Chapter 17
Shell Morphology and Growth
In Chapter 15, detailed studies of the parametrically defined fractional spirals exposed their geometry and potential application. Spiral behavior is seen in many areas of natural science. In this chapter, we study the applicability of the fractional trigonometric spirals to the morphology and growth of shelled sea animals. The spiral shape is nearly ubiquitous in shelled sea animals. The beauty and large variety of these creatures have been the basis of numerous scientific and common interest studies. These include an encyclopedic book on shells by Dance [25], a natural history study by Vermeij [120], and an excellent X-ray study by Conklin [24], which nicely expose the inner structure of sea shells. Here, a detailed study of the shell of the Nautilus pompilius as well as a simple morphological study of six additional animal shells are presented.
17.1 Nautilus pompilius
This section studies the morphology and evolutionary growth of the Nautilus pompilius based on the fractional R1-trigonometry. This study, Lorenzo [86], was performed prior to the development of the fractional meta-trigonometry. It is likely that the fractional meta-trigonometric functions would show somewhat improved results over those based on the R1-trigonometry. Morphological models based on the fractional trigonometry are shown to be superior to those of the commonly assumed logarithmic spiral. The R1-trigonometric functions further infer fractional differential equations (FDEs) which, together with power law parametric functions, will be used to develop a fractional growth equation modeling evolution of the Nautilus from conception to maturity.
17.1.1 Introduction
One of the earliest studies concerning the mathematical morphology of the N. pompilius shell was the classic work of Thompson [118]. Use of the logarithmic or equiangular spiral to model this shell has been forwarded by several authors. From a scientific point of view, there is the book by Thompson [118] and the later effort by McMahon and Bonner [93]. In books to popularize mathematics and its application, there are contributions by Land [61] and Hargittai [43]. Because the Nautilus is considered endangered, there has been significant scientific interest in its growth rate. Intensive studies in this area include Landman et al. [62, 63], Cochran [23], and Westermann et al. [122]. Studies on the fractal nature of the Nautilus [21] have been done by Castrejón et al.
The objective here is to create a model for the Nautilus morphology based on the fractional trigonometry or more specifically on the R1-fractional trigonometry, for comparison to the earlier modeling based on the logarithmic spiral. Beyond the issues of morphology, the fractional trigonometric approach allows modeling of the growth rate of the animal and the creation of a fractional growth rate equation.
The shells of four N. pompilius were studied. Three shells were sectioned along the plane of symmetry for use in the morphology studies. A fourth tri-cut shell was used to model the locus of the siphuncles. Details on the photography and measurements used in this study may be found in Lorenzo [86]. The results of these measurements are contained in Table 17.1. The computational and fitting procedures are summarized in the following section on morphology. The following sections are adapted from Lorenzo [86] with permission of ASME.
Table 17.1 Shell dimensions
| Shell number | Lmax maximum dimension (diameter) mm |  max. radius mm |
Number of septa | Lsp spiral length mm | c recession mm/° |
| 1 | 170.5 | 108.0 | – | – | n.a. |
| 2 | 180.5 | 112.5 | 33 | 640.4 | 1 mm/8° |
| 3 | 174.0 | 111.0 | 35 | 618.0 | 1 mm/23° |
| 4 | 175.0 | 111.5 | 33 | 622.5 | 1.5 mm/25° |
Source: Lorenzo 2011b [86]. Reproduced with permission of ASME.
Note: c is the maximum tip recession from plane in mm/number of degrees over which it tapers to zero.
17.1.2 Nautilus Morphology
Of the four shells studied, shell 1 was used to study the siphuncle morphology. For shells 2–4, logarithmic and R1-fractional spirals were chosen to best fit the photographed shell spiral morphology. The fitting process was performed manually for all cases.
The R1-fractional spirals are computed based on the following forms:
where 
and b are scaling factors determined in the fitting process along with the parameters q, v, a, and 
. Note that 
is the formal parameter of the fractional spiral, and it is a spatial parameter rather than a temporal parameter.
