EVOLUTION & DEVELOPMENT 14:6, 501–514 (2012) DOI: 10.1111/ede.12003 Quantification of ontogenetic allometry in ammonoids Dieter Korn Museum f ¨ ur Naturkunde, Leibniz Institute for Research on Evolution and Biodiversity, Berlin Author for correspondence (email: dieter.korn@mfn-berlin.de) SUMMARY Ammonoids are well-known objects used for studies on ontogeny and phylogeny, but a quantification of on- togenetic change has not yet been carried out. Their planispi- rally coiled conchs allow for a study of “longitudinal” ontoge- netic data, that is data of ontogenetic trajectories that can be obtained from a single specimen. Therefore, they pro- vide a good model for ontogenetic studies of geometry in other shelled organisms. Using modifications of three cardi- nal conch dimensions, computer simulations can model ar- tificial conchs. The trajectories of ontogenetic allometry of these simulations can be analyzed in great detail in a theoret- ical morphospace. A method for the classification of conch ontogeny and quantification of the degree of allometry is proposed. Using high-precision cross-sections, the allomet- ric conch growth of real ammonoids can be documented and compared. The members of the Ammonoidea show a wide variety of allometric growth, ranging from near isometry to monophasic, biphasic, or polyphasic allometry. Selected ex- amples of Palaeozoic and Mesozoic ammonoids are shown with respect to their degree of change during ontogeny of the conch. INTRODUCTION Conceptual background Ammonoids are extinct ectocochleate cephalopods (i.e. cephalopods with an outer shell) that have long been cele- brated for the opportunity to study their ontogeny. Their ac- cretionary growth with conservation of juvenile stages allows for the investigation of complete ontogenetic transforma- tions of conch geometry, ornament details, and septal char- acters. This is in striking contrast to other organisms such as vertebrates (which reabsorb and rebuild their skeletons) or arthropods (which grow with moulting events). Therefore, ammonoids are particularly suitable for studying ontogenetic allometry (for a distinction of the three levels of allometry, static, ontogenetic, and phylogenetic, see Cock 1966; Gould 1966, 1977; Cheverud 1982; Klingenberg 1998). The geometry of spirally coiled conchs of cephalopods has attracted researchers for a long time. Robert Hooke al- ready in the 17th century had been interested in the regular conch growth of the Living Nautilus (posthumously pub- lished 1705). Pioneering mathematical analyses of the conchs of Recent Nautilus and of ammonites were published during the mid-19th century (e.g. von Buch 1832; Moseley 1838; Naumann 1840; M ¨ uller 1850; Sandberger 1855, 1856). More than 100 years later, David Raup (Raup and Michel- son 1965; Raup 1966, 1967) outlined the theoretical frame- work for further investigations on the geometry of coiled conchs. He analyzed various types of molluscs in terms of their geometrical properties and defined principal morpho- logical conch parameters, the so-called Raupian parameters, conch radius, whorl width, and whorl expansion rate (the opening rate of the whorl spiral). In the 1970s, the working group of Kullmann and Kant (e.g. Kullmann and Scheuch 1970; Kant 1973) began to quan- tify ontogenetic change in Palaeozoic ammonoids. These studies can be seen as a consequence of a long tradition of describing Palaeozoic ammonoids with particular use of their conch ontogeny (for a review, see Korn and Klug 2012). Kullmann and Kant produced a large number of conch cross- sections, particularly of Carboniferous material, and were the first authors to describe ontogenetic trajectories in greater detail. In several classic articles, Huxley (1924, 1932) and Huxley and Teissier (1936) have outlined the principles of allometry (reviewed in great detail by Gould 1966 and Klingenberg 1998). Simple ontogenetic allometry can be expressed by the power function y = bx a , where y is a variable (e.g. the width of an ammonoid conch), x is the reference parameter (e.g. the diameter of an ammonoid conch), b is a convenient for the calculation (e.g. the ratio between conch width and conch di- ameter), and a measures the deviation from isometry. Values greater than 1 (positive allometry) or lower than 1 (negative allometry) document different ontogenetic trajectories of x and y, isometry occurs where a = 1. Klingenberg (1998) pointed out that there are two prin- cipal kinds of data for studies of ontogenetic allometry; (a) the often analyzed “cross-sectional data,” in which each individual specimen is measured at a single stage, and (b) the more rarely used “longitudinal data,” in which each in- dividual is measured multiple times during growth. Only the C  2012 Wiley Periodicals, Inc. 501