From the Golden Ratio to Fibonacci Phyllotaxis Spirals

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.

Normalizing Topologically 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 triangulation.

Normalizing Topologically 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.

Normalizing Topologically 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.