure1The study of biological variation is based on the comparison of features (e.g., composition, structure) of one or more organisms using distinct methodologies (e.g., spectrometry, morphometry) to characterize it (Adams et al., 2004). In some cases, this variation is expressed in the morphology of different components, showing modifications in development within a species (e.g., sexual dimorphism) or through the evolutionary history of a group (Claude, 2008). In ectocochlate cephalopods, the need to increase the descriptive power to study the morphological variation of the conch has prompted the development of several morphometrical models (Table 1). In general, these methods employ linear measurements as quantitative descriptions of the conch morphology combined with statistical analyses to find biological patterns (e.g., ecological constraints, and evolutionary tendencies, Adams et al., 2004). Early attempts to assess the conch morphological variation used simple linear measurements of the conch collected from cross or longitudinal sections (Moseley, 1838; Sandberger, 1851, 1853, 1857; Trueman, 1940). More recently, Raup (1967) and Raup and Michelson (1965) refined this technique, focusing on the study of the ammonoid conch geometry and adapting the Raupian model, previously defined for gastropods (Raup, 1961, 1966; Urdy et al., 2010). The Raupian model consists of four geometric parameters (S: the shape of the generating curve, W: the whorl expansion rate, D: the position of the generating curve relative to the coiling axis, and T: the rate of whorl translation), calculated from five linear measurements (Figure 1). Korn (1997, 2010) adopted most of these linear measurements, incorporating derived parameters, indices, and rates (e.g., the imprint zone rate), centering on the evaluation of the ontogenetic trajectories in Palaeozoic ammonoids (Figure 1). Later, this modified version of the Raupian model was proposed as the ‘standard’ to characterize the conch morphology for all (Devonian to Cretaceous) ammonoids (Klug et al., 2015). During the last decade, several morphometric models were proposed to study the conch morphology and morphogenesis, showing an increasing degree of complexity as more factors were considered. These models have been successfully applied to explain the origin of distinct ribbing patterns and their relationship with other parameters (e.g., Erlich et al., 2016, Table 1) to test the paleoecology of ectocochleate cephalopods (for a review see Naglik et al., 2015) or to define ecomorphospaces (e.g., Westermann, 1996, 2013; Ritterbush, 2015; Tendler et al., 2015). Recently, these methods have been integrated to machine learning testing the potential for future automated ammonoid taxonomic assignment (Foxon, 2021).

Although morphometric models are widely accepted as a suitable tool to study ectocochleate cephalopods, Gerber (2017) noted that these methods possess underlying constraints that can hinder the complete description of the conch morphological variation, suggesting that the landmark-based framework (geometric morphometric, GM) could be more suitable. For example, the complex shape of the whorl cross-section cannot be accurately characterized by simple linear measurements, and it is usually typified by descriptive terms that cannot be statistically compared between samples (Figure 1, Smith, 1986). Previous applications of GM methods in ectocochleate cephalopods include the evaluation of the suture line and conch morphology in ammonoids using outline-based methods (Korn and Klug, 2012; Klein and Korn, 2014; Allen, 2016; Wegerer et al., 2018), and landmark-based methods to analyze the morphologic variation on specific ammonoid taxa (Neige, 1999; Courville and Crônier, 2016; Bischof et al., 2021). Results of these studies showed the potential of GM methods to study the biologic variation in ectocochleate cephalopods. In this work, we introduce an alternative technique to study the shape of whorl cross-sections using semilandmarks, derived from the virtual modelling methodology described in a previous study (Morón-Alfonso et al., 2020). We use the term “semilandmarks” instead of “landmarks”, given that the homology between these points cannot be determined precisely from the curves that define the whorl shape of ammonoids. As follows, we apply this method to study the whorl cross-section of 50 ammonoid specimens representing 49 species, defining their semilandmark configurations and generating the corresponding virtual models.


