Universal models of biological growth and form are rare. One such model is the logarithmic spiral, which has been purported to explain the growth of biological structures such as teeth, claws, horns, and beaks. However, the logarithmic spiral only describes the path of the structure through space, and cannot generate these shapes. Here we show a new universal model based on a power law that explains the extreme diversity of tooth shapes in vertebrates, including humans, mammoths, saber-toothed cats, tyrannosaurs and giant megalodon sharks. This power cascade model can be used to predict the age of mammals with ever-growing teeth, including elephants and rodents, and determine sex in elephants. We view this as the third general model of tooth development, along with the patterning cascade model for cusp number and spacing, and the inhibitory cascade model that predicts relative tooth size. Beyond the dentition, this new model also explains the growth of claws, horns, antlers and beaks of vertebrates, as well as the fangs and shells of invertebrates, and thorns and prickles of plants. This model operates independently of the logarithmic spiral, and is present throughout these diverse biological systems. The power cascade provides a mechanistic basis for the generation of these pointed structures across the tree of life.