Because the fractional trigonometric functions are defined by infinite series and are not periodic as are the circular functions, evaluation for large arguments requires high-precision computations. Thus, the unscaled functions are evaluated in the symbolic computer program Maple®. In the Maple® code, a minimum of 100 digits of precision was used to allow the functions to be determined for the number of rotations needed to define the shell morphologies. The output of the Maple® code is linked to code written in Matlab® where the results are scaled by k, and rotated to match the shell orientation in the photographs. The results are then plotted atop the shell image together with the logarithmic spiral that is computed in Matlab®.
The fitting process for both the R1-fractional spirals and the logarithmic spirals was done manually. The logarithmic or equiangular spirals are defined by the polar equation
The results presented are based on the equivalent rectangular parametric forms:
where 
is the angle of rotation. Thus, only two parameters are required to fit the logarithmic spirals, 
the logarithmic scaling constant, and 
the logarithmic spiral growth rate parameter.
Figures 17.1–17.3 present the results of the morphological fitting of shells 2–4, respectively. For each shell, four image panels are given: (a) with the logarithmic spiral, (b) with the R1-trigonometric spiral, (c) with both spirals on the same plot, and lastly, (d) various enlarged views of panel (c) to show important inner details. The numerations on the axes of these figures are in pixels and a slightly different (pixel/mm) calibration applies for each figure.
Figure 17.1 Shell number 2. Morphological models: (a) Logarithmic spiral, (b) R1-trigonometric spiral, (c) both spirals, (d) both spirals enlarged; see Table 17.2 for parameters. Axes are in pixels.
Source: Lorenzo 2011b [86]. Reproduced with permission of ASME. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
Figure 17.3 Shell number 4. Morphological models: (a) Logarithmic spiral, (b) R1-trigonometric spiral, (c) both spirals, (d) both spirals enlarged; see Table 17.2 for parameters. Axes are in pixels.
Source: Lorenzo 2011b [86]. Reproduced with permission of ASME. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
The calibration factors for shells 2–4 are 13.30, 13.20, and 13.16 pixels/mm, respectively.
The axes of the plots are nominally 
for the x-axis and 
for the y-axis. The negation is required because spiral rotation of each (half) shell is clockwise. The axis orientation of each shell is not known prior to the photography and fitting process. Thus, an angular offset is applied during the fitting process for the fractional spirals. These offsets rotate the plots in the clockwise direction. For shells 2–4, the angular offsets were found to be 85°, 84°, and 90°, respectively. Thus, the axes are reversed for shell 4 and nearly so for shells 2 and 3.
The morphological results for the three shells showed a remarkable similarity. Curiously, the geometrical centers of the shells as inferred by both the logarithmic and the fractional spirals were displaced (∼1–2 mm) from the physical center of the shell. In all cases, the geometric center was slightly below and to the left of the physical center (i.e., the void center). This can be seen in the enlargement panels of the figures. A common center was found to best fit both the logarithmic and fractional spirals for each shell.
The logarithmic spiral parameters for all three shells were found to be C2 = 75 (in pixels) and growth rate 
. The logarithmic spirals with these parameters fit the shell morphologies quite well after approximately a half revolution of the shell spiral (see enlargements, panel d, of Figures 17.1–17.3). With only two parameters, the fitting process for the logarithmic spirals was simple and direct. However, the fit of the logarithmic spiral to the Nautilus morphology became progressively worse approaching the origin from the one-half revolution point. And, of course, the infinite number of rotations around the origin from the half revolution point inward does not match at all.
The resulting R1-trigonometric spiral parameters required to fit the three shells are presented in Table 17.2. As mentioned earlier, the resulting parameters are remarkably similar. The following observations are made. As expected, the parameter q controls the growth rate of the R1-spirals and the maximum difference in q (growth rate) is only 0.0010 measured against an average q = 1.1220. Furthermore, the fitting process was very sensitive to the magnitude of q. The parameter v was found to be of relatively weak influence and could easily have been selected at v = −0.2 with an attendant adjustment of the scaling factor k. Similarly, because the spiral bar was so small, the quency-related parameter a could have ranged by over an order of magnitude with little apparent change in the form of the fractional spiral; again however, the scale factor k would require adjustment. Thus, the morphological fits are not unique. The parameter b is a scale factor on the logarithmic spiral spatial growth rate and also the increment between the plotted points. Finally, the value of 
(the spatial parametric variable of the fractional spiral) increases from zero at the origin of the plots to a maximum at the tip. The last datum point did not necessarily coincide with the maximum radius point of the shell.