s figure2The use of virtual modelling has improved our understanding of the hydrodynamic and hydrostatic properties of ectochocleate cephalopod shells (e.g., Peterman et al., 2019a, 2019b, 2020; Hebdon et al., 2020). First described by Peterman et al. (2019a), this method consists of the iteration of a virtual whorl cross-section following a pathway in a three-dimensional space, allowing it to generate a virtual model of the cephalopod conch for experimentation. Based on this approach, Morón-Alfonso et al. (2020) described a modelling process to construct complex planispiral ammonoid conchs noting that one of the advantages of the method was the flexibility to model non-elliptical whorl cross-sections. We go one step further, standardizing this modelling technique to study the shape of the whorl cross-section implementing a geometric morphometric approach.s figure3 This new method consists of the modification of a simple plane using the subdivision surface modifier included in the open-source program Blender 2.92 (Blender Online Community, 2021) based on the Catmull and Clark (1978) algorithm (Figure 2). Subdivision surface is a widespread procedure used in computer graphics to generate complex three-dimensional shapes from simple geometries. This algorithm increases exponentially the number of vertices, edges, and faces of a mesh through simple rules (see Catmull and Clark, 1978; Sabin, 2002). Every time the algorithm is applied, the surfaces of polygonal geometries become smoother at the cost of a smaller size (Figure 2). This process can be modified by increasing the number of edges (and hence the number of vertices) to model more complex two- and three-dimensional shapes (Sabin, 2002). Despite that there are infinite ways to model whorl cross-sections of cephalopod conchs using this technique, based upon the standard dimensional linear measurements used in ammonoids (Figure 1), we can divide the whorl cross-section of a theoretical specimen into six curves and four regions defined by 18 vertices/semilandmarks (Figure 3). The distance between these vertices can be extrapolated to the standard parameters used for the cephalopod conchs (Figure 3). This basic configuration can be modified including more features (e.g., ornamentation) to obtain more detailed models. In some evolute ammonoids, the imprint and contact zones are nonexistent or reduced. In this special case, certain semilandmarks can share the same coordinates (i.e., merging the vertices; Figure 3).

This configuration can also be used to model the whorl cross-section of other ectocochleate cephalopods such as heteromorph ammonoids and nautiloids (Figure 3), or be applied to the conchs of endocochleates like modern Spirula and, potentially, to other organisms with spirally coiled hard parts like gastropods or foraminifera. A key advantage of this new technique to define semilandmarks with respect to other GM methods is the capacity to directly translate the semilandmark configurations to complex virtual models using the modelling program. Consequently, any theoretical or transformed semilandmark configuration (e.g., descaled) can be rendered to a virtual model for comparison. Hereafter, we will refer to this operation as ‘reverse modelling’.


Once the basic theoretical model and the number of semilandmarks are determined, images of real ammonoid cross-sections can be used as stencils exporting them as empties in the Blender 2.91 workspace. To test the reliability and constraints of the method, we used images of 50 ammonoid specimens to construct a basic morphospace and study the patterns obtained from it. The images consisted of orthoslices collected from physical-optical, medical, and micro-CT data, coupled with photographs and illustrations of the specimens with a proper orientation found in repositories and the literature. The number of species is relatively low (due to a lack of available conch cross-sections), and represents 49 species belonging to 48 genera ranging from the Late Devonian to Late Cretaceous, showing a wide spectrum of morphologies and stratigraphic age ranges (Table 2). The repository information, landmark configurations, and more detailed data of the specimens can be found in Appendix 1, and for a video tutorial of the virtual modelling method applied to these samples see Appendix 2. For partially eroded specimens only the best-preserved flank was modelled and then mirrored to obtain a complete virtual model of the whorl cross-section (hereafter designated as virtual whorl cross-section). For this work, we performed four levels of subdivisions as the standard to generate the virtual models.

Statistical Analyses

The data including the semilandmark coordinates were exported from the modelling software to a spreadsheet using PAST 4.06 (Hammer et al., 2001). These data were then processed to generate a Clarion TopScan (TPS) file required for the analysis. Additional data containing the ID of the specimens, the maximum diameter, and the mean stratigraphic age range (calculated as an average between the maximum age and minimum age of the taxon) were included in a table separately. Following, this data was loaded to an R 4.04 environment and a Generalized Procrustes Analysis (GPA) and statistical analysis were performed using the Morpho and Geomorph Packages (Adams and Otarola-Castillo, 2013; Schlager, 2017). The GPA translates all the semilandmark configurations to the origin, scales them to unit-centroid size, and optimally rotates them (using a least-squares criterion) until the coordinates of corresponding points align as closely as possible. The resulting aligned Procrustes coordinates represent the shape of each specimen and are found in a curved space related to Kendall's shape space (Kendall, 1984). The multivariate analysis consisted of a Principal Component Analysis (PCA) and a Multivariate Linear Regression (MLR). PCA is used for reducing the dimensionality of complex datasets, increasing interpretability, and generating predictive models, but at the same time minimizing information loss (Jolliffe and Cadima, 2016). The MLR was performed using one independent variate (the mean stratigraphic age range or the diameter) and n dependent variates (Principal Components). This analysis fits each dependent variate separately to the independent variate using simple linear regressions. Further, an overall Multivariate Analysis of Variance (MANOVA) test of the multivariate regression significance is provided. The data obtained from the PCA such as the predictive semilandmark configurations were submitted to the reverse modelling to generate the matching virtual whorl cross-section for each specimen.

