Bernat Espigulé

Complex trees are two-dimensional objects living in the complex plane, but they can be represented in 3D by adding a certain depth at each branching level. The specimens selected for this mathematical arboretum are complex trees with a plane-filling tipset that can be used to tile the entire plane. From left to right trees in the bottom are T{-0.3376 + i 0.5623, -0.1226 - i 0.7449}, the tame twin dragon tree T{(1+i√7)/4,-(1+i√7)/4}, and the twin dragon tree T{1/2+i/2,-1/2-i/2}. In the front row we have a pair of ternary complex trees with a Rauzy fractal tipset on the left, and the fludgeflake tree T{i/√3,1/2-i/2√3,-1/2+i/2√3} on the right.

This animation explores the parameter space of a newly discovered family of tipset connected ternary complex trees related to the Sierpinski triangle. The parameter space for this family is defined in the annulus {z=x+iy : 1/4 < |z| < 1}. Starting at t=0 and ending in t=2Pi, each frame M_c(t) shows in black the parameters z that generate ternary complex trees T{z,1/2,1/4z} with at least a tip point landing in c(t):=1+Exp(i t)/2. The fast algorithm to generate M_c(t) that made this animation possible was recently discovered by the author. Because of the asymptotic similarity between M_c(t) and self-similar sets F{z,1/2,1/4z} with parameter z in M_c(t), this animation is a novel way to illustrate what is out there in the blink of an eye.

"The Beauty of Roots" unveils the hierarchical algebraic structure of the connectivity locus M for a family of self-similar sets studied over the past 35 years. Some mathematicians who studied M include Michael Barnsley, Thierry Bousch, Christoph Bandt, Boris Solomyak, John Baez, Bill Thurston, Danny Calegari, Sarah Koch, Alden Walker, Pablo Shmerkin, Stefano Silvestri, and the author himself Bernat Espigule. This diagram shows all roots of polynomials up to degree 12 with coefficients 1, -1, and 0, always starting with 1. The radius and color-coding of each algebraic number comes from the degree m of the lowest degree polynomial who has it as a root. This original picture came to the author after proving a theorem related to complex trees.