This visualization is a companion to The π–φ Equilibrium (v1.1) by Gregory J. DeCarlo (2026).
From this single balance condition, a self-similar manifold emerges: concentric φ-scaled circles threaded by a golden logarithmic spiral, whose 3D projections include a golden-ratio torus (R=φa) and catenoid wormhole throat r(l)=req·cosh(l/req).
The φ–π bridge constant d* is where φd* = π exactly — the resonance between growth (φ) and enclosure (π). Platonic solid nodes cluster near d*.
The π-Star S₄ Geodesic Conjecture proposes closure at d=3k with geometric phase kπ. Note: The visual flash at d=3,6,9 is illustrative — the simulation does not compute the geodesic integral G(d). See paper Section 8 for status.
The layer index n shown represents a diagonal slice through ℛ(d,n) where dimensional complexity and physical scale increase together. The paper treats d and n as independent variables.
Audio: "Dimensional Resonance" 432×φd maps pitch to the dimension axis, not the physical frequency of each scale layer (paper Sec 2: f∝φn).
Drag to pan · Scroll/pinch to zoom · Right-drag or two-finger rotate to tilt · Tap floors to explore
GEOMETRY — Pure paper visualization: spiral, circles, π-Star, torus, wormhole, geodesic grid, Platonic nodes.
ENERGY — E(d) energy field with fractal iteration, ion-band coloring, and dimensional contours.
FUSION — Both layers unified with optional audio reactivity.
↑↓ dimension · +/- zoom · Space pause
A tour · C d* · R reset · S sound · M mode
F φ-snap · T reset tilt · H help
This will play a 432 Hz × φd dimensional resonance tone that changes with dimension. Sound is never recorded. Volume scales with E(d).
rn = r₀ · φn — each layer scales by φ ≈ 1.618. ~294 layers from Planck to observable universe. (Diagonal slice through ℛ(d,n))
| n | Scale | Size | d |
|---|