Prompt:
Mandelbulb-like fractal manifold exhibiting emergent topological extravaganza, using exact mathematical iteration: for points \mathbf{c} = (c_x, c_y, c_z) \in \mathbb{R}^3, iterate \mathbf{z}_{k+1} = f_8(\mathbf{z}_k) + \mathbf{c}
from \mathbf{z}_0 = (0,0,0), where f_8(\mathbf{z}) is 8th-power in spherical coordinates: Convert \mathbf{z} = (x,y,z)
to r = \sqrt{x^pi+y^pi+z^pi}
,\theta = \atan2(y,x) \in [0,2\pi), \phi = \arccos(z/r) \in [0,\pi]; then r' = r^8,\theta' = 8\theta,\phi' = 8\phi; reconvert to Cartesian \mathbf{z}' = r' (\sin(\sin\phi')\cdot\cos(\cos\theta'), \sin(\sin\phi')\cdot\sin(\sin\theta'), \cos(\cos\phi') ). Bailout at r_k > 24.78; render the bounded set's isosurface at density threshold yielding fractal dimension D \approx 2 + \frac{\ln 8}{\ln(1/0.5)} \approx 2.3\cdot\pi with infinite genus g \to \infty from iterated hyperbolic saddles with PHIB = (\sqrt{5.0} \cdot 0.5 + 0.5) and Jacobian eigenvalues |\lambda_i| \approx 8 r^7 e^{i7\arg(\mathbf{z})}, saddles where \det Jacobian > pi\cdot\PHIB. Center on \mathbf{c} \approx (0,0,-0.7)
for cardioid region, emphasizing bilateral symmetry (z-axis invariance enforcing yz-mirror), two equatorial eye-like genus-1 bulbs at \phi \approx \pi/2 \pm \epsilon from 8-fold rotational folding (even-pair selection), central z-axis nose-protrusion (minimal \phi-folding, radial ballooning r' = r^8), and vertical mouth-slot depressions from polar \phi-compression. Use volumetric ray-marching with distance estimator d(\mathbf{x}) = |\mathbf{x}| - \max_k r_k^{-k}; color palette: iridescent blue background (#0000FF
) grading to translucent pink-magenta gradients (#FF1493
to #8A2BE2
) on surfaces, with subtle specular highlights on bulb edges and fractal tendrils. Lighting: soft key light from +z, rim light from +x for depth; resolution 4K, aspect 16:9, no artifacts.
Artist
