meneghiniana, Arachnoidiscus ornatus, and Aulacodiscus oregonus than higher symmetric taxa with the exception of Arachnoidiscus ehrenbergii (2) (Figure 2.23). Rescaled random instability was more evident in less symmetric taxa Actinoptychus splendens, Cyclotella meneghiniana, and Aulacodiscus oregonus as well, in contrast to higher symmetric taxa, with the exception of the Arachnoidiscus ehrenbergii forming valve (Figure 2.24). Overall, less symmetry is associated with higher chaotic and random instability (Figures 2.23 and 2.24).
Figure 2.18 Asterolampra marylandica (SanDBay02_03dx1100) as a sequence of 24 cumulative concentric rings simulating valve formation from annulus to valve margin.
2.4 Discussion
From average symmetry per taxon for all valves, triangular and non-regular structured surface circular taxa exhibited less symmetry than highly regularly structured circular and four-edged or greater polygonal taxa (Figure 2.12). By inspection, the result is evident, and the method of using entropy to determine symmetry states has explicit utility. Additionally, although we measured rotational symmetry, other symmetries are present as well (Figure 2.3). Separate tests for each type of symmetry could be made, and quantification of each type of symmetry would need to be instituted to determine the contribution of a particular symmetry throughout centric diatom development.
Figure 2.19 Plots of symmetry vs. 24-valve formation simulation steps from annulus to valve margin for eight centric diatom taxa. Order from least to highest symmetry curve per taxon is read left to right in legend. Symmetry increases exponentially from the annulus to the completed valve.
Figure 2.20 Valve formation steps for eight taxa as a probability density function of entropy values resulting in a gamma distribution.
Figure 2.21 Valve formation steps for eight taxa as a cumulative density function of entropy values.
Figure 2.22 Valve formation steps for eight taxa as a skew-right normal prior probability density function of entropy values.
Table 2.2 Valve formation simulation for eight centric diatom taxa and characterization of instability and stability from Lyapunov exponents. Positive sums indicate chaotic instability, negative sums indicate stability, and Lyapunov exponents from Kolmogorov-Sinai (KS) entropy indicate random instability.
| Taxon* | File name | Chaotic Instability: Sum of + Lyapunov exponents | Stability: Sum of - Lyapunov exponents | Random Instability: Lyapunov exponent from K-S entropy |
| Actinoptychus senarius | motewx1500_2 | 5.0775 | -429.424 | 4399.39 |
| Actinoptychus splendens | MisBy5_1hx2000 | 4.987 | -388.348 | 3079.95 |
| Arachnoidiscus ehrenbergii (forming valve) | halfmax500 | 4.96527 | -390.747 | 3186.1 |
| Arachnoidiscus ehrenbergii (1) | provbay5_12ix400 | 5.36916 | -379.328 | 2359.42 |
| Arachnoidiscus ehrenbergii (2) | ProvBay5_12lx450 | 5.04099 | -398.893 | 4656.9 |
| Arachnoidiscus ornatus | PoiDumox700 | 5.18264 | -454.589 | 2669.33 |
| Asterolampra marylandica | SanDBay02_03dx1100 | 5.15695 | -383.425 | 3354.2 |
| Aulacodiscus oregonus | PoiDumjx600www | 5.1158 | -424.086 | 4033.01 |
| Coscinodiscus sp. | MisBayhx500 | 5.00421 | -465.124 | 3277.96 |
| Cyclotella meneghiniana + | SClemtbx1800 | 5.09775 | -422.176 | 2448.34 |
* All taxa are external valves unless otherwise noted.
+ Symmetry value is the average of all rotations test results.
The results for external valves was somewhat different from those for all valves combined. Average symmetry per taxon revealed that most of the triangular taxa or those with three-part valve features were the least symmetric in contrast to the most symmetric taxa, including species from Aulacodiscus, taxa with four-part valve features or shape, Glyphodiscus stellatus, Eupodiscus radiatus, Amphitetras antediluviana, and Triceratium pentacrinus fo. quadrata). Actinoptychus senarius, Spatangidium arachne, Asterolampra marylandica, Arachnoidiscus ehrenbergii, and Arachnoidiscus ornatus were notable exceptions to the four-part rule in the highly symmetric group (Figure 2.13). As expected, species of Asteromphalus were nearer in symmetry to Asterolampra grevillei and less symmetric than Asterolampra marylandica.
