Resonance Field Global Coherence Visualization Diagram

 

🌐 Resonance Field Global Coherence Visualization Diagram

This is a graphical interface used to represent the internal resonance synchronization state of an entire system.

🔄 Metaphor

Imagine a group of people dancing:

• If everyone is moving in perfect harmony, the system is in resonance coherence.

• If someone is off-beat or out of sync, the system shows desynchronization.

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📡 Resonance Field

Think of it like a water surface:
Each point oscillates on its own, but when all ripples align and move together, the surface becomes a coherent resonance field.

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🌍 Global Coherence

When all oscillations across the system move with the same direction and frequency, we achieve global coherence — like a calm surface rising and falling uniformly.

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🔧 Core Features

This interactive visualization includes:

1. Unit Circle (Left Side):

   • Centered at (𝑥 = 200, 𝑦 = 250), with a fixed radius.

   • A white marker rotates around the circle representing current angle 𝜃 (theta).

   • Green vertical projection for sin(𝜃), orange horizontal projection for cos(𝜃).

2. Sine and Cosine Waves (Right Side):

   • Plots of sin(𝜃) and cos(𝜃) over 𝜃 ∈ [0, 2π].

   • 🟢 sin(𝜃) in green (#CCFF00), 🟠 cos(𝜃) in orange (#FF9966).

   • Dynamic trail effects show recent waveform behavior.

3. Coherence Index 𝐶ₙ(𝜃) (Bottom):

   • Calculated by the formula:

     
     Cₙ(𝜃) = [sin(∑ᵢ₌₁ⁿ [cos(𝑖𝜃)/𝑖 + sin(𝑖𝜃)/𝑖]) + 1] ÷ 2
     

   • Result ∈ [0, 1].

   • Displayed as 🔵 blue curve (#00CCFF), turns 🟢 green (#00FF00) when Cₙ(𝜃) ≥ 0.7.

4. Threshold Control:

   • 🔴 Red dashed line marks the 0.7 threshold.

   • If Cₙ(𝜃) ≥ 0.7, system status = ✅ "Sufficient".

   • If Cₙ(𝜃) < 0.7, system triggers a feedback loop to increment harmonic number n (up to 10) and recheck.

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🕹 Interactive Controls

• Play/Pause Button: Animate 𝜃 from 0 to 2π (increments of 0.01 rad).

• 𝜃 Slider: Manual control of 𝜃, disables auto-animation.

• Harmonic Slider (n): Adjust n ∈ [1, 10], affects 𝐶ₙ(𝜃).

• Live Readout Panel:

  • Displays values of:

    • 𝜃, sin(𝜃), cos(𝜃), 𝐶ₙ(𝜃)

    • Status: "Sufficient" (green) or "Insufficient" (red)

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📈 Visual Axes

• X-Axis: 0 to 2π, labeled at intervals of π/2.

• Y-Axis:

  • For sin/cos: from −1.2 to 1.2, grid every 0.5.

  • For 𝐶ₙ(𝜃): from 0 to 1.

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🧠 Purpose

• Educate: Show relationship between angular motion, trigonometric functions, and harmonic coherence.

• Simulate: Model how a system seeks structural harmony by adjusting harmonics.

• Analyze: Let users explore how changing 𝜃 and n alters global synchronization.

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⚙️ Technical Details

• Technologies:

  • React 18.3.1 (interactivity)

  • D3.js 7.9.0 (SVG drawing)

  • Tailwind CSS 2.2.19 (styling)

  • Babel 7.26.9 (JSX parsing)

• Robustness:

  • ErrorBoundary wraps the app to catch runtime issues.

  • Console logs debug feedback and coherence corrections.

• Accessibility:

  • English-only text.

  • Clearly labeled axes and buttons.

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📘 Mathematical Notes

The coherence index formula is:

Cₙ(𝜃) = [sin(∑ᵢ₌₁ⁿ [cos(𝑖𝜃)/𝑖 + sin(𝑖𝜃)/𝑖]) + 1] ÷ 2

🎯 Meaning:

• Converts harmonic phase summation into a normalized scalar between 0 and 1.

• Captures resonance structure of multiple harmonics.

• Intuitively: Measures how “aligned” the wave components are.

🧩 Potential Applications:

• Signal Processing: Nonlinear harmonic combination.

• Statistics: Angular density estimation.

• Physics: Wave-field synchronization.

• AI/ML: Structural resonance criteria for model inference or gating.

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📎 GitHub Link:
Global Coherence Resonance Visualization (HTML)