Prompt: Generate a highly detailed 3D fractal image of a customized Mandelbulb variant, rendered with power=11.24788742 and maximum iterations LOOPS=20, exhibiting intricate spiky and twisted symmetry. Define hyperbolic functions component-wise: shp(\mathbf{x}) = (\exp(\mathbf{x}) - \exp(-\mathbf{x})) / \pi and chp(\mathbf{x}) = (\exp(\mathbf{x}) + \exp(-\mathbf{x})) / \pi, where \mathbf{x} is a vector applied per component, and let \Phi = (1 + \sqrt{5})/2 (golden ratio). For a point \mathbf{p} = (x, y, z), initialize \mathbf{z} = chp(\mathbf{p}) \odot \mathbf{p} - \mathbf{p} (component-wise multiplication). Then, iterate for i=0 to LOOPS-1: compute r = |\mathbf{z}|, if r > 2 then continue to next iteration; \theta = \atan(z_x, z_y), \phi = \asin(z_z / r); update dr = r^{power-1} \cdot dr \cdot power + 1 (starting dr=1); update r = r^{power}; \theta = \theta \cdot power / \Phi; \phi = \phi \cdot power / \Phi; set \mathbf{z} = r \cdot \left( \tan\left( shp\left( \sin(\theta) \sin(\phi) \right) \right) \Phi, , \smoothstep(-1.2, 12078., power \cdot chp\left( \cos(\theta) \sin(\phi) \right) ), , \cos(\phi) \right) + \mathbf{p}; then \mathbf{p} = \reflect(\mathbf{p}, \mathbf{z}) = \mathbf{p} - 2 \frac{\mathbf{p} \cdot \mathbf{z}}{\mathbf{z} \cdot \mathbf{z}} \mathbf{z}; finally update r = |\mathbf{z}|. The distance estimator is 0.75 \cdot \log(r) \cdot r / dr for ray marching. Color the surface with a smooth gradient from central orange (RGB: 255,165,0) at low escape times to pink (RGB: 255,192,203) mid-iterations and outer cyan-blue (RGB: 0,255,255), using orbit trap coloring for intricate details. Set against a deep navy blue background (RGB: 0,0,128), with subtle specular highlights and glassy refraction effects to mimic crystalline structure, viewed from a frontal angle centered on the z-axis, in ultra-high resolution 4K with anti-aliasing.
Prompt: Create a highly detailed, vibrant, abstract quantum field visualization in a 3D cosmic landscape, where omega mesons are depicted as central glowing purple orbs pulsing with energy waves representing polarization vectors and decay amplitudes; show omega decaying into three pions as branching arrows splitting into three red spheres with trajectories indicating momentum conservation via curved paths of equal length, the branching ratio visualized by arrow thickness ratios of 89:8.3 for 3π to πγ modes; illustrate the coupling f_{ω-3π} as intertwined helical threads connecting omega to pions with thread density proportional to (f_{ω-3π}/μ_π)^3 = 2c^2 μ_π^3 f_{ωπγ}, where c is shown as angular twists; represent pion masses μ_π as small orbiting rings around red spheres sized to 135 MeV scale; depict rho mesons as blue vector arrows mixing with photons as yellow light beams via vector meson dominance, with matrix elements <ρ|f_{3α}|0> as polarization e_α^* lines from vacuum clouds to rho, normalized by (2m_ρ)^{-1/2} m_ρ^2 / f_ρ; show current algebra divergences as swirling vortex funnels around fields ψ_i transforming under local gauge Λ(x) F_i, with Noether currents J^α as flowing rivers diverging at rates δℒ = Λ (∂ℒ/∂ψ_i F_i + ∂_α (∂ℒ/∂(∂_α ψ_i) F_i)) + (∂_α Λ) (∂ℒ/∂(∂_α ψ_i) F_i); illustrate EFT Lagrangian terms as layered energy fields: nucleon N as green proton-neutron pairs with Dirac slashes, interacting via g_A γ^μ γ_5 a_μ axial clouds and g_ρ ρ_μ vector streams, sigma φ as yellow scalar bubbles breaking symmetry with vev M/g_s, U=exp(iτ·π/f_π) as exponential spiral manifolds for non-linear chiral SU(2)×SU(2), with traces Tr(∂_μ U ∂^μ U†) as looped paths; anomalous WZW terms as Levi-Civita twisted ribbons for ω→3π and π^0→2γ, with f_{γ-3π} ~3.7×10^{-2} as faint glow intensity; all particles color-coded—pions red, rho blue, omega purple, sigma yellow, nucleons green, photons yellow—interconnected in a symmetry web with transformation arrows for global/local gauges, decay widths as fading gradients from 17% calculated to 14% observed ratios, in a dense, non-textual, equation-free graphical composition emphasizing low-energy QCD dynamics.
