From the Golden Ratio to Fibonacci
We explain why the assumption of
growth governed by the Golden Ratio in phyllotaxis leads to a Fibonacci
number of spirals. We also explain why two families of spirals in
opposite directions are usually evident, both Fibonacci in number.
Finally, we explain why flat phyllotaxis patterns, such as in
sunflowers, seem to have different numbers of spirals depending on how
far from the center you look.
Minimal Surfaces I: Global to Local Index
We show that in any triangulated
3-manifold, every index $n$ topologically minimal surface can be
transformed to a surface which has local indices (as computed in each
tetrahedron) that sum to at most $n$. This generalizes classical
theorems of Kneser and Haken, and more recent theorems of Rubinstein
and Stocking, and is the first step in a program to show that every
topologically minimal surface has a normal form with respect to any
Minimal Surfaces II: Disks
We show that a topologically minimal disk
in a tetrahedron with index $n$ is either a normal triangle, a normal
quadrilateral, or a normal helicoid with boundary length $4(n+1)$. This
mirrors geometric results of Colding and Minicozzi.
Minimal Surfaces III: Bounded Combinatorics
We show that there are a finite number of
possible pictures for a surface in a tetrahedron with local index $n$.
Combined with previous results, this establishes that any topologically
minimal surface can be transformed into one with a particular normal
form with respect to any triangulation.