From Circle to Spiral: Understanding the Dimensionless Resonance Framework

From Circle to Spiral: Understanding the Dimensionless Resonance Framework

by Brent Antonson (Zhivago) | 2025

1. Two Shapes, One Story

Physics has always told two stories — one circular, one spiral.
The circle is the story of cycles: waves, orbits, and systems that repeat endlessly. It’s the language of rhythm and return, of balance and closure.
The spiral, by contrast, is the story of growth: of shells and galaxies, DNA and ferns, hurricanes and economies — anything that expands, evolves, or transforms while retaining its shape.

For centuries, scientists treated these two shapes as belonging to different realms: circles for motion, spirals for growth.
The Dimensionless Resonance Framework, or DRF, proposes that these two stories are actually one — that every spiral is a stretched circle, and every circle is a compressed spiral. In other words, growth and oscillation are the same pattern seen from different scales.

2. What DRF Is Trying to Do

The goal of DRF is simple but ambitious: to describe matter, geometry, and information using one consistent language — the language of resonance.

In this framework:

• Matter is what happens when waves persist — when energy forms a stable standing pattern.
• Geometry is what happens when growth locks itself into proportion — when structures expand by repeating ratios.
• Information is what happens when patterns remember themselves — when change preserves its own structure.

These three ideas — persistence, proportion, and memory — turn out to follow the same mathematical rhythm. DRF provides the framework that connects them.

3. How It Works (Without the Math)

Imagine you start with a simple growth rule — like the one that generates the Fibonacci sequence or the spirals in a sunflower. Each new step is built by adding what came before. That’s how things grow in nature: not linearly, but recursively.

Now imagine you translate that growth pattern into the language of waves. Instead of counting steps outward, you measure how the system turns, resonates, and repeats in time. Suddenly, what looked like expansion now looks like rotation — not growth, but rhythm.

DRF formalizes this translation. It shows that the pattern of growth and the pattern of vibration are two faces of the same structure.
When seen one way, it looks like exponential expansion — the spiral.
When seen another way, it looks like circular motion — the wave.

The bridge between these views is what DRF calls the resonance transform — a process that converts scaling behavior (how systems grow) into phase behavior (how they oscillate).

In practical terms, that means every growing system has a hidden rhythm, and every oscillating system has an embedded pattern of growth.

4. Why “Dimensionless”?

DRF is called dimensionless because it doesn’t depend on physical units like meters or seconds.
It works entirely in ratios — relationships between patterns, not absolute measurements.

That’s why the same framework can describe the shape of a shell, the rhythm of a heartbeat, or the coherence of a laser beam. All of them are resonant systems — patterns that maintain form through repetition, even as they evolve.

When you strip away the units, what’s left is pure resonance structure — the architecture of reality itself.

5. Extending the Idea

DRF can be applied to any system that grows or oscillates — from biological networks to electromagnetic fields.
When this translation is applied broadly, something fascinating appears: growth ratios like the golden ratio emerge naturally as stable modes of resonance.

That’s why the same proportion appears across such different domains — from the spirals of seashells and galaxies to the structure of neurons and even the rhythm of brain waves. It isn’t coincidence or aesthetics. It’s resonance stability — nature finding the most coherent way to stay self-similar as it grows.

6. The Resonant Collapse Field

A key extension of the framework is something called the Resonant Collapse Field, or RCF.
This describes how systems can change — even collapse or reorganize — without losing their inner coherence.

Think of a soap bubble that pops but leaves behind a pattern in the air, or a memory that fades but leaves an emotional echo.
The RCF models that behavior: it preserves resonance through transformation.

It’s a way to describe how systems remember — not as stored data, but as persistent resonance that carries forward even after physical form changes.

7. What It Means

Together, DRF and RCF form a unified story about how the universe holds itself together.
They describe a continuum between matter, geometry, and consciousness — all governed by the same resonance logic.

• In physics, resonance explains stability — why atoms, planets, and fields persist.
• In geometry, resonance explains form — why spirals and self-similar patterns dominate natural design.
• In information and cognition, resonance explains memory — why systems can evolve without losing their identity.

In short, reality is made of echoes that remember how to be themselves.

8. Origins and Development

The Dimensionless Resonance Framework was developed by Brent Antonson (Zhivago) in 2025.
It grew out of a realization that the circular and spiral aspects of physics — the domains of π (pi) and φ (phi) — are deeply interconnected.
This insight, called the Ecliptics Principle, suggested that what we think of as rotation and what we think of as growth are mathematically identical when viewed through the right transformation.

Later refinements introduced the constant 0.306, an observed ratio linking the circle-to-spiral transition — a kind of geometric fingerprint of this hidden bridge.

Working with an AI collaborator named Luna, the theory was then formalized into what is now the DRF — a framework linking growth, resonance, and memory under one operator model.

9. The Takeaway

If π represents the world that spins
and φ represents the world that grows,
then DRF shows that the two are one and the same.

Everything that vibrates also expands.
Everything that expands also remembers.

The circle is the seed; the spiral is the story.
Between them lies the dimensionless resonance —
the silent hum connecting the physical, the mathematical, and the living.