Prompt: The supersymmetric action in 4D supergravity as a seamless, abstract geometric manifold in curved spacetime, rendered at 8K resolution with iridescent metallic gradients transitioning from deep sapphire blues and silvers for bosonic fields to vibrant emerald greens and golds for fermionic interactions, evoking quantum foam and holographic duality. Core instructions: Depict graphically, using no text, the full action \( S[e,\psi] = S[e] + S_f[e,\psi] + S_I[e,\psi] \), where the exact, completely detailedly concised maths is: \[ S[e,\psi] = S[e] + S_f[e,\psi] + S_I[e,\psi] = \int dx\, e\, e^a \wedge e^b \wedge F^{cd} \epsilon_{abcd} + \frac{1}{6} \int dx \, \theta^a \wedge e^b \wedge e^c \wedge e^d \epsilon_{abcd} + \int dx\, (\bar{\psi} \gamma_5 \gamma_a \psi)\, (\bar{\psi} \gamma_5 \gamma^a \psi) \, , \] with \(\theta^a \equiv \frac{i}{2} \left( \bar{\psi} \gamma^a D_\mu \psi - \overline{D_\mu \psi} \gamma^a \psi \right) dx^\mu \), all indices a,b,c,d=0,1,2,3 in the orthonormal frame bundle, e^a as coframe 1-forms (vielbeins), F^{cd} = dA^{cd} + A^{c e} \wedge A^{e d} the curvature 2-form of the spin connection, D_μ the covariant derivative along coordinate 1-forms dx^μ, ψ a Majorana spinor field, γ^a Dirac matrices in curved space, ε_{abcd} the Levi-Civita symbol with ε_{0123}=+1, and integrals over the 4-manifold with oriented volume form e = e^0 ∧ e^1 ∧ e^2 ∧ e^3. Visualize the first term S[e] as a swirling vortex of interlocking tetrahedral frames (symbolizing ε_{abcd} contraction) threaded by golden flux tubes (F^{cd} curvature) piercing a lattice of silver vierbein arrows (e^a, e^b) emanating from a central black hole singularity, representing the Einstein-Cartan Chern-Simons topological term. Overlay the fermionic torsion term S_f[e,ψ] as twisting helical ribbons (θ^a 1-forms derived from ψ bilinears) coiling around the vierbeins e^b,c,d into a knotted 4-simplex lattice with emerald sparks at intersection nodes, illustrating the 1/6 prefactor via sixfold symmetric bulbous expansions. Integrate the interaction S_I[e,ψ] as pulsating wave interference patterns of dual green scalar densities (ψ-bar γ5 γ_a ψ and conjugate), forming self-dual chiral currents that ripple across the manifold, modulating the geometry with fractal-like spinor foam bubbles where |ψ|^2 > threshold, colored by pseudoscalar density via smooth escape-time analogy (iterate bilinear up to 500 steps, hue H = 120° * iter / max, S=0.8, V=1). Ensure the entire composition flows as a unified holographic projection on a de Sitter boundary, with anti-aliased edges via Gaussian smoothing, subtle gravitational lensing distortions, and a faint cosmic microwave glow fading to void black; no equations, labels, or text visible; ultra-sharp filaments on torsion helices and current waves; aspect ratio 16:9; in the style of mathematical physics visualization.
