The ecological context of bilateral symmetry of organ and organisms

A particular phenotype is generated through numerous networks of interactions from within the cell to the whole ecosystem. The level of the environmental noise and the ability of a given genotype to render different phenotypes under different environmental conditions, called phenotypic plasticity, determine survival or death at individual and/or population level. It is important to highlight that non-lethal environmental changes are important for generating genetic variability and promoting biological adaptations. However, when the level of environmental noise starts to be stressed, the developmental stability (DS) of the organism can be perturbed. The DS has been analysed through the symmetry deviations in organs or organisms with bilateral structure. The symmetry deviations occur due to inability to contain disorders from environmental or endogenous conditions during its development. This deviation is called fluctuating asymmetry (FA) when is a non-directional deviation in the symmetry of a bilateral structure normally distributed in a population. Low FA has been associated with greater DS. The analysis of FA has often been used to measure the effects of environmental perturbations. In this review, I discuss the concept of plant stress and phenotypic plasticity connecting both to the generation of an asymmetry phenotype, highlighting the usefulness of FA as an indicator of the level of stress which the organism is subjected to. Further, although this review explores mainly the connection between FA and stress in plants, the ecological context of symmetry in animals and plant-insect interaction is also discussed. Finally, I provided some methodologies used to detect symmetry variations in organs or organisms with bilateral structure.

Comparative performances of new and existing indices of crown asymmetry:an evaluation using tall trees of Eucalyptus pilularis(Smith)

Over the past 50 years,crown asymmetry of forest trees has been evaluated through several indices constructed from the perspective of projected crown shape or displacement but often on an ad hoc basis to address specifi c objectives related to tree growth and competition,stand dynamics,stem form,crown structure and treefall risks.Although sharing some similarities,these indices are largely incoherent and non-comparable as they diff er not only in the scale but also in the direction of their values in indicating the degree of crown asymmetry.As the fi rst attempt at devising normative measures of crown asymmetry,we adopted a relative scale between 0 for perfect symmetry and 1 for extreme asymmetry.Five existing crown asymmetry indices(CAIs)were brought onto this relative scale after necessary modifi cations.Eight new CAIs were adapted from measures of circularity for digital images in computer graphics,indices of income inequality in economics,and a bilateral symmetry indicator in plant leaf morphology.The performances of the 13 CAIs were compared over diff erent numbers of measured crown radii for 30 projected crowns of mature Eucalyptus pilularis trees through benchmarking statistics and rank order correlation analysis.For each CAI,the index value based on the full measurement of 36 evenly spaced radii of a projected crown was taken as the true value in the benchmarking process.The index(CAI 13)adapted from the simple bilateral symmetry measure proved to be the least biased and most precise.Its performance was closely followed by that of three other CAIs.The minimum number of crown radii that is needed to provide at least an indicative measure of crown asymmetry is four.For more accurate and consistent measures,at least 6 or 8 crown radii are needed.The range of variability in crown morphology of the trees under investigation also needs to be taken into consideration.Although the CAIs are from projected crown radii,they can be readily extended to individual tree crown metrics that are now commonly extracted from LiDAR and other remotely sensed data.Adding a normative measure of crown asymmetry to individual tree crown metrics will facilitate the process of big data analytics and artifi cial intelligence in forestry wherever crown morphology is among the factors to be considered for decision making in forest management.

Symmetrical Distribution of Primes and Their Gaps

Primes are of great importance and interest in mathematics partially due to their hard-to-predict distribution. A corollary of the Goldbach Conjecture is that two primes are equally distanced from a mid-point integer. Here the authors demonstrate that most primes are bilateral symmetrically distributed on the both sides of the halves of super products (or their integer multiples) of primes. This pattern suggests that greater primes may be obtained more efficiently by subtracting smaller ones from constants equal to super products (or their integer multiples) of primes.

An elliptical blade is not a true ellipse,but a superellipse–Evidence from two Michelia species

The shape of leaf laminae exhibits considerable diversity and complexity that reflects adaptations to environmental factors such as ambient light and precipitation as well as phyletic legacy.Many leaves appear to be elliptical which may represent a‘default’developmental condition.However,whether their geometry truly conforms to the ellipse equation(EE),i.e.,(x/a)^(2)+(y/b)^(2)=1,remains conjectural.One alternative is described by the superellipse equation(SE),a generalized version of EE,i.e.,|x/a|^(n)+|y/b|^(n)=1.To test the efficacy of EE versus SE to describe leaf geometry,the leaf shapes of two Michelia species(i.e.,M.cavaleriei var.platypetala,and M.maudiae),were investigated using 60 leaves from each species.Analysis shows that the majority of leaves(118 out of 120)had adjusted root-mean-square errors of<0.05 for the nonlinear fitting of SE to leaf geometry,i.e.,the mean absolute deviation from the polar point to leaf marginal points was smaller than 5%of the radius of a hypothesized circle with its area equaling leaf area.The estimates of n for the two species were<2,indicating that all sampled leaves conformed to SE and not to EE.This study confirms the existence of SE in leaves,linking this to its potential functional advantages,particularly the possible influence of leaf shape on hydraulic conductance.

Corneal functional optical zone under monocular and binocular assessment

Background:In this retrospective randomized case series,we compared bilateral symmetry between OD and OS eyes,intercorneal differences and Functional Optical Zone(FOZ)of the corneal aberrations.Methods:Sixty-seven normal subjects(with no ocular pathology)who never had any ocular surgery were bilaterally evaluated at Augenzentrum Recklinghausen(Germany).In all cases,standard examinations and corneal wavefront topography(OPTIKON Scout)were performed.The OD/OS bilateral symmetry was evaluated for corneal wavefront aberrations,and FOZ-values were evaluated from the Root-Mean-Square(RMS)of High-Order Wavefront-Aberration(HOWAb).Moreover,correlations of FOZ,spherical equivalent(SE),astigmatism power,and cardinal and oblique astigmatism for binocular vs.monocular,and binocular vs.intercorneal differences were analyzed.Results:Mean FOZ was 6.56±1.13 mm monocularly,6.97±1.34 mm binocularly,and 7.64±1.30 mm intercorneal difference,with all strongly positively correlated,showing that the diameter of glare-free vision is larger in binocular than monocular conditions.Mean SE was 0.78±1.30 D,and the mean astigmatism power(magnitude)was 0.46±0.52 D binocularly.The corresponding monocular values for these metrics were 0.78±1.30 D and 0.53±0.53 D respectively.SE,astigmatism magnitude,cardinal astigmatism component,and FOZ showed a strong correlation and even symmetry;and oblique astigmatism component showed odd symmetry indicating Enantiomorphism between the left and right eye.Conclusions:These results confirm OD-vs.-OS bilateral symmetry(which influences binocular summation)of HOWAb,FOZ,defocus,astigmatism power,and cardinal and oblique astigmatism.Binocular Functional Optical Zone calculated from corneal wavefront aberrations can be used to optimize refractive surgery design.