🌌 Segmented Spacetime: Exponential Triangle Visualizer

Mathematical Framework

Exponential Growth Model:

The radius in segmented spacetime follows the exponential law:

r(N) = r₀ · e^(λN)

Parameters:

• r₀ = initial radius at N = 0 (reference scale)
• λ = growth rate parameter (controls expansion speed)
• N = continuous segment number (real-valued, not discrete)
• e ≈ 2.71828 (Euler's number, base of natural logarithm)

Continuous vs. Discrete Nature:

While N is treated as a continuous variable in the mathematical model, the physical segmentation arises through φ-quantization 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.

Connection to φ-segmentation:

The discrete segment radii follow r_n = r₀ · φ^n. Taking logarithms:

ln(r_n) = ln(r₀) + n · ln(φ)

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.

φ-Quantization: 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.