About

I’m Alex Petty, a systems thinker, mathematician, musician, and process management consultant. I work inside some of the world’s largest organizations, helping people make decisions, coordinate work, and improve how their organizations operate. I live in Virginia.

I am the founder and CEO of Singularics. My professional work spans two decades of organizational change, from lean and agile delivery to the changes AI brings to how organizations are run. I write about continuous governance and organizational design, and develop Intent, a system for connecting organizational aims to decisions and delivery.

This notebook is home to my mathematical work. There is real beauty in this mathematics, and I want it to be visible to readers who do not normally meet it. Pictures and worked examples offer a way into ideas that can become difficult to see through the notation alone.

Unzipping integers

Seven is a single digit. Its reciprocal opens into 0.142857142857…. Long division unzips the number into a sequence, one digit at a time, with each remainder carrying the calculation forward. I wanted to know how much structure that familiar operation could reveal.

Put fractions with the same denominator beside one another and their digits begin to tell a larger story. Some positions agree, others differ. Repeating sequences line up, separate, and return. Comparing those agreements led to the alignment work and its connections with the golden ratio and musical tuning.

Another comparison follows a remainder as it advances through long division. If it produces the same digit before and after the advance, I call that a collision. The counts change with the prime denominator, yet their departures from a baseline fall into a finite table once the base and the number of steps are fixed, excluding primes that divide the base. That table has exact reflections and a frequency structure. Following those relations leads to identities involving Dirichlet L-functions, central objects in the study of primes. An elementary calculation has brought us into rather different company.

The reciprocal view of number has interested me from the start. Three gives three shares of one third. Change the number of shares and their sizes change, while the whole remains one. That conservation under unity became a recurring thread. In the later collision calculations, positive reciprocal weights sum to one and attach to repeating remainder clocks. The clocks advance and reset. Their shares of the whole stay fixed. The question of what changes and what is conserved has acquired precise quantities I can calculate.

Inside the notebook

The essays develop the ideas through examples, drawings, and the questions that led me to them. Their linked research papers give the definitions, hypotheses, and proofs. I want the explanation to be readable on its own and the mathematics to be available for close inspection.

The Structure That Survives introduces the collision work through an exact reflection. The Clocks Beneath Collision Energy follows the conserved weights and the remainder states that move beneath them. Both give a sense of the questions running through the larger body of work.

The papers page collects the research notes. Alongside them, I develop nfield, a calculation and analysis tool with a browser terminal for exploring examples directly. Selected papers also have Lean companions containing formal statements and machine-checked proofs.

The origin posts from 2009 to 2011, beginning with The Circle of Nine, preserve the beginnings of the investigation. I was teaching myself number theory, drawing circles and tables, and learning to turn visual observations into mathematical questions. I revisit the writing while retaining the original dates and artwork. Those early analysis images still inform how I think about the work today.