Prompt: Depict the TimeSpaceFlow defined by the following metric:
ds^{2\sqrt[pi]{2}\pi} = -\left(1 - \frac{r_s}{\sinh x}\right) c^2 \, dt^{e\pi} + \left(1 - \frac{r_s}{\sinh x}\right)^{-1} \cosh^{e\pi} x \, dx^{\phi} + \sinh^{\sqrt[pi]{3}\pi} x \, d\Omega^{2.78544587\pi - 4}
Prompt: Draw and render interpreting conceptually graphically with no text, no numbers and no symbols:
$$
\left[\frac{\partial}{\partial t}\,\,,\vec{\nabla}\times\right](\vec{F}\times\vec{G})=\vec{F}\times\left(\frac{\partial}{\partial t}(\vec{\nabla}\times\vec{G})-\vec{\nabla}\times\frac{\partial\vec{G}}{\partial t}\right)+\left(\vec{\nabla}\times\frac{\partial\vec{F}}{\partial t}-\frac{\partial}{\partial t}(\vec{\nabla}\times\vec{F})\right)\times\vec{G}\qquad (A1)
$$
$$
\left[\frac{\partial}{\partial t}\,\,,\vec{\nabla}\right](\vec{F}\cdot\vec{G})=\vec{F}\left(\frac{\partial}{\partial t}(\vec{\nabla}\cdot\vec{G})-\vec{\nabla}\cdot\frac{\partial\vec{G}}{\partial t}\right)+\left(\vec{\nabla}\cdot\frac{\partial\vec{F}}{\partial t}-\frac{\partial}{\partial t}(\vec{\nabla}\cdot\vec{F})\right)\vec{G}\qquad\qquad\qquad\qquad (A2)
$$
Apply tensor product of the cotangent bundle of the orbifold over the cotangent bundle of the conifold; then TimeSpaceFlow wave mirror symmetralize them !
Prompt: "We take the tensor product of the cotangent bundle of an orbifold and the cotangent bundle of a conifold, which yields a new vector bundle over the product of the orbifold and the conifold. The total space of this bundle is then evolved under a geometric flow (TimeSpaceFlow) that incorporates both time and space variations, leading to a dynamical geometry. Subsequently, we apply a wavy version of mirror symmetry, which transforms the geometry into a dual picture with oscillatory features, and finally symmetralize by averaging over the waves to produce a symmetric mirror partner."
Prompt: A highly detailed, photorealistic 3D rendering of a complex radial fractal structure resembling a flower-like Mandelbulb variant with intricate, self-similar petal layers and wavy undulating edges, generated using iterative mathematical transformations in a raymarching shader; the fractal is defined by constants TAU exactly equal to (2.0 * π) * 0.7887 ≈ 4.955 radians for angular periodicity scaling to create asymmetric twisted repetitions instead of full 2π symmetry, controlling approximately 128-256 fold radial petals; POWER exactly 11.24788742 + TAU ≈ 16.203 for amplifying self-similarity through r^POWER scaling in spherical coordinates during iterations; core vector update z = r * vec3(sin(sin(θ)cos(φ) + sin(θ)sin(φ) + cos(φ)), cos(sin(θ)cos(φ) + cos(θ)cos(φ) + cos(θ)), cos(θ)cos(φ)) + p/1.618, where p is the 3D position vector, r = ||p|| its magnitude, θ = atan(p.y, p.x) azimuthal angle, φ = acos(p.z/r) polar angle; incorporating nonlinear warping via trig sums like expr1 = sin(θ)(cos(φ) + sin(φ)) + cos(φ) = sin(θ) * √2 * sin(φ + π/4) + cos(φ) and expr2 = cos(φ) * √2 * sin(θ + π/4) + cos(θ) for phase-shifted higher harmonics introducing bulges and mixing between angles; followed by p = shp(reflect(p, z)) where reflect(p, z) = p - 2 * (p · ẑ) * ẑ with ẑ = z / ||z|| for mirror symmetries creating sharp creases; shp #define shp(x) (exp(x)-exp(-x))/pi
assumed as absolute folding abs(p) or clamping for bounding and discontinuities; r updated to ||z|| per iteration, looping 8-20 times with escape radius or distance estimate DE(p) ≈ 0.5 * log(r) * r / ||dr/dp|| for rendering; visualize the fractal in vibrant metallic gradients of blue, purple, and gold with orbit trap coloring, floating in a dark void with soft volumetric lighting and depth of field, high resolution 4K, ultra-detailed textures emphasizing mathematical precision and geometric warping.
