SEGMENTED_SPACETIME/Slider.html at main · error-wtf/SEGMENTED_SPACETIME · GitHub

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<!DOCTYPE html>
<html lang="de">
<head>
  <meta charset="UTF-8">
  <title>Segmented Spacetime: Exponential Triangle (λ → N)</title>
  <style>
    body {
      font-family: sans-serif;
      text-align: center;
      padding: 2rem;
      background: linear-gradient(135deg, #667eea 0%, #764ba2 100%);
      color: white;
      min-height: 100vh;
    }
    .container {
      background: rgba(255,255,255,0.1);
      backdrop-filter: blur(10px);
      border-radius: 20px;
      padding: 2rem;
      max-width: 800px;
      margin: 0 auto;
      box-shadow: 0 8px 32px rgba(0,0,0,0.3);
    }
    h1 {
      margin-bottom: 1rem;
      font-size: 1.8rem;
    }
    .theory-note {
      background: rgba(255,255,255,0.15);
      padding: 1.5rem;
      border-radius: 10px;
      margin: 1rem 0;
      font-size: 0.9rem;
      line-height: 1.5;
    }
    svg {
      margin-top: 1.5rem;
      background: rgba(255,255,255,0.9);
      border-radius: 15px;
      box-shadow: 0 4px 16px rgba(0,0,0,0.2);
    }
    label {
      display: block;
      margin-top: 1.5rem;
      font-weight: bold;
      font-size: 1.1rem;
    }
    input[type=range] {
      width: 400px;
      height: 8px;
      margin: 0.5rem 0;
    }
    .value-display {
      font-family: 'Courier New', monospace;
      background: rgba(255,255,255,0.2);
      padding: 0.5rem 1rem;
      border-radius: 8px;
      display: inline-block;
      margin: 0.5rem;
      font-size: 1.1rem;
    }
    #equationResult {
      font-family: 'Courier New', monospace;
      font-size: 1.2rem;
      margin: 1.5rem 0;
      background: rgba(255,255,255,0.2);
      padding: 1rem;
      border-radius: 10px;
    }
  </style>
</head>
<body>
  <div class="container">
    <h1>🌌 Segmented Spacetime: Exponential Triangle Visualizer</h1>

    <div class="theory-note" style="text-align: left; line-height: 1.8;">
      <h3 style="margin-top: 0; text-align: center;">Mathematical Framework</h3>

      <p><strong>Exponential Growth Model:</strong></p>
      <p>The radius in segmented spacetime follows the exponential law:</p>
      <p style="text-align: center; font-size: 1.1rem; margin: 1rem 0;">
        <strong>r(N) = r₀ · e^(λN)</strong>
      </p>

      <p><strong>Parameters:</strong></p>
      <ul style="margin: 0.5rem 0;">
        <li><strong>r₀</strong> = initial radius at N = 0 (reference scale)</li>
        <li><strong>λ</strong> = growth rate parameter (controls expansion speed)</li>
        <li><strong>N</strong> = continuous segment number (real-valued, not discrete)</li>
        <li><strong>e</strong> ≈ 2.71828 (Euler's number, base of natural logarithm)</li>
      </ul>

      <p><strong>Continuous vs. Discrete Nature:</strong></p>
      <p>While N is treated as a <em>continuous</em> variable in the mathematical model, the physical
      segmentation arises through <strong>φ-quantization</strong> where φ = (1+√5)/2 ≈ 1.618034 (golden ratio).
      Each physical segment scales by factor φ, but the underlying growth trend is captured by the
      continuous exponential function.</p>

      <p><strong>Connection to φ-segmentation:</strong></p>
      <p>The discrete segment radii follow r_n = r₀ · φ^n. Taking logarithms:</p>
      <p style="margin-left: 2rem;">ln(r_n) = ln(r₀) + n · ln(φ)</p>
      <p>This shows that φ-based growth is equivalent to exponential growth with λ = ln(φ) ≈ 0.481211,
      establishing the mathematical bridge between discrete φ-steps and continuous exponential expansion.</p>
    </div>

