A Size-Independent Revision of the Fractal Step Method for Ammonite Sutures

: The novel coronavirus has presented specimen-access challenges to geoscientific researchers, including paleobiologists interested in fossil ammonoids. Ammonoid sutures are geometric patterns formed by the intersection of the septa and the shell wall, and have long been a diagnostic tool for ammonite researchers for such applications as species identification, taxonomic relationships, ontogenetic change, functional and evolutionary morphology, and other aspects of ammonoid paleobiology. Without access to specimens in museum and institutional collections, researchers must rely on previously published illustrations and photographs of ammonoid sutures. However, many of these illustrations were published decades ago without an index of scale. Suture tracings lacking a scale bar are not usable by researchers interested in applying the quantitativeness of fractal geometry to the interpretation of septal complexity. Additonally, distortion of the marginal elements of suture geometry occurs due to shell curvature near the venter and umbilicus. The revised approach described here eliminates the problem of missing scale information in fractal analysis of ammonite sutures, making use of just the lateral lobe and adjacent saddle (lateral lobe-saddle pairs). Our revised method ’s non-requirement of a full hemisuture also facilitates comparisons among sutures within an ontogenetic sequence, or sutures from multiple ammonite taxa.


Introduction
Without access to collections due to widespread and prolonged shutdown of institutions, reliance on ammonoid suture tracings which were published prior to the onset of the COVID-19 pandemic has become one of the primary ways in which those studying suture complexity can continue their work. Suture complexity is an amalgamation of the degree of infolding of the suture, and the geometric arrangement of the folds. [1] Suture geometry, a component of suture complexity, has potentially significant but incompletely-understood implications for ammonoid paleobiology, including; habitat preferences, particularly with respect to water depth; phylogeny, and ontogeny.
Historically, attempts to quantify ammonoid suture complexity have often involved viewing suture patterns as self-similar entities approximating fractals [2,3,4,5,6,7,8,9,10,11,12]. Fractals are mathematical constructs in which an infinite number of subdivisions can occur on an object, such that each subdivision is a perfect identical miniature

1) The Richardson
Step Method.

The Richardson
Step Method described by [10] requires set step lengths to be chosen at a number of millimeters, such as 10mm. It is important to consider that longer step lengths, such as 25mm, provide only general information, and shorter step lengths, such as 5mm, provide greater accuracy ( Figure 2). Conversely, as step length is increased, accuracy and resolution are lost. In ontogenetic studies, shorter step length provides the highest resolution possible, which is necessary due to the limited range of fractal values.  Figure 31E].
One must choose a beginning and end point on the ammonite suture, usually a hemisuture, which is bounded by suture's midpoint, the external lobe, E, and one of its end points, one of the first umbilical lobes, U1 [21]. Step Method as described by [12] using fractions of the hemisuture length as step lengths. Green is 1/5, magenta is 1/10, and blue is 1/15. Eogaudryceras numidum suture redrawn by KM from [20, Figure 31E].
Beginning at the center of the External Lobe, one pivots their ruler until 10mm on the ruler intersected the suture line. This is the first step. The second step begins at the end point of the first step, and this will continue until U1 is reached. The number of whole steps on the suture are counted. Partial steps are not included.
The number of steps and the number of millimeters in the step length used are plugged into Equation 1:
The fractal value for each suture which is measured can then be recorded. In digitized specimens where scale is unavailable, the ontogeny can be represented by the whorl number.
2) The Revised (Covid) Step In the Revised Step Method, step length is equal to a given fraction of the length between the base of the lateral lobe (L) and its adjacent saddle on the umbilical-facing side. In Figure 4, the magenta line is the full length of the lateral lobe-saddle pair, and the green line shows a step length of 1/10 the length of the lateral lobe-saddle pair. One-tenth hemisuture lengths were used by [12].
. The number of additional steps measured in the revised method versus the Richardson method varies across genera (see Table 1). Because the number of steps in the revised method for most lytoceratids is almost always exactly twice The general geometry of sutures, more than the number of subdivisions, appears to be the driving force behind the variation in conversion values across taxa seen in Table 1. As a result, in some unresolved groups, such as primitive lytoceratids Holcolytoceras and Psiloceras [20,26], Conversion Values may be indicators of taxonomy.
The Revised fractal equation then becomes: , C(f) describes converted fractal value.
If the number of divisions of the measured section is always assumed to be ten, the revised method eliminates the division step, and unifies the measurement of all ammonite sutures relative to the ammonite to which they belong. For comparing ontogenies of two or more ammonites, it should be clearly stated which ontogenetic stage contains the first suture in the measured sequence. This can be done by denoting the whorl of the first included suture in data where shell diameter is unavailable.   Table 1, using the program PAST [23]. The blue lines are 95% confidence. Suture geometry has generally always been considered a primary diagnostic tool when assigning taxonomy to a specimen [24]. The mean fractal value of all ammonite sutures is accepted to approximate 1.37 [12,25]. This paper heavily relies on suture tracings published by other authors, and actively sought, in vain, suture ontogenies that had previously been quantified fractally (Figure 7) [25,26]. However, it is impossible to make further comparisons from the 1990 publication, as the step length is not specified. Furthermore, the approach taken by Garcia-Ruiz et al. [12,25] did not account for the ontogenetic stage in which the sutures were measured, and the authors noted that the method did not quantify the rate of change between sutures in the ontogenetic sequence [12].Our revised method facilitates the ability to graph entire ontogenies, such as those rarely published before [26] without access to the full undistorted hemisuture. It also enables a higher resolution of fractal measurement than the Richardson method. In the Richardson Step Method, different or unknown step lengths can be used, so measurements of even the same suture may vary if different step lengths were deployed.

Preprints
Comparisons between fractal dimensions of previously published taxa are not reusable unless the step length is given in the publication, which it generally is not, and is the same as the step length used for other ammonites being compared. Comparing ammonites based on their own respective ontogenies rather than standard units of measurement that are arbitrary to them circumvents this issue.