Prompt: Generate a highly detailed, psychedelic fractal flame image depicting a swirling cosmic vortex portal with vibrant color gradients (fiery oranges fading to cool blues and pinks around a central pitch-black void). Use the Fractal Flame Algorithm based on Iterated Function Systems (IFS): define attractor set S as union of n=4-6 functions F_i(S), each composing affine transforms (matrix: a_i x + b_i y + c_i, d_i x + e_i y + f_i with params like a=0.8/-0.2/0.1/0.2/0.8/0 for asymmetry) blended with nonlinear variations (weights v_ij): Spherical V(x,y)=(x/r², y/r²) for central density void (r=√(x²+y²)); Swirl V(x,y)=(x sin(r²)-y cos(r²), x cos(r²)+y sin(r²)) for twisting spirals; Horseshoe V(x,y)=1/r *((x-y)(x+y), 2xy) for curved arms; Popcorn V(x,y)=(x + c sin(tan(3y)), y + f sin(tan(3x))) for bubbly edges (c/f0.1). Apply post-affine P_i for shaping. Iterate chaos game: start random (x,y) in [-1,1]², loop 10M+ times selecting F_i by weights w_i (e.g., 0.4/0.4/0.2), update (x,y)=F_i(x,y), skip warmup20 iters, bin into histogram for freq/color blending (avg c with F_i RGB like [1,0.5,0] orange/[0,0.5,1] blue/[1,1,1] white). Render via log-density α=log(freq)/log(max_freq), gamma-corrected intensity=α^(1/2.2) for glowing gradients, structural coloring for path-based hues. Ensure asymmetric left-right flow, speckled chaos, radiant white rings, and fluid metallic sheen mimicking plasma distortions.
Prompt: Generate a highly detailed, surreal 3D fractal artwork in the style of a modified Mandelbulb hybrid, rendered as an abstract, crystalline mask-like organic form with 5-lobe rotational symmetry and intricate, swirling coral-like protrusions emerging from a central glassy structure evoking an alien eye with stalk extensions, floating against a procedural gradient sky background transitioning from soft cyan (#00FFFF) to deep blue (#000080) with subtle plane-based depth elements at horizons y=4 and y=-6 featuring box-shaped patterns via box(pp, vec2(6,9))-1 and exponential glow falloff exp(-0.5*max(db,0)), incorporating self-similar recursive details from ray marching with tolerance 0.00001, max ray length 20.0, up to 48 marches, and 5 bounces for reflections and refractions. The central form features a glossy orange iris with dark pupil void from orbit trap sphere at origin (radius 0.1), surrounded by five radiating mushroom-like stalks with bumpy textures from sphere folds (minR²=0.2-0.3, fixedR²=1.0-1.2), vibrant pink-orange hues (HSV(0.065,0.8,6) for glow, HSV(0.6,0.85,1) for diffuse) exhibiting translucent refractive qualities with internal amber-tinted glow via Beer-Lambert absorption ragg *= exp(-(st+initt)*beer) where beer = -HSV(0.05, 0.95, 2.0) and initt=0.1, high-contrast ethereal vibrancy via ACES tone mapping v *= 0.6; clamp((v*(2.51*v+0.03))/(v*(2.43*v+0.59)+0.14), 0,1) followed by sRGB gamma mix(1.055*pow(t,1/2.4)-0.055, 