Fluctuating asymmetry in plant leaves is a widely used measure in geometric morphometrics for assessing random deviations from perfect symmetry. In this study, we considered the concept of fluctuating asymmetry to improve the prototype leaf shape of the functional-structural plant model L-Cucumber. The overall objective was to provide a realistic geometric representation of the leaves for the light sensitive plant reactions in the virtual plant model. Based on three-dimensional data from several hundred in situ digitized cucumber leaves comparisons of model leaves and measurements were conducted. Robust Bayesian comparison of groups was used to assess statistical differences between leaf halves while respecting fluctuating asymmetries. Results indicated almost no directional asymmetry in leaves comparing different distances from the prototype while detecting systematic deviations shared by both halves. This information was successfully included in an improved leaf prototype and implemented in the virtual plant model L-Cucumber. Comparative virtual plant simulations revealed a slight improvement in plant internode development against experimental data using the novel leaf shape. Further studies can now focus on analyses of stress on the 3D-deformation of the leaf and the development of a dynamic leaf shape model. Full article

Let A be an $n\times n$ complex matrix. Assume the determinantal curve ${\mathbb{V}}_A=\{[(x,y,z)]\in {\mathbb{CP}}^2:F_A(x,y,z)=\det (x\Re (A)+y\Im (A)+z{I}_n)=0\}$ is a rational curve. The Fiedler formula provides a complex symmetric matrix S satisfying $F_S(x,y,z)=F_A(x,y,z)$ . It is also known that every Toeplitz matrix is unitarily similar to a symmetric matrix. In this paper, we investigate the unitary similarity of the symmetric matrix S and the matrix A in the Fiedler theorem for a specific parametrized family of $4\times 4$ nilpotent Toeplitz matrices A. We show that there are either one or at least three unitarily inequivalent symmetric matrices which admit the determinantal representation of the ternary from $F_A(x,y,z)$ associated to the specific $4\times 4$ nilpotent Toeplitz matrices. Full article