  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.