Evaluation of Morphometric Patterns

s figure4There have been described several patterns of covariation in ammonoids related to the morphology of the whorl cross-section (De Baets et al., 2015; Monnet et al., 2015b, 2015c). Therefore, a valuable comparison is testing if these patterns, described from morphometric analyses, are observed using GM methods as well. This evaluation can be done approximating a theoretical whorl-section to the standard model and applying transformations related to the variation of the whorl cross-section; in this case, the aperture height (ah), the whorl width (ww), and the whorl height (wh, Figure 4). The variation of the whorl width will be associated with scaling in the x-axis implying an expansion or compression in the horizontal plane (Figure 4A-B), and the variation of the aperture height and whorl height are linked to translations in the y-axis (Figure 4C-D). The whorl height is recognized as an elongation or contraction, while the aperture height is associated with changes in the size of the imprint zone. Both vary depending on the degree of involution (Figure 4C-D). Comparison between covariation patterns can be difficult to assess solely from the transformation vectors. Therefore, we used simple graph bars to illustrate the change in position (translation) in each semilandmark (Figure 4). As follows, the total variation will be a combination of the x and y vectorial components.

Special Cases

To complement the discussion and to portray some of the possible applications of the methodology, an examination of two special cases was incorporated in the study. The first case consists of the inclusion of two Maorites species in the statistical analyses, M. seymourianus (Kilian and Reboul, 1909) and M. densicostatus (Kilian and Reboul, 1909), both from the López de Bertodano Formation (Campanian-Maastrichtian) of the James Ross Basin, Antarctica. These species have a well-established stratigraphical framework (Tobin et al., 2012; Witts et al., 2016), and previous research of their ontogenetic trajectories showed a possible case of speciation involving paedomorphism (Morón-Alfonso, 2019), likely registered in the fossil record due to the unusually high average sedimentation rate determined for this formation (0.1 to 0.2 mm per year, Witts et al., 2016; Scasso et al., 2020). Evaluation of the variation of the whorl cross-section between these species elucidated the processes implicated during an evolutionary turnover.

The second special case involved the elaboration of a complementary PCA and PGA including additional whorl cross-sections at different diameters of an exceptionally well-preserved Proleymeriella schrammeni (Jacob, 1907) specimen used in a previous study (Hoffmann et al., 2021). The resulting transformations between the PC-scores were used to analyze the change of the whorl cross-section through ontogeny. For this case, despite that the whorl cross-section does not define the general geometry of the conch at a given diameter it could indicate the transition between different ontogenetic stages, signalling periods in which the morphology of the whorl cross-section is similar among different taxa (e.g., morphologic phylotypic periods; Levin et al., 2016).


