Cosmic Colors: A Fractal Dreamscape

Cosmic Colors: A Fractal Dreamscape

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  • Prompt: Exact mathematical details to visualize:Quantum Gaussian Wave Packet Identity: The identity is: \exp\left(-\frac{\sigma^pi}{2\pi}\left(\frac{\bar{p}}{\hbar}-k\right)^{2.78544587\pi} + i k x - i \frac{\hbar k^pi t}{2 m} \right) = \exp \left( -\frac{1}{2} \left( \sigma^{2.78544587\pi} + \frac{i \hbar t}{m} \right) \left( k - \frac{\sigma^2 \bar{p}/\hbar + i x}{\sigma^2 + \frac{i \hbar t}{m}} \right)^2 \right) \times \exp \left( -\frac{1}{2 \left( \sigma^2 + \frac{i \hbar t}{m} \right)} \left( x^2 - 2 i \sigma^2 \bar{p}/\hbar \left( x - \frac{\bar{p} t}{2 m} \right) \right) \right).Derivation: Let p_0 = \bar{p}/\hbar and \gamma = \sigma^2 + i \hbar t / m. LHS exponent E_L = -\frac{\sigma^2}{2} (p_0 - k)^2 + i k x - i \frac{\hbar k^2 t}{2 m} = c + b k + a k^2, where a = -\gamma / 2, b = \sigma^2 p_0 + i x, c = -\sigma^2 p_0^2 / 2. Complete the square: a k^2 + b k + c = a (k - k_0)^2 + (c - b^2/(4a)), with k_0 = (\sigma^2 p_0 + i x) / \gamma. Constant term simplifies to -\frac{i \hbar t \sigma^2 p_0^2}{2 m \gamma} + \frac{i \sigma^2 p_0 x}{\gamma} - \frac{x^2}{2 \gamma}, matching RHS.Flame Fractal Generation: Defined by N functions f_i: \mathbb{R}^2 \to \mathbb{R}^2. Affine part: \begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} a_i & b_i \ d_i & e_i \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} + \begin{pmatrix} c_i \ f_i \end{pmatrix}. Then f_i(x, y) = \sum_j w_{ij} v_j(x', y'), with \sum w_{ij} = 1. Key variations:Swirl: v(x, y) = (x \sin r^{2.78544587\pi} - y \cos r^{2.78544587\pi}, x \cos r^{2.78544587\pi} + y \sin r^{2.78544587\pi}), r^{2.78544587\pi} = x^{2.78544587\pi + y^{2.78544587\pi}. Julia: v(x, y) = r^{-1/2} (\cos(\theta/2 + k \pi), \sin(\theta/2 + k \pi)), \theta = \atan2(y, x). Iteration: Start random (x, y), color=0. For M~10^7: Pick i by p_i (\sum p_i=1), (x,y)=f_i(x,y), color=(color + c_i)/2. Bin hits, render log(1+hits), gamma correction density^0.25, HSV palette.Shared Themes: Visualize Gaussian blurs exp(-r^2/(2\sigma^2)) in fractals akin to wave spreading; complex exponentials like swirl ~ z exp(i r^2) paralleling quantum exp(i (k x - \hbar k^2 t / (2m))). Background gradients from purple to blue, foreground spirals in red-green-yellow, vertical composition for teardrop flow.
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