Table 17.2 Parameters for morphological fit using R1-trigonometric fractional spirals
| Shell no. | Figure no. | b | q | v | a | k pix mm | Offset (°) | tm at max. radius |
| 2 | 17.1 | 200,000 | 1.1215 | −0.3 |  |
0.180 0.01353 |
85° | 94 |
| 3 | 17.2 | 200,000 | 1.1220 | −0.3 |  |
0.167 0.01265 |
84° | 93 |
| 4 | 17.3 | 200,000 | 1.1225 | −0.3 |  |
0.169 0.01284 |
90° | 91 |
Source: Lorenzo 2011b [86]. Reproduced with permission of ASME.
Figure 17.2 Shell number 3. Morphological models: (a) Logarithmic spiral, (b) R1-trigonometric spiral, (c) both spirals, (d) both spirals enlarged; see Table 17.2 for parameters. Axes are in pixels.
Source: Lorenzo 2011b [86]. Reproduced with permission of ASME. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
Fractional Differential Equations
The morphological data presented in Table 17.2 may be used together with the Laplace transforms of Table 6.1 to determine the FDEs representing the Nautilus fractional spiral morphology. It may be readily shown that
and
Thus, the simultaneous FDEs representing the morphology (in pixels) are
and
where 
is the unit impulse function. The scaling and functional form of 
must be determined from additional data (see Section 17.1.5). Note that the order of the FDE is twice the trigonometric order. The solutions to equations (17.4) and (17.5) are
The R1-fractional spiral fits were viewed to be excellent over nearly the entire range of rotation, from the origin to the point of maximum radius. Of course, Nautilus, similar to all natural creatures, have differing degrees of perfection of form, and minor deviations are found. It is observed that the logarithmic spiral and the R1-fractional spiral become (approximately) asymptotic to each other after about one-half to one revolution of the fractional spiral for all cases studied. It can be seen that near the shell origin, the logarithmic spiral continues to rotate and does not match the morphology of the Nautilus and has an infinite error relative to the actual shell morphology! The maximum deviation of the fractional spiral from the shell spiral occurred over the range from the origin to the next plotted point; see Figures 17.1d and 17.3d. The logarithmic spiral, of course, spirals forever around the mathematical origin with ever decreasing radius. Clearly, the logarithmic spiral cannot represent the Nautilus over this important range and thus cannot be used as a basis for further detailed scientific study of its evolution. Notice that the two FDEs are independent of each other except for the common synchronizing impulse that starts the dynamic process. The addition of cross-coupling terms to equations (17.4) and (17.5) may allow improved modeling near the origin but is beyond the scope of this study.
It is of interest to take a detailed view of the R1-fractional spirals used to fit the shells. The data shown in Figures 17.4 and 17.5 apply to shell 4, which is taken as typical. It is in the range 
, where the important deviation from the logarithmic spiral occurs. This deviation from exponential behavior is also seen in the plot of radius versus 
. Figure 17.4 shows the x and y components of the fractional spiral shown in Figure 17.3b. In the figure, the components are scaled to match those of Figure 17.3. The angular rotation of the spiral required to match the image is accounted for. It is also noted that the y component is proportional to 
and the x component is proportional to 
. Figure 17.5 shows the spiral radius (in pixels) for the logarithmic and fractional spirals versus spiral angle theta (in radians). In Figure 17.5, exponential behavior of both the logarithmic and the R1-spirals for theta greater than approximately 2.5 radians is seen. It is in the range 
, where the important deviation of the logarithmic spiral from the actual morphology occurs. This deviation from exponential behavior is also seen in the plot of radius versus 
. Behavior in this range is important to possible understanding of the conceptional period of shell growth.
Figure 17.4 Shell 4, x, y, and radius components of R1-trigonometric spiral shown in Figure 17.3b versus fractional spiral rotation 
in radians.
Source: Lorenzo 2011b [86]. Reproduced with permission of ASME. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
Figure 17.5 Expanded view spiral radii (in pixels) for logarithmic and fractional spirals versus spiral angle theta (in radians).
