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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. Although, fewer taxa were analyzed with respect to forming valves, a similar result emerged when compared to external valves. Four-part valve or greater featured or shaped taxa were the most symmetric (Figure 2.14). Taxa with higher forming than external valve symmetry were Triceratium favus and Triceratium pentacrinus fo. quadrata (Figure 2.14). Different species of Asteromphalus forming valves were found to be similar in symmetry to those of Asterolampra marylandica, contrary to the situation with external valves.

Figure 2.23 Valve formation simulation comparison of symmetry and chaotic instability. See Table 2.2 for numerical values.

Figure 2.24 Valve formation simulation comparison of symmetry and random instability. See Table 2.2 for numerical values.

A comparison of symmetric and asymmetric external valve surfaces resulted in higher symmetry for Arachnoidiscus and Aulacodiscus over Asteromphalus (Figure 2.15). Steeper slopes for the symmetric genera of 1.276 and 2.2925 also indicate more symmetry change among species than Asteromphalus species with a slope of 1.1878. Generally, rotationally asymmetric diatom surfaces exhibit less symmetry than morphologically similar taxa such as Asterolampra marylandica (Figure 2.13). However, asymmetric surfaced Spatangidium arachne has higher symmetry than Asteromphalus species (Figure 2.13). The five rays of Spatangidium arachne are thin, straight and regularly spaced and sized, which may be recovered as more dihedrally symmetric than Asteromphalus species which are more variable in these surface features [2.146].

Normal and abnormal valves for Cyclotella meneghiniana were compared in terms of symmetry (Figure 2.16). Normal valves had a steeper slope than abnormal valves, indicating a greater change in symmetry. That is, abnormal valves have less symmetry or greater asymmetry than normal valves of Cyclotella meneghiniana.

Initial valves were compared to normal vegetative valves of Cyclotella meneghiniana. The average symmetry of the initial valves was much higher than average symmetry for external and forming vegetative valves (Figure 2.17). The difference in symmetry of the central area was the key to the difference between initial and vegetative valves. The undulated central area of the vegetative valve has less symmetry than the symmetry of the more uniform convex shape in initial valves.

In the valve formation simulation, least symmetric taxa ranged from Actinoptychus splendens to the highest symmetric taxon, Actinoptychus senarius (Figures 2.23 and 2.24). The more distinct and regular the valve surfaces features were present, the more symmetric was the taxon. Less symmetric taxa had the highest chaotic and random instability (Figures 2.23 and 2.24). This relation between asymmetry and instability is supported for other organisms [2.54]. The main difference in our analysis is that the sources of instability are different for different degrees of symmetry.