# The twist and turns of curved objects - Oxford Mathematics research investigates the stability and robustness of everted spherical caps

Everyday life tells us that curved objects may have two stable states: a contact lens (or the spherical cap obtained by cutting a tennis ball, see picture) can be turned ‘inside out’. Heuristically, this is because the act of turning the object inside out keeps the central line of the object the same length (the centreline does not stretch significantly). Such deformations are called ‘isometries’ and the ‘turning inside out’ (or everted) isometry of a thin shell is often referred to as mirror buckling.

However, mirror buckling is only strictly an isometry for objects with a vanishing thickness: an object with small, but finite thickness bends and stretches slightly at its outer edge (see second figure). Depending on its size, this bent region can even prevent the object from having two stable states – if the shell is too ‘shallow’, it will not stay in the everted shape but will ‘snap’ back to the natural state.

The rapid snapping between these two states is used to create striking children’s toys, while the Venus Flytrap plant uses an analogous mechanism to catch flies unaware. Surprisingly, however, the conditions under which two stable states exist has not been characterized, even for a spherical shell. In a recent study, Oxford Mathematicians Matteo Taffetani and Dominic Vella with colleagues from Boston University investigated when a spherical shell may exist in this everted state, together with the robustness of the everted state to poking. One surprising result of their analysis is that, though bistability is possible only for shells that are ‘deep enough’, the transition can be understood quantitatively using a mathematical model that exploits the shallowness of a shell.

The study of when the everted state exists provides one perspective on mirror buckling. However, it is also known that very thin shells (which are expected to remain close to isometry) can form polygonal buckles on being poked (think of a ‘broken’ ping pong ball). To gain new understanding of this instability, and how it interacts with snap-through, the authors then studied how robust the everted state is to poking: will it buckle or snap-through first? They found that even when once buckled polygonally, the purely axisymmetric theory gives a good account of when snap-through occurs, suggesting that the underlying mirror buckled solution, while not ultimately attained in this limit, heavily influences the stability of the whole shell structure.