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ResonanT⁴ Demo

Interactive visualization of modular orbits on 𝕋³, projected into 3D space via a transverse geometric phase.

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ResonanT⁴ demo showing a closure-compatible orbit rendered in 3D — A multi-strand braided structure (n=321, k=36) rendered on a 3D grid. The looping, intertwined paths are composed of evenly-spaced colored points transitioning through cyan, green, yellow, orange, pink, and purple. Topology: 107×3 (three strands of 107 points each). The right panel shows parameter controls: Geometry & Lattice (n, k, radial/vertical amplitude, R), Phases & Motion (ω, ξ, ψ), and Visual settings (tau speed, particle size, knot step). Bottom readout displays lattice structure, detuning formula (n = q·k + δ → 321 = 8·36 + 33), and frequency ratios.

What This Demo Visualizes

The demo renders modular orbits defined on the 3-torus 𝕋³ and projects them into 3D space using a fourth transverse phase τ. You can explore how different parameter combinations produce closure-compatible or drifting configurations.

Parameters

Geometry & Lattice

n (index count): Number of discrete phase indices around the orbit. Higher values produce denser sampling.
k (winding number): How many times the orbit winds before closing. Coprime n and k produce full coverage.
a (radial amplitude): Amplitude of radial modulation in the toroidal embedding.
h (vertical amplitude): Amplitude of vertical oscillation. Can be locked to |a| for symmetric configurations.
R (major radius): Major radius of the torus embedding.

Phases

ω (omega): Angular frequency multiplier for the first intrinsic phase.
ξ (xi): Angular frequency multiplier for the second intrinsic phase.
ψ (psi): Phase offset. Can be locked to τ for synchronized transverse sweep.

Visual

τ speed: Speed of the transverse phase sweep animation.
Index marker size: Size of the particles marking discrete phase indices.
Connection generator s: Connects indices by t → t+s (mod n), revealing different orbit structures.
Curve density: Number of interpolated points between indices for smooth curves.
Curve color / Hue offset: Toggle gradient coloring and shift the color palette.

To explore the mathematics behind this visualization, read the papers:

Read Papers