Source: Lorenzo 2011b [86]. Reproduced with permission of ASME. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
17.1.3 Spiral Length
The length of an arc expressed in parametric form is given by (equation (15.9))
The relationships 
and 
from equations (6.36) and (6.40) are applied to an R1-spiral in the form of equation (17.1) to provide the length from origin to tip as
This equation was numerically integrated for each of the shells, using the various parameters as provided in Table 17.2, to give the computed lengths for shells 2–4, respectively, as 658.1, 612.3, and 596.0 mm with a maximum error of 4.3% of the measured results indicated in Table 17.1. The singularity of the 
was avoided by starting the integration at 
, the integration was stopped at the datum point corresponding to the largest radius; see, for example, Figure 17.4.
17.1.4 Morphology of the Siphuncle Spiral
The morphology associated with the locus of siphuncles of shell 1 is considered in this section. To expose most of the siphuncles, a shell sectioned with two lateral cuts created a 14 mm thick slice that contained the center portion of the shell used for this study. Figure 17.6a shows an image of the shell slice without modeling data. The calibration factor for this image is 13.03 pixels/mm. Table 17.3 contains the parametric constants used for the R1-trigonometric spiral seen in Figure 17.6b–d. The enlargement shown in panel c indicates a high-quality match between the model and the locus. The dark triangles shown in the further enlargement (panel d) extrapolate the model inward presumably modeling the inner siphuncles that are hidden by the outer shell of the animal in this cut.
Figure 17.6 Shell number 1. Siphuncle morphological models: (a) Image without model, (b) R1-trigonometric spiral of observed locus, (c) panel (b) enlarged, (d) spirals enlarged further with model of hidden locus indicated. See text and Table 17.3 for parameters.
Source: Lorenzo 2011b [86]. Reproduced with permission of ASME. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
Table 17.3 Parameters for morphological fit of shell 1 siphuncle locus using the R1-trigonometric fractional spirals
| Shell no. | Figure no. | b | q | v | a | k pix mm | Offset (°) | tm at max. radius |
| 1 | 17.6 | 200,000 | 1.13575 | −0.3 |  |
0.12 0.00921 |
−120° | 59 |
Source: Lorenzo 2011b [86]. Reproduced with permission of ASME.
Notice that the siphuncle locus has a more rapid divergence than seen for the main spirals of shells 2–4. This is validated by the fact that the order parameter q for the siphuncle is q = 1.1375 as opposed to 
for shells 2–4.
17.1.5 Fractional Growth Rate
There has been considerable interest in the growth rate of the Nautilus. Landman et al. [63] have used radiometric methods to determine the apertural growth rates in
immature N. pompilius of 0.19–0.31 and 0.12 mm/day in submature animals. Cochran and Landman [23] report apertural growth rates of 0.07–0.14 mm/day for Nautilus belauensis for growth between the last two septa for the mature animal and a minimum of 10 years to reach maturity. Westermann et al. [122], using X-ray techniques to study the shell growth of the N. pompilius, indicate apertural growth of 0.11–0.18 mm/day for juvenile and early adolescents and estimates 2673 days (7–8 years) to reach maturity. Detailed analysis of these results and more are beyond the scope of this work.
The motivation of these studies has been to develop an understanding that will contribute to the survival of this last existing cephalopod. This study attempts to use the results of these investigators to forward a possible growth model for the Nautilus. As a crude guideline for the analysis to follow, we arbitrarily accept growth rates of 0.20–0.40 mm/day for immature Nautilus at age of about 2–3 years and 0.10–0.14 mm/day for mature Nautilus at maturity of 8–10 years. In this section, we consider the issue of growth of the Nautilus as inferred by fractional spiral morphology.
The parametric fractional trigonometric functions that morphologically match the Nautilus spirals contain the formal spatial parameter 
. It is clear that 
can be virtually any continuous function of actual time 
and still match the morphology. The problem is to determine 
to fit the constraints.
The fractional morphology of the Nautilus is expressed as the pair of scaled simultaneous equations (equation (17.1). This description differs from that of the logarithmic spiral in that its basis derives from the fundamental FDE (Hartley and Lorenzo [45]). Might the dynamic differential equations associated with the fractional spirals in fact define the evolution of the Nautilus shell and by implication the animal growth?
