On the perfect hexagonal packing of rods

In most cases the hexagonal packing of fibrous structures or rods extremizes the energy of interaction between strands. If the strands are not straight, then it is still possible to form a perfect hexatic bundle. Conditions under which the perfect hexagonal packing of curved tubular structures may exist are formulated. Particular attention is given to closed or cycled arrangements of the rods like in the DNA toroids and spools. The closure or return constraints of the bundle result in an allowable group of automorphisms of the cross-sectional hexagonal lattice. The structure of this group is explored. Examples of open helical-like and closed toroidal-like bundles are presented. An expression for the elastic energy of a perfectly packed bundle of thin elastic rods is derived. The energy accounts for both the bending and torsional stiffnesses of the rods. It is shown that equilibria of the bundle correspond to solutions of a variational problem formulated for the curve representing the axis of the bundle. The functional involves a function of the squared curvature under the constraints on the total torsion and the length. The Euler?Lagrange equations are obtained in terms of curvature and torsion and due to the existence of the first integrals the problem is reduced to the quadrature. The three-dimensional shape of the bundle may be readily reconstructed by integration of the Ilyukhin-type equations in special cylindrical coordinates. The results are of universal nature and are applicable to various fibrous structures, in particular, to intramolecular liquid crystals formed by DNA condensed in toroids or packed inside the viral capsids

A formula for the minimal coordination number of a parallel bundle

An exact formula for the minimal coordination numbers of the parallel packed bundle of rods is presented based on an optimal thickening scenario. Hexagonal and square lattices are considered

Writhe formulas and antipodal points in plectonemic DNA configurations

The linking and writhing numbers are key quantities when characterizing the structure of a piece of supercoiled DNA. Defined as double integrals over the shape of the double-helix, these numbers are not always straightforward to compute, though a simplified formula exists. We examine the range of applicability of this widely-used simplified formula, and show that it cannot be employed for plectonemic DNA. We show that inapplicability is due to a hypothesis of Fuller theorem that is not met. The hypothesis seems to have been overlooked in many works

Equilibrium Shapes with Stress Localisation for Inextensible Elastic Mobius and Other Strips

We formulate the problem of finding equilibrium shapes of a thin inextensible elastic strip, developing further our previous work on the Möbius strip. By using the isometric nature of the deformation we reduce the variational problem to a second-order one-dimensional problem posed on the centreline of the strip. We derive Euler–Lagrange equations for this problem in Euler–Poincaré form and formulate boundary-value problems for closed symmetric one- and two-sided strips. Numerical solutions for the Möbius strip show a singular point of stress localisation on the edge of the strip, a generic response of inextensible elastic sheets under torsional strain. By cutting and pasting operations on the Möbius strip solution, followed by parameter continuation, we construct equilibrium solutions for strips with different linking numbers and with multiple points of stress localisation. Solutions reveal how strips fold into planar or self-contacting shapes as the length-to-width ratio of the strip is decreased. Our results may be relevant for curvature effects on physical properties of extremely thin two-dimensional structures as for instance produced in nanostructured origami

Characterisation of cylindrical curves

We employ moving frames along pairs of curves at constant separation to derive various conditions for a curve to belong to the surface of a circular cylinder

Equilibrium shapes with stress localisation for inextensible elastic möbius and other strips

© Springer Science+Business Media Dordrecht 2015. We formulate the problem of finding equilibrium shapes of a thin inextensible elastic strip, developing further our previous work on the Möbius strip. By using the isometric nature of the deformation we reduce the variational problem to a second-order onedimensional problem posed on the centreline of the strip. We derive Euler-Lagrange equations for this problem in Euler-Poincaré form and formulate boundary-value problems for closed symmetric one-and two-sided strips. Numerical solutions for the Möbius strip show a singular point of stress localisation on the edge of the strip, a generic response of inextensible elastic sheets under torsional strain. By cutting and pasting operations on the Möbius strip solution, followed by parameter continuation, we construct equilibrium solutions for strips with different linking numbers and with multiple points of stress localisation. Solutions reveal how strips fold into planar or self-contacting shapes as the length-to-width ratio of the strip is decreased. Our results may be relevant for curvature effects on physical properties of extremely thin two-dimensional structures as for instance produced in nanostructured origami

Forceless Sadowsky strips are spherical

© 2018 American Physical Society. We show that thin rectangular ribbons, defined as energy-minimizing configurations of the Sadowsky functional for narrow developable elastic strips, have a propensity to form spherical shapes in the sense that forceless solutions lie on a sphere. This has implications for ribbonlike objects in (bio)polymer physics and nanoscience that cannot be described by the classical wormlike chain model. A wider class of functionals with this property is identified

Forceless folding of thin annular strips

Thin strips or sheets with in-plane curvature have a natural tendency to adopt highly symmetric shapes when forced into closed structures and to spontaneously fold into compact multi-covered configurations under feed-in of more length or change of intrinsic curvature. This disposition is exploited in nature as well as in the design of everyday items such as foldable containers. We formulate boundary-value problems (for an ODE) for symmetric equilibrium solutions of unstretchable circular annular strips and present sequences of numerical solutions that mimic different folding modes. Because of the high-order symmetry, closed solutions cannot have an internal force, i.e., the strips are forceless. We consider both wide and narrow (strictly zero-width) strips. Narrow strips cannot have inflections, but wide strips can be either inflectional or non-inflectional. Inflectional solutions are found to feature stress localisations, with divergent strain energy density, on the edge of the strip at inflections of the surface. ‘Regular’ folding gives these singularities on the inside of the annulus, while ‘inverted’ folding gives them predominantly on the outside of the annulus. No new inflections are created in the folding process as more length is inserted. We end with a discussion of an intriguing apparent connection with a deep result on the topology of curves on surfaces

Ecomorphology reveals Euler spiral of mammalian whiskers

Whiskers are present in many species of mammals. They are specialised vibrotactile sensors that sit within strongly innervated follicles. Whisker size and shape will affect the mechanical signals that reach the follicle, and hence the information that reaches the brain. However, whisker size and shape have not been quantified across mammals before. Using a novel method for describing whisker curvature, this study quantifies whisker size and shape across 19 mammalian species. We find that gross two‐dimensional whisker shape is relatively conserved across mammals. Indeed, whiskers are all curved, tapered rods that can be summarised by Euler spiral models of curvature and linear models of taper, which has implications for whisker growth and function. We also observe that aquatic and semi‐aquatic mammals have relatively thicker, stiffer, and more highly tapered whiskers than arboreal and terrestrial species. In addition, smaller mammals tend to have relatively long, slender, flexible whiskers compared to larger species. Therefore, we propose that whisker morphology varies between larger aquatic species, and smaller scansorial species. These two whisker morphotypes are likely to induce quite different mechanical signals in the follicle, which has implications for follicle anatomy as well as whisker function