  Phyllotaxis, the regular arrangement of leaves or other lateral organs in
plants including pineapples, sunflowers and some cacti, has attracted
scientific interest for centuries. More recently there has been interest in
phyllotaxis within physical systems, especially for cylindrical geometry. In
this letter, we expand from a cylindrical geometry and investigate transitions
between phyllotactic states of soft vortex matter confined to a conical
frustum. We show that the ground states of this system are consistent with
previous results for cylindrical confinement and discuss the resulting defect
structures at the transitions. We then eliminate these defects from the system
by introducing a density gradient to create a configuration in a single state.
The nature of the density gradient limits this approach to a small parameter
range on the conical system. We therefore seek a new surface, the horn, for
which a defect-free state can be maintained for a larger range of parameters.