Now, equation (17.9)
provides an expression for the scaled length of an R1-fractional spiral in pixels. We now let 
then 
and 
. Then,
The growth rate of the spiral is the derivative of this expression, namely
To determine 
to satisfy the observed growth rate constraints of the Nautilus, several linear, exponential, and logarithmic forms were considered for 
. A power law relation for 
was found to reasonably satisfy the constraints, namely
and
Then,
Scaling this result to millimeters, and selecting 
and 
, the growth rate versus 
is presented in Figure 17.7. The final point on the graph corresponds to t = 3650 days or approximately 10 years. At that point, the growth rate is 0.13 mm/day. The growth rate when t = 300 days is approximately 0.2 mm/day. It is noted that t = 0 relates to the time of conception of the Nautilus and the times estimated here relate to that starting point. Thus, the beginning of life is seen to be, both literally and mathematically, a singular event. The power law form appears to be reasonable model for the Nautilus evolution.
Figure 17.7 Shell number 4, Growth rate in mm/day versus t – time (days), (
).
Source: Lorenzo 2011b [86]. Reproduced with permission of ASME.
Clearly, the mapping 
from equation (17.13), together with morphological equations (17.4) and (17.5), then represent a complete inferred dynamic description of the evolution of the Nautilus.
Of importance beyond this particular application is the fact that the aforementioned process provides a possible methodology for converting fractional trigonometric morphological information into a dynamic model using auxiliary data.
17.1.6 Nautilus Study Summary
The morphology and growth rate of the N. pompilius has been studied based on the R1-fractional trigonometry. Very good fits of the Nautilus spiral were shown for the three shells studied. The fits did not appear to be unique in that various ranges of some of the parameters would allow equivalent quality fits of the Nautilus spiral. Beyond roughly one-half revolution of the shell's spiral, the logarithmic and fractional spiral fits were both very good. However, the morphology fits of the shells using the fractional trigonometric spirals were superior to those of the logarithmic spiral in that they matched the much more closely near the origin and allowed the modeling of growth rate.
The growth rate associated with the evolution of the Nautilus was also studied. It was found that a power law relationship between the morphological parameter 
and actual time t provided a reasonable approximation to observed growth rates. An important feature of the growth rate model is that it provides growth estimates from conception to maturity.
A fractional growth rate equation (equation (17.12) based on the R1-fractional trigonometry was determined. A further generalization of the fractional growth equation (17.15) is easily obtained by replacing the R1-trigonometric functions by the more general the fractional meta-trigonometric of Chapter 9. Because of the importance of growth rate of agricultural animals and because of the occurrence of spiral (and helical) morphology in other animals, plants, and microbiological life forms, the breadth of application of such an equation may be very wide.
This study is considered preliminary in that superior fits of the morphology may be possible with the more general fractional meta-trigonometry of Chapter 9. Also, the inclusion of coupling terms between the x(t) and y(t) terms may further improve modeling near the origin. In addition, more detailed analysis of the 
relationship with observed growth may yield superior growth rate relationships.
17.2 Shell 5
- Super Family: Strombacea
- Family: Strombidae
- Genus: Strombus alatus
- Common name: Florida fighting conch
In this and the following sections, the morphology of a variety of shells is studied. The identifications of the shells from this point forward have been made on the image based approach of Dance [25], in which only the characteristics of the shell are considered, ignoring the soft anatomy of the animal (see also Harasewych [41]). They are fitted with parametrically defined fractional meta-trigonometric spirals defined by
The Laplace transform for 
are given in equations (9.66) and (9.67), respectively. From these transforms, a pair of spatial simultaneous FDEs may be inferred
and
where 
and 
. Note that the right sides of equations (17.17) and (17.18) may be significantly simplified by using selected Laplace transforms from Section 12.4.
Rather than repeat these FDEs for these shells, we simply list the defining parameters for each case which in turn define the constants in equations (17.17) and (17.18). The reader should also be aware that the spiral plots are x and y biased and rotated as required to align with the images.
