By introducing mechanics, we begin to explain the mechanisms that lead to the observed patterns. Some years ago, Dublin-born Alan Newell and Patrick Shipman at the University of Arizona applied elasticity theory to continuum models of growing cacti shoots. The formation of phylla was represented by buckling of the growing surface and the resulting pattern, corresponding to a minimum of the elastic energy, took the form of superimposed waves.

The observed pattern involved triad interactions, the interactions between three waves which satisfy criteria called resonance conditions. For a resonant triad, the wavenumbers of two waves must sum to that of the third, K1 + K2 = K3. This is similar to the recurrence relation for the Fibonacci numbers, F(n) = F(n-1) + F(n-2) providing a link between the wavenumbers and the arithmetic of the Fibonacci sequence.

But mechanics could not answer all the questions. For a more complete picture, biochemistry was needed. Recently, Newell and Matt Pennybacker (2013) have shown that the observed patterns emerge from a pattern-forming front arising from a combination of a biochemical and mechanical instabilities. Growth is stimulated by a plant hormone called auxin. The growth front is a solution of a nonlinear partial differential equation (PDE) for u, the concentration of auxin

Auxin-PDEThe formation of primordial seeds is driven by high auxin concentrations. The growing seeds exert forces on each other, creating geometric patterns, and the geometry can trigger the production of auxin, leading to a feedback loop. Thus, biochemistry, mechanics and geometry all play a role in generating the observed patterns.