s figure5The variation for the virtual whorl cross-sections is explained by 29 principal components (PCs, Appendix 3). The first three PCs explain around 90% of the variation found in the sample (Table 3). Further, predicted semilandmark configurations were calculated for each of these PCs and the reverse modelling method was applied to illustrate the principal transformations (Figure 5, Figure 6, Figure 7). Going from negative to positive values, the transformations in PC1 (when compared to the expected patterns of variation) are consistent with an expansion and contraction of the whorl cross-section, showing an increase of the involution degree (Figure 5).s figure6 The semilandmarks that display the highest variation are related to the imprint zone (Figure 5). In contrast, the transformation in PC2 adjusts to a compression of the whorl cross-section localized in the peripheric semilandmarks, and an overall elongation alongside an increase in the involution degree (Figure 6). Accordingly, PC2 determines the general outline of the whorl cross-section going from ellipsoidal to lanceolate shapes.s figure7 Following, the transformations in PC3 do not adjust completely to any of the expected covariation patterns (Figure 7), and most of the variation is observed on semilandmarks 3, 6, 12, and 15, which are related to the location of the whorl width parameter with respect to the horizontal axis. Consequently, PC3 encompasses changes between rectangular to sub-triangular shapes. When plotted together, these PCs result in a complex three-dimensional morphospace. To ease the interpretations, PC1 was plotted against PC2 and PC3 separately (Figure 8, Figure 9). s figure8From this three-dimensional morphospace, homeomorphy and clustering can be distinguished (e.g., Figure 8, Figure 9, between species 29, 31, and among species 26, 37, and 47). To evaluate other biological patterns the PCs were plotted against the maximum diameter and the mean stratigraphic age range (Figure 10). s figure9The results are compatible with previous patterns found in ammonoids (i.e., the reduction of the involution through phylogeny as size increased through time). To this effect, and following the polarity of the transformations used in this work, PC1 shows an inverse relationship with the diameter and a direct relationship with the mean stratigraphic age range, showing compression and a decrease in involution as size increases, and the opposite morphologic pattern going from older to younger taxas figure10 (Figure 10A-B). Contrary, PC2 shows no significant relationship with the diameter and a weak relationship with thes figure11 mean stratigraphic age range (Figure 10C-D). MANOVA results for the MLR show the same pattern for the diameter and mean stratigraphic age range showing no significant relationship with the diameter and a weak but significant relationship with the mean stratigraphic age range (Table 4). Finally, we analyzed the ontogenetic changes of P. schrammeni compared to the morphospace obtained from the first PCA. Results are coherent with the transformations obtained for the initial analyses, indicating a predominance of PC1 during early ontogeny (juvenile phase), and PC2 and PC3 reaching adulthood (Figure 11).


Virtual Modelling Technique

s figure12The results demonstrate that the semilandmark-based method using virtual modelling is a viable option to study the biological variation of the whorl shape in ammonoids. Moreover, the virtual modelling technique can be combined with the methodology explained in Morón-Alfonso et al. (2020) to generate basic segments required for virtual models employed in hydrodynamic and hydrostatic studies (Figure 12). The modelling of complex whorl shapes with special ornamentation, such as furrows, keels or spines, remain difficult. Their topology could be adjusted in a simplified version of the original whorl cross-section, and the ornamentation can be described separately as a particular feature of the studied sample (Figure 12B). The alternative option is to add more semilandmarks (Figure 12C), which could be important to study the variation in specific taxa. The second constraint is related to the lack of interplay between the modelling platform and the statistical programs used in geometric morphometrics. On this subject, the modelling method can be done quickly, but exporting the data (i.e., the semilandmark coordinates) must be done manually to keep the same arrangement on each sample. Finally, there have been several sophisticated models for studying the ammonoid conch, but models based on linear measurements are still predominant in the field because of their practicality and simplicity (e.g., Tajika and Klug, 2020). Therefore, we note that the virtual modelling technique described here is not designed to replace these morphometric models...actually, the results obtained from both methodologies could be used to evaluate the allometric variation in the specimens (e.g., in this work we evaluate the morphologies against the diameter) or be incorporated to other semilandmark-based methodologies for more complete descriptions (e.g., Gerber, 2017).

Biological Patterns

With our dataset, we were able to elaborate a morphospace showing several possible biological patterns (Figure 9, Figure 10). The transformations in PC1 seem to be related to a covariation pattern, originally obtained from the standard Raupian model, involving an increase in the compression and changes in the involution degree through time. This macroevolutionary tendency may be asserted by increasing the number of species spanning a longer period of time (for a review see Monnet et al., 2015b). However, PC1 only explains around 57% of the variation, which indicates that there is still a considerable range of variation that has not been covered (Table 3). On this subject PC2 (21%) and PC3 (11%) seem to be strongly linked to another pattern rarely assessed quantitatively denominated as the First Buckman’s Rule of Covariation (De Baets et al., 2015; Monnet et al., 2015c). This rule involves a relationship between the degree of involution, compression, and an unaccentuated ornamentation, and it is observed in both the intra- and inter-specific variation of closely related taxa (Monnet et al., 2015c). A relevant case is observed in the Maorites species, M. densicostatus and M. seymourianus. These species share a very similar discoidal and subinvolute conch morphology (following Klug et al., 2015 terminology), and discrimination between both is based on the ornamentation and differences in the whorl width and whorl height in adults (see Morón-Alfonso, 2019). In the morphospace, these species share similar PC-scores for PC1, and most of the variation is explained by PC2 and PC3. An evolutionary tendency towards the more compressed form M. densicostatus was suggested in previous studies, but besides the increase in compression, no other evidence suggested an explanation to this speciation case (Macellari, 1986; Morón-Alfonso, 2019). Because PC2 and PC3 are related to the degree of compression and the shape of the ventral region, these PCs define the general outline of the conch and are likely important to determine its hydrodynamic performance. Further, if this last statement is confirmed using hydrodynamic analyses in complete virtual models of the conch of these species, it could explain the cases of homeomorphy and clustering observed in the morphospace (e.g., Figure 9, specimens 29 and 31), implying a convergent evolution to a potential optimal ecomorphology related to their size.