Shell 5 is identified as a Florida fighting conch based on its knobbed spire and coloration; however, the thin lip does not match that species which has a much thicker lip (see Figure 17.8f and g). The shell has been cut to expose an inner spiral; this cut, perpendicular to the axis of coiling, is shown in Figure 17.8a. The prominent spire viewed from the side is also seen here. The spire as viewed from above is shown in Figure 17.8b and c. In panel c the projected spiral, in pixels, is fitted with the parameters
Figure 17.8 Shell 5. (a) Normal view of cut shell, (b) top view of spire, (c) spire with spiral fit, (d) enlarged view of spire, (e) enlarged view with spiral fit, (f) view of body interior, (g) body interior with spiral fit. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
The calibration factor for this image is 391.0 pixels/cm. Note in panel c, because of the many rotations of the physical spiral, the fractional spiral (center) is extended by a logarithmic spiral after approximately three rotations (see numerical issues 14.3). This is more clearly seen in the magnified views of panels (d) and (e) where the dark gray data points are the logarithmic extension. A few overlapping points are shown. Further note that at the origin (panel d), it appears that two spirals start the growth process and then merge into a single spiral form. The logarithmic spiral is described by
with 
and appropriate bias and rotation.
Of greater interest for this shell is the body whorl (volume below the spire). The cut through this volume exposes the spiral shown in Figure 17.8f and g. This spiral is well fit with the parameters
The calibration factor for this image is 480.4 pixels/cm. Observe that there is an inward closure of the shell away from the fractional spiral near the end of growth. The reason for this is unexplained. However, this behavior is also seen on the remaining shells. As mentioned earlier, the cuts through the body whorl were perpendicular to the axis of coiling, slightly different results would likely be obtained if the cut were made parallel to the inclination of the final rotation of the spire.
Finally, it is interesting to note that a spiral with approximately 10 rotations exists for the spire spiral and one with approximately 
rotations fills nearly the same space in the body whorl leading to the conclusion that spiral growth is fundamental to these creatures.
17.3 Shell 6
- Family: Conus
- Genus: tinianus or vexillum
This shell (Figure 17.9) is characterized by its shallow spire and rounded shoulder. The coloration seen in this specimen is different from that seen in reference books (Dance pp. 214–215). The spiral of the spire is indistinctly defined and, therefore, not modeled. However, it appears to have approximately six or more rotations.
Figure 17.9 Shell 6. (a) Cut shell side view, (b) shell top view, (c) view showing interior fractional spiral fit, (d) magnified view. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
The cut through the body whorl exposes the spiral shown in Figure 17.9c and d. This spiral is well fit with the parameters
The calibration factor for this image is 480.4 pixels/cm. Observe that there is an inward closure of the shell away from the fractional spiral near the end of growth.
17.4 Shell 7
- Super Family: Cypraeacea
- Family: Cypraea Tigris
- Genus: Cypraea linnaeus
- Common name: Tiger Cowry
This is the first of two different-sized cowries to be studied, Figure 17.10. Viewed from the outside, the shape of these shells does not suggest a spiral structure. Figure 17.10a shows the shell of the larger cowry with a cut perpendicular to its length. The cut through the body exposes the spiral shown in Figure 17.10c and d. This spiral is well fit with the parameters
Figure 17.10 Shell 7. (a) External view of cut, (b) cut section of shell, (c) fractional spiral fit, (d) magnified view of spiral fit. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
The calibration factor for this image is 300 pixels/cm. Observe that there is an inward closure of the shell away from the fractional spiral near the end of growth.
Figure 17.10c shows the fractional spiral fit to the shell cross section, and Figure 17.10b provides a reference without the analytical result. Figure 17.10d gives a magnified view of the fit near the origin.
17.5 Shell 8
- Super Family: Strombacea
- Family: Strombidae
- Genus: Tibia insulaechorab
- Common Name: Arabian Tibia
The lip of this shell (Figure 17.11a) is normally thicker and has several blunt projections, Dance [25] pp. 76–77. However, the shell has been damaged and shows possible attack wounds. This shell has been fit with two fractional spirals: the inner spiral beginning at the origin and an outer spiral starting a common point (black circle), which ends the inner spiral and becomes the origin of the outer spiral. Panels (b) and (d) show the internal spiral structure with the common point shown on the magnified view. Panels (d) and (e) show the inner and outer spirals, respectively.