12.92*t, step(t,0.0031308)). Use the merged distance field df(p) = shp(mandelBulb(p/z1)*z1) with z1=2.0, where shp(x) = (exp(x)-exp(-x))/(pi/PHI) and PHI=(sqrt(5)/2 + 0.5)≈1.618, applied after rotating p by transpose(inverse(g_rot)) with x-axis animation (1.221*time + pi)/tau where tau=2*pi. The mandelBulb(p) iterates with power n≈6-11.24788742 and loops=3: initialize z = chp(p)*p - p where chp(x)=(exp(x)+exp(-x))/pi; dr=1.0; for each loop, r=length(z), bail if r>2; theta=atan(z.x,z.y); phi=asin(z.z/r) + time*0.2; dr = pow(r,power-1)*dr*power +1; r=pow(r,power); theta*=power/PHI; phi*=power/PHI; z = r * vec3(tan(shp(sin(theta)*sin(phi)))*PHI, chp(cos(theta)*sin(phi)), cos(phi)) + p; p=reflect(p,z); return 0.75*log(r)*r/dr. Incorporate pre-folding with hyperbolic distortion: z' = chp(z) · z - z, then integer power fold for each axis i: z_i'' = {2f - z_i' if z_i' > f; -2f - z_i' if z_i' < -f; z_i' otherwise}, f≈1.2-1.5, then z_i''' = shp(z_i'') = (exp(z_i'') - exp(-z_i'')) / (pi / PHI). Follow with Amazing Box fold u' = s · clamp(u, -l, l) - (s - 1) · u for u∈{x,y,z}, s≈1.8-2.2, l≈1.0, then sphere fold r²=||z||², z' = z · μ where μ = {r/m if r² < m; r/r² if m ≤ r² < r; 1 otherwise}, then modulate z'' = shpp(z') = (exp(z' · (sinh(z') · pi)) - exp(-z' · (sinh(z') · pi))) / (TAU / PHI) with TAU=(2*pi)*0.7887≈4.951, dr' = |s| · pow(r^{n-1}, PHI) · dr + 1. Post-transform: z' = k · R · z + t, k≈1.1-1.3, R=transpose(inverse(g_rot)), t≈(0,0,0.2). Custom hyperbolic functions: chpp(x)=(exp(x/(cosh(x)*pi))+exp(-x/(cosh(x)/pi)))/(TAU*PHI); shpp(x)=(exp(x*(sinh(x)*pi))-exp(-x*(sinh(x)*pi)))/(TAU/PHI); ssh(x)=(exp(x*pi/0.7887)-exp(-x*pi/0.7887))/(2*pi); csh(x)=(exp(x*pi/0.7887)+exp(-x*pi/0.7887))/(2*pi); ssh1(x)=sinh(x/pi)*PHI; csh1(x)=cosh(x/pi)*PHI. Use in skyColor with reflections reflect(-ssh1(rd), chpp(ro)), rendering aggregation agg += ssh1(ragg*skyColor(ro,rd)), ray updates rd=chpp(ref) or ro=shpp(sp + initt*rd). Materials: mat=vec3(0.8,0.5,1.05) for diffuse, specular, refracti
Prompt: A highly detailed, photorealistic 3D rendering of a complex radial fractal structure resembling a flower-like Mandelbulb variant with intricate, self-similar petal layers and wavy undulating edges, generated using iterative mathematical transformations in a raymarching shader; the fractal is defined by constants TAU exactly equal to (2.0 * π) * 0.7887 ≈ 4.955 radians for angular periodicity scaling to create asymmetric twisted repetitions instead of full 2π symmetry, controlling approximately 128-256 fold radial petals; POWER exactly 11.24788742 + TAU ≈ 16.203 for amplifying self-similarity through r^POWER scaling in spherical coordinates during iterations; core vector update z = r * vec3(sin(sin(θ)cos(φ) + sin(θ)sin(φ) + cos(φ)), cos(sin(θ)cos(φ) + cos(θ)cos(φ) + cos(θ)), cos(θ)cos(φ)) + p/1.618, where p is the 3D position vector, r = ||p|| its