    <label for="lambda">Lambda (λ): <span class="value-display" id="lambdaValue">0.200</span></label>
    <input type="range" id="lambda" min="0" max="1" step="0.001" value="0.2">

    <p>Segment number N (continuous): <span class="value-display" id="nValue">?</span></p>

    <svg width="400" height="600" viewBox="0 0 400 600">
    <polygon id="upperTriangle" fill="none" stroke="black" stroke-width="3" />
    <polygon id="lowerTriangle" fill="none" stroke="black" stroke-width="3" />
    <text id="upperText" x="180" fill="#1e40af" font-weight="bold" font-size="16">λN</text>
    <text id="lowerText" x="180" fill="#1e40af" font-weight="bold" font-size="16">λt</text>

    <!-- Oberes Dreieck Labels -->
    <text x="90" y="255" fill="#374151" font-size="14">r₀e</text>
    <text x="290" y="255" fill="#374151" font-size="14">r(N)</text>
    <text x="140" y="270" fill="#059669" font-size="14">x</text>
    <text x="260" y="270" fill="#059669" font-size="14">z</text>

    <!-- Unteres Dreieck Labels -->
    <text x="90" y="355" fill="#374151" font-size="14">r₀e</text>
    <text x="290" y="355" fill="#374151" font-size="14">r(t)</text>
    <text x="140" y="370" fill="#059669" font-size="14">x</text>
    <text x="260" y="370" fill="#059669" font-size="14">z</text>
  </svg>

  <p id="equationResult">r(N) = r₀ · e^λN = ???</p>

  <div class="theory-note" style="margin-top: 2rem; font-size: 0.8rem;">
    <strong>φ-Quantization:</strong> The actual discrete structure arises through φ = (1+√5)/2 ≈ 1.618034,
    where each segment scales by the factor φ. The continuous representation shows the
    underlying exponential growth trend.
  </div>
  </div>

  <script>
    const lambdaSlider = document.getElementById('lambda');
    const lambdaValue = document.getElementById('lambdaValue');
    const nValue = document.getElementById('nValue');
    const upperTriangle = document.getElementById('upperTriangle');
    const lowerTriangle = document.getElementById('lowerTriangle');
    const upperText = document.getElementById('upperText');
    const lowerText = document.getElementById('lowerText');
    const equationResult = document.getElementById('equationResult');

    function updateTriangles() {
      const lambda = parseFloat(lambdaSlider.value);
      const r0 = 1;

      // IMPORTANT: N is now a REAL NUMBER, no longer an integer simplification!
      // The continuous segment number reflects the true nature of the theory
      const n = 4 + 6 * lambda; // Continuous from 4 to 10
      const rN = r0 * Math.exp(lambda * n);

      // Display with higher precision for real numbers
      lambdaValue.textContent = lambda.toFixed(3);
      nValue.textContent = n.toFixed(4); // 4 decimal places for continuous N

      // Coordinates adjusted for larger SVG (viewBox: 400×600)
      // Center at x=200, upper triangle at y≈250, lower at y≈350
      const scaleFactor = 30; // Reduced for better visibility
      const upperY = 250 - lambda * n * scaleFactor;
      const lowerY = 350 + lambda * n * scaleFactor;

      // Triangles centered at x=200, width ±100
      upperTriangle.setAttribute('points', `200,${upperY} 100,250 300,250`);
      lowerTriangle.setAttribute('points', `200,${lowerY} 100,350 300,350`);

      upperText.setAttribute('y', upperY - 10);
      upperText.setAttribute('x', 180); // Centered
      lowerText.setAttribute('y', lowerY + 25);
      lowerText.setAttribute('x', 180); // Centered

      // Result with higher precision
      equationResult.textContent = `r(N) = r₀ · e^(λN) = ${rN.toFixed(6)} (N = ${n.toFixed(4)})`;
    }

    lambdaSlider.addEventListener('input', updateTriangles);
    window.onload = updateTriangles;
  </script>
</body>
</html>