Fig. 1

Generative models for shapes of teeth and other pointed structures, showing the effects of relative growth rates on shape. Logarithmic spiral (top): if the rates of growth of the two sides of the structure (rA and rB) are equal, a symmetrical structure such as a cone is produced (a). If the rate of growth on one side is lower (e.g. rA < rB), then the structure curves to follow a logarithmic spiral (black curved line) (b). Power cascade (left): when the power growth rate of the distance from the tip (rD) is equal to the growth rate of the radius (rR), then a cone is produced (a). When rR is less than rD, a power cone is generated (c). Here, rD = 2rR, generating a paraboloid. Both of these inequalities in growth rates can be combined to form a power cone curving along a logarithmic spiral, or a power spiral (d)