Everything I work with turns out to be a dynamic system. Dynamic systems have geometry.
The same forms appear at every scale: cycles nesting within cycles, structures that persist through motion, shapes that emerge wherever periodic processes interact. This page is about what that geometry reveals.
A cycle within a cycle is a torus. Earth spins once a day while orbiting the sun once a year. Any point on its surface traces a toroidal path through space. The sphere we call home is that torus with time removed.
Nested periodicity is everywhere. Hearts beat within breaths. Breaths cycle within circadian rhythms. Electrons orbit within vibrating molecules. Wherever systems persist through time, cycles nest inside cycles, and that nesting generates toroidal geometry at every scale.
Three ways to see a dynamic system: the snapshot freezes time and loses the motion. The fractal captures everything and dissolves into detail. The torus sits between, at the scale where dynamics become legible. KAM theory, the Hopf fibration, Arnold tongues, Ruelle-Takens: the mathematics shows toroidal structure isn't occasional but inevitable, and that fractals are what toruses become under stress, coupling, or further nesting. Two stages of one thing.
When complex systems learn, whether neural networks, organisms, or ecosystems, their trajectories through state space have measurable geometric properties. Training dynamics exhibit strange attractors, fractal structure, and bounded chaos. I'm interested in how data complexity shapes those trajectories, and what the fractal dimension of a training path reveals about what the system is actually doing.
The senses aren't passive receivers. They're active systems that create the geometry of experience. The nervous system processes nested periodic signals; it is itself toroidal, cycling at multiple timescales at once. I'm drawn to how what we feel relates to the shapes dynamics make.
From cardiac-respiratory coupling to planetary orbits, the same toroidal architecture appears wherever periodic processes embed within larger ones. Where a system sits in that hierarchy, and what pushes it from one stage to the next, is a question that cuts across disciplines.
Hexagonal patterns emerge wherever equal oscillators couple on a surface: Bénard convection, cymatics, the compound eyes of insects. They aren't imposed from outside. They're the mathematically inevitable result of equal-frequency processes interacting. I'm interested in how simple coupling rules generate complex spatial structure.
Computers in high school. Web design with databases and custom apps. Philosophy and economics at Binghamton. Graphic design. Years in construction. Bodywork. Two decades of yoga and meditation underneath all of it. What connects them is attention to how systems actually behave: not the idealized version, but the felt, dynamic, sometimes chaotic reality. The research formalizes what the code and the hands already knew.
Based in Nicasio, California. If any of this resonates, as a researcher, collaborator, or fellow pattern-noticer, I'd welcome a conversation.