The cusp of an apple

Singularities are common in diverse physical systems1 and lead to universal structures2,3. This universality suggests that they should also naturally arise in biological systems, where active growth, autonomous motion, kinesis and taxis focus deformations in spacetime, as exemplified in the morphogenetic processes determining biological size and shape4. A familiar example of a morphogenetic singularity is seen in the humble apple, which forms in the neighbourhood of the stalk as the apple grows. Here we study the geometry and morphogenesis of the cusp of an apple by combining observations of fruit growth with a simple theory, finite element simulations and controlled swelling experiments using a physical gel simulacrum. Our observations show that the axisymmetric cusp develops into a self-similar form, which can be understood in terms of a mechanical theory for the inhomogeneous growth of a soft sphere. Physical experiments using local inhibition in swelling gels corroborate our theoretical predictions. These experiments further show that axisymmetric cusps can lose stability and become lobed. We use simulations to show that the number of cuspidal lobes depends on the ratio of the size of the stalk to the size of the sphere, as well as the amplitude and periodicity of perturbations that mimic the role of fruit anatomy, consistent with observations of multi-cusped fruits. A study of growing apples shows that the singular cusp at the stalk has a universal form that arises due to the differential growth of a soft solid. Although the cusps are usually symmetric, they can lose stability to form lobes that depend on the geometry of the fruit.