Fractal geometry of ammonoid sutures

Abstract

Interior chamber walls of ammonites range from smoothly undulating surfaces in some taxa to complex surfaces, corrugated on many scales, in others. The ammonite suture, which is the expression of the intersection of these walls on the exterior of the shell, has been used to assess anatomical complexity. We used the fractal dimension to measure sutural complexity and to investigate complexity over evolutionary time and showed that the range of variation in sutural complexity increased through time. In this paper we extend our analyses and consider two new parameters that measure the range of scales over which fractal geometry is a satisfactory metric of a suture. We use a principal components analysis of these parameters and the fractal dimension to establish a two-dimensional morphospace in which the shapes of sutures can be plotted and in which variations and evolution of suture morphology can be investigated. Our results show that morphospace coordinates of ammonitic sutures correspond to visually perceptible differences in suture shape. However, three main classes of sutures (goniatitic, ceratitic, and ammonitic) are not unambiguously discriminated in this morphospace. Interestingly, ammonitic sutures occupy a smaller morphospace than other suture types (roughly one-half of the morphospace of goniatitic and ceratitic sutures combined), and the space they occupied did not change dimensions from the Jurassic to the late Cretaceous.We also compare two methods commonly used to measure the fractal dimension of linear features: the Box method and the Richardson (or divider) method. Both methods yield comparable results for ammonitic sutures but the Richardson method yields more precise results for less complex sutures.

Shell features, main colonized environments, and fractal analysis of sutures in Late Jurassic ammonites

“…Several approaches to the quantification of sutural complexity in ammonites and its relationship with shell geometry and/or the occupation of morphospace have been published during the last few years (Boyajian & Lutz 1992;Dommergues et al 1996;Lutz & Boyajian 1995;01oriz & Palmqvist 1995;Saunders 1995;Saunders & Swan 1984;Saunders &Work 1996;Swan & Saunders 1987).…”

“…the characteristic shape of the curve appears over a wide range of sizes). Accordingly, the sutures can be regarded as fractal curves (quasi-fractals semu Seilacher & LaBarbera 1995) in a formal and even morphogenetic sense (Bayer 1985;Boyajian 1990, 199 1;Boyajian & Lutz 1992;Damiani 1984Damiani , 1986Damiani , 1990Checa & Garcia-Ruiz 1996;Garcia-Ruiz et al 1990;Garcia-Ruiz & Checa 1993;Guex 1981;Lutz & Boyajian 1995;Seilacher 1988Seilacher , 1991Slice 1993). Cranial sutures of mammalian skulls can also show high levels of complexity (although below those of ammonitic sutures), especially in stags and bovids whose skulls have to withstand stress (Long 1985;Gibert & Palmqvist 1995;Hartwig 1991).…”

“…1990;Lutz & Boyajian 1995;Oloriz & Palmqvist 1995;01oriz et al 1996). However, fractal analysis is weak in characterizing the differences between the three main suture types (i.e.…”

Morphostructural constraints and phylogenetic overprint on sutural frilling in Late Jurassic ammonites

“…1 shows the fractal dimension (D f ) values of ammonite sutures in those specimens inhabiting normal neritic-to-epeiric seas (n = 239) and epioceanic fringes (n = 41) in relation to whorl height (W h , log 10 -transformed to avoid departures from normality) corresponding to the sutures analysed (see details on the methodology used for estimating D f values in Lutz & Boyajian (1995), Oló riz & Palmqvist (1995 and Oló riz et al (1997Oló riz et al ( , 1999). …”

Wavelet analysis of ammonoid sutures

“…The debate regarding the "ammonoid suture problem" has been based on the following indices that represent the geometric properties of the suture line: Index of Suture Complexity (Westermann, 1971;Ward, 1980;Ballantine, 2007), Intersutural Span (Jacobs, 1990), Sutural Amplitude Index (Batt, 1991), Sutural Complexity Index (Saunders, 1995;Saunders andWork, 1996, 1997;Saunders et al, 1999) and fractal dimension (García-Ruiz et al, 1990;Boyajian and Lutz, 1992;Lutz and Boyajian, 1995;Olóriz and Palmqvist, 1995;Checa and García-Ruiz, 1996;Olóriz et al, 1997Olóriz et al, , 2002Pérez-Claros et al, 2002, among others. However, these indices are somewhat ambiguous because disparate suture forms are commonly represented by similar index values (Allen, 2006).…”