Research images, animations, and visualization outputs.
Folding of an unstretchable circular helicoid into a Mobius band through an isometric mapping.Three identical helicoids with appropriate twists can be glued to construct a three-fold symmetric Mobius band.Energetically stable symmetric Mobius bands from folding of unstretchable helicoids.Energetically stable figure-eight Mobius bands from folding of unstretchable helicoids.Energetically saddle knotted Mobius bands from folding of unstretchable helicoids.Energetically saddle unknotted Mobius bands from folding of unstretchable helicoids.Single-bead and triangular three-bead models of Covid-19 spike proteins on a spherical core.Experimental coronavirus core images and prolate/oblate core models.Measured spike charge heterogeneity and a charged rigid-bead rod spike model.Two charged elastic loops on a rigid sphere can adopt multiple equilibrium configurations.Increasing inter-loop charge density strengthens repulsion and squeezes loops apart.Increasing intra-loop charge density opens each loop.Bifurcation diagram showing that the first unstable mode depends on loop length and interaction strengths.
