Singularities are common in diverse physical systems¹ and lead to universal structures^2,3. 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.

奇点在不同的物理系统中很常见¹并导致通用结构^2,3。这种普遍性表明它们也应该自然地出现在生物系统中，在这些系统中，主动生长、自主运动、运动和滑行聚焦时空变形，如决定生物大小和形状的形态发生过程所示⁴. 在不起眼的苹果中可以看到形态发生奇点的一个常见例子，随着苹果的生长，它在茎的附近形成。在这里，我们通过将水果生长的观察与简单的理论、有限元模拟和使用物理凝胶模拟物的受控膨胀实验相结合来研究苹果尖端的几何形状和形态发生。我们的观察表明，轴对称尖点发展成自相似形式，可以根据软球不均匀生长的力学理论来理解。在溶胀凝胶中使用局部抑制的物理实验证实了我们的理论预测。这些实验进一步表明，轴对称的尖瓣会失去稳定性并变成裂片状。