What does the tree of life look like as it grows? Evolution and the multifractality of time

By unifying three foundational principles of modern biology, we develop a mathematical framework to analyze the growing tree of life. Contrary to the static case, where the analogy between phylogenetic trees and the tree that grows in soil is drawn, our framework shows that the living tree of life is analogous to a Cantor dust where each branch is a distinct fractal curve. The system as a whole is therefore multifractal in the sense that it consists of many unique fractals. The three foundational principles for the mathematical framework are that phylogeny is nested, phylogeny is dualistic (i.e., transitive between singularities and populations), and phylogeny is stochastic. Integrating these three principles, we model the dynamic (i.e., living) tree of life as a random iterated function system that generates unique convexly related sequences of branching random variables (visualized in Animation 1). The multifractal nature of this dynamic tree of life implies that, for any two living entities, the time interval from their last common ancestor to the present moment is a distinct fractal curve for each. Thus, the length of a time interval along each distinct branch is unique, so that time is also multifractal and not an ultrametric on the tree of life.