Regarding the variation of the whorl cross-section in the ontogeny of P. schrammeni, this specimen shows a complex morphological change between whorls. In the present case, PC1 seems to dictate the transformations early in the ontogeny, showing a decrease in the involution degree and expanding the area of the whorl (Figure 11). Contrary, PC2 and PC3 transformations are dominant reaching adulthood, involving a change from a sub-elliptical to a sub-triangular shape. The first observation suggests that PC1 transformations may be related to a process consistent with an increase in inhabitable area, maybe related to a more active lifestyle (creating the space necessary for the musculature and the mantle cavity), which would explain the macroevolutionary tendency presented at the beginning of this section as well. Contrary, the transformations of PC2 and PC3 in this specimen may point to a more complex picture regarding Buckman’s first rule of covariation, implying that these transformations could be related to the development of reproductive structures, maturity, or sexual dimorphism, which would also explain why this pattern is relevant in closely related taxa such as the two species of Maorites. However, currently, there is no clear explanation for Buckman’s rules of covariation suggesting either an ecologic or morphogenetic driver (Monnet et al., 2015c), or even if it is a frequent pattern or is restricted to specific taxa. The transformations included in PC2 and PC3 may be important to determine the origin of this covariation pattern as more data become available, particularly from studying the ontogeny of closely related taxa.

In summary, the basic morphospace constructed in this work provides a first overview of the wide variation found in the ammonoid whorl shapes. Further, the biological patterns exhibit here should be validated by increasing the number of specimens (around 300 to 600 to be compared with other morphospaces), and further research is required to evaluate how the changes of the whorl-cross section determine the geometry of the conch, and as follows, their ecological properties. About the modelling technique applied to ammonoids, high-resolution images of cross-sections and tomographic data are the most suitable. Yet, this data is still scarce, and heavily skewed among groups, as different authors prioritized other features and traditional morphometrics to study the ammonoid conch (Bardin et al., 2014). Accordingly, if possible, these empirical data are preferred over diagrammatic sketches to reduce cognitive biases (Blanco, 2017) and should be added to the species description to generate accurate models.


Results demonstrate that the virtual modelling method described herein is a viable option to study the shape of the whorl cross-section in ammonoids using geometric morphometrics.

The variation of the whorl cross-section in ammonoids is mainly explained by three principal components that comprise a complex morphospace, exhibiting several cases of homeomorphy and clustering. The patterns of variations found indicate morphological changes associated to long-term evolutionary trends, and independent secondary changes observed in closely related taxa and/or maturity.


The authors thank M.J. Royo y Gómez (Servicio Geológico Colombiano, Bogotá, Colombia) for giving access to some of the specimens used in this study. M.C. Rodríguez Amenábar (Instituto Antártico Argentino, Buenos Aires, Argentina) and M. Tanuz (Universidad de Buenos Aires, Buenos Aires, Argentina) are thanked for the loan of specimens under their care. Thanks to the Clínica La Sagrada Familia (Buenos Aires, Argentina) and its staff, where some of the CT-scan studies were accomplished. Special thanks to the cephalopod research community for the free access to additional data required for the analyses. We also thank the anonymous reviewers for their commentaries that significantly improved the manuscript. This is the contribution R-382 to the Instituto de Estudios Andinos ‘Don Pablo Groeber’ (IDEAN, UBA-CONICET).


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