symmetry analysis. Values ranged from 0.0000946 to 0.0184 measurement error with 0.0172 to 1.994% bias. The percentages fall within the range of values that do not affect the measurement of true variation [2.103]. For symmetry analysis, the entropy values measured are not biased by measurement error.
An image of Biddulphia sp. was used to illustrate measurement of dihedral symmetry. Entropy and symmetry values were 1.94041 and 6.96162 with a measurement error of 0.0000614 and a 0.00663% bias. Biddulphia sp. dihedral symmetry measurement was accomplished using rotation and flipping the masked image about the x and y axes in the x-y plane.
Valve formation was simulated in a simple model by sequentially adding concentric rings of valve structure from an annulus to the valve margin to cumulatively create a valve face surface. Although diatom valve formation is much more complicated than this, the simulation provided a glimpse into an intuitive way to view growth over time. Ten circular-shaped centric diatom images were chosen to illustrate the method for Actinoptychus senarius, Actinoptychus splendens, Arachnoidiscus ehrenbergii, Arachnoidiscus ornatus, Asterolampra marylandica, and Aulacodiscus oregonus. Twenty-four concentric rings were arbitrarily selected on each image. Each ring was of an arbitrary width and added cumulatively, starting with an annulus at the center and ending with a ring at the valve margin (Figure 2.18).
Figure 2.8 Typical entropy vs. symmetry plot. The example is image ProvBay5_12lx450, Arachnoidiscus ehrenbergii. The first datum is the original masked tilt-corrected image.
Figure 2.9 Number of rotations vs. tilt-corrected final entropies for Cyclotella meneghiniana external valves using 3 to 64 rotations.
Figure 2.10 Histogram and overlain probability density function for all tilt-corrected final entropies resulting in a Gaussian distribution.
Figure 2.11 Histogram and overlain cumulative density function for all tilt-corrected final entropies.
Figure 2.12 For all images, average symmetry per taxon ranging from Triceratium crenulatum with the lowest symmetry to Actinoptychus senarius with the highest symmetry.
Valve formation simulation for the eight taxa was plotted against symmetry, and the resultant curves were exponential depicting the change in symmetry from annulus to valve margin (Figure 2.19). The plot includes a second Arachnoidiscus ehrenbergii external valve and a forming valve of this taxon. For valve formation simulation, the distribution of average entropy values was depicted in a histogram and overlain PDF (Figure 2.20) as a gamma distribution and is a maximum entropy distribution. A histogram and overlain CDF of average entropy values were also constructed (Figure 2.21). Standardizing the PDF produced a skewed-right normal prior probability distribution (Figure 2.22), establishing the relation between available information and its probability of occurrence. The prior distribution is also a maximum entropy distribution as a minimization of prior information given via the PDF.
Figure 2.13 Average taxon symmetry for all external centric diatom valves ranging from Triceratium crenulatum with the lowest symmetry to Glyphodiscus stellatus with the highest symmetry.
Figure 2.14 Vertical valve formation comparison: average external and forming valve symmetry for ten centric diatoms.
Figure 2.15 Symmetry values for Asteromphalus, Arachnoidiscus, and Aulacodiscus external valves.
2.3.2 Valve Formation—Stability and Instability Analyses
Stability analysis was performed using symmetries calculated from the valve formation simulation of cumulative concentric rings from annulus to valve margin for eight taxa. Lyapunov exponents were calculated to determine the behavior of symmetry changes. Initial conditions for valve formation are vValve formation (0) = vannulus. Each valve formation simulation produced a series of Lyapunov exponents comprising a Lyapunov spectrum [2.166]. The largest real Lyapunov exponent was positive and characterized the overriding behavior of symmetry changes during valve formation. For all taxa, sum of the positive Lyapunov exponents signified chaotic asymmetric behavior characterizing instability (Table 2.2). The remainder of the real Lyapunov exponents in descending order were very small (large negative values), summed, and signified stability (Table 2.2). For Lyapunov exponents calculated from KS entropy, some random instability was present during valve simulation as well (Table 2.2).
Figure 2.16 Cyclotella meneghiniana symmetry for normal and abnormal valves.
Figure 2.17 Cyclotella meneghiniana symmetry values for normal forming and external vegetative and initial valves.
The change in symmetry was compared with the change in chaotic instability for taxa used in the valve formation simulation. Symmetry increased monotonically in the order of least symmetric to most symmetric: Actinoptychus splendens, Cyclotella meneghiniana, Arachnoidiscus ornatus, Aulacodiscus oregonus, Coscinodiscus sp., Arachnoidiscus ehrenbergii (1), Arachnoidiscus ehrenbergii forming valve, Asterolampra marylandica, Arachnoidiscus ehrenbergii (2), and Actinoptychus senarius (Table 2.2). Rescaling symmetry and chaotic instability values, a comparative bar graph showed that chaotic instability was more evident in less symmetric taxa Actinoptychus splendens, Cyclotella