Figure 17.11 Shell 8. (a) External view of cut, (b) magnified view near origin, (c) view of cut section, (d) inner spiral fit, (e) outer spiral fit. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
The outer spiral is well fit with the parameters
The inner spiral is well fit with the parameters
The calibration factor for this image is 374 pixels/cm.
17.6 Shell 9
- Super Family: Cypraeacea
- Family: Cypraea Tigris
- Genus: Cypraea linnaeus
- Common name: Tiger Cowry
This is the second and smaller of the cowries studied. Figure 17.12a shows the shell of the smaller cowry with a cut perpendicular to its length. The cut through the body exposes the spiral shown in Figure 17.12b and c. This spiral is well fit with the parameters
Figure 17.12 Shell 9. (a) External view of cut, (b) fractional spiral fit, (c) magnified view. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
The calibration factor for this image is 368 pixels/cm. Observe that, again, there is an inward closure of the shell away from the fractional spiral near the end of growth. It is important to note that in spite of the size change of approximately 0.5×; the parameters are nearly identical with those of Shell 7. Figure 17.12b and c shows the fractional spiral fit and a magnified view. Evidence of some chipping, from the cut, is seen in the magnified view.
17.7 Shell 10
[
- Super Family: Cardiacea
- Family: Cardiidae
- Genus: Trachycardium magnum or Acrosterigma burchardi
It is of some interest to know if the shells of bivalves, which differ greatly from the shells previously studied, showed any evidence of fractional geometry. Figure 17.13a–c shows the top view of the uncut and cut shell and side view of cut shell. The side view (Figure 17.13d) and the magnified view (Figure 17.13e) show an excellent fit of a fractional meta-trigonometric spiral defined by the parameter set
Figure 17.13 Shell 10. (a) Top view, (b) location of cut, (c) side view of cut, (d) fractional spiral fit. (e) magnified view. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
The calibration factor for this image is 347.6 pixels/cm.
The important difference between this shell and the predecessors is the relatively large value of the fractional trigonometric order parameter q, which defines the spiral divergence rate. Clearly, fractional trigonometrically defined geometry also applies to the shell of this creature.
17.8 Ammonite
An ammonite is the fossil shell of a sea creature (cephalopod) likely dating back to the Mesozoic age. The ammonite studied here (Figure 17.14) was found in Madagascar and was estimated to be about 100,000 years old. Unfortunately, there was some grinding of the outer perimeter, possibly 1–2 mm in places, presumably to remove sharp edges of the fossil. Figure 17.14b and d shows a very good fit using the fractional meta-trigonometric functions
Figure 17.14 Ammonite fossil. (a) External view of fossil, (b) fractional spiral fit, (c) magnified view, (d) magnified view of fit. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.
The calibration factor for these images is 125.0 pixels/cm, and the spiral has been rotated 32° to align with the image. This ammonite fossil appears to be quite similar to the Nautilus, and it is interesting to note that the ammonite has 75 clearly defined septa, while the N. pompilius studied in Section 17.1 had only 33–35. However, the growth rate parameters were quite similar, with for the Nautilus and for the ammonite. The fractional meta-trigonometric functions used here allowed a very good fit over much of the spiral.
17.9 Discussion
With the need to fit six parameters for each fractional spiral fit, the fidelity of the manually fit fractional spirals is far from optimal. Several hours were dedicated to each fitting. It is likely that greatly improved fidelity may be achieved with either more dedicated time or application of some automated optimization process.
It seems clear that the geometric form of these creatures (shells) is well defined by the fractional trigonometric spirals. As mentioned previously, the cuts made on the shell were approximately perpendicular to the axis of coiling. It would be of significant interest to study the spirals that would occur if the cuts were made through the body whorl but parallel to the last spire whorl.
Surprisingly, even the shell of the Cowry is internally of spiral form, while it does not appear to be so when viewed externally. An important common feature of all, but shell 10, is that the spiral growth rate as indicated by the value of q, is found to be limited to the narrow range of 1.089 ≤ q ≤ 1.185. Does this indicate some optimum in form for strength or efficient use of space? Or, does it possibly indicate the limits of nourishment available to support the growth rate?
Finally and quite importantly, for the Nautilus, we have seen that when the temporal growth can be related to the spiral geometry, dynamic fractional growth equations may be inferred.