CNLS and T-Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
While the statics of many simple physical systems reproduce the striking number-theoretical patterns found in the phyllotaxis of living beings, their dynamics reveal unusual excitations: multiple classical rotons and a large family of interconverting topological solitons. As we introduce those, we also demonstrate experimentally for the first time Levitov's celebrated model for phyllotaxis. Applications at different scales and in different areas of physics are proposed and discussed.