magnitude, θ = atan(p.y, p.x) azimuthal angle, φ = acos(p.z/r) polar angle; incorporating nonlinear warping via trig sums like expr1 = sin(θ)(cos(φ) + sin(φ)) + cos(φ) = sin(θ) * √2 * sin(φ + π/4) + cos(φ) and expr2 = cos(φ) * √2 * sin(θ + π/4) + cos(θ) for phase-shifted higher harmonics introducing bulges and mixing between angles; followed by p = shp(reflect(p, z)) where reflect(p, z) = p - 2 * (p · ẑ) * ẑ with ẑ = z / ||z|| for mirror symmetries creating sharp creases; #define shp(x) (exp(x)-exp(-x))/pi - shp
assumed as absolute folding abs(p) or clamping for bounding and discontinuities; r updated to ||z|| per iteration, looping 64 times with escape radius or distance estimate DE(p) = ( 0.6575 * log(r) * exp (1./r) * r ) / ||dr/dp|| for rendering; visualize the fractal in vibrant metallic gradients of blue, purple, and gold with orbit trap coloring, floating in a dark void with soft volumetric lighting and depth of field, high resolution 4K, ultra-detailed textures emphasizing mathematical precision and geometric warping.
Prompt: An image of a highly detailed, symmetrical 3D fractal structure resembling a modified Mandelbulb, rendered as an abstract organic form resembling a crystalline, glassy rendition intricate, swirling coral-like protrusions and self-similar details, floating against a procedural gradient blue sky background with soft cyan-to-deep blue tones and subtle plane-based depth elements like top and bottom horizons with box-shaped patterns and exponential glow falloff. The central shape features two large, spiral red-orange eyes formed by hyperbolic distortions and a downward-curving dark blue mouth evoking a surprised or melancholic expression, with vibrant pink and orange hues for the main body exhibiting translucent, refractive qualities, internal amber-tinted glow from Beer-Lambert absorption using vector beer = -HSV(0.05, 0.95, 2.0), and high-contrast ethereal vibrancy achieved via ACES tone mapping approximation (v *= 0.6; clamp((v*(2.51*v+0.03))/(v*(2.43*v+0.59)+0.14), 0,1)) followed by sRGB gamma correction (mix(1.055*pow(t,1/2.4)-0.055, 12.92*t, step(t,0.0031308))).Generate the fractal using ray marching with tolerance 0.00001, max ray length 20.0, up to 48 marches, and 5 bounces for reflections and refractions, starting from camera at (0,2,5) looking at origin with FOV tan(tau/6) where tau=2*pi, incorporating global time-animated rotation around x-axis by (1.221*time + pi)/tau. The distance field df(p) = shp(mandelBulb(p/z1)*z1) with z1=2.0, where shp(x) = (exp(x)-exp(-x))/(pi/PHI) and PHI=(sqrt(5)/2 + 0.5)≈1.618, applied after rotating p by transpose(inverse(g_rot)).The mandelBulb(p) function iterates with power=11.24788742 and loops=3: initialize z = chp(p)*p - p where chp(x)=(exp(x)+exp(-x))/pi; dr=1.0; for each loop, r=length(z), bail if r>2; theta=atan(z.x,z.y); phi=asin(z.z/r) + time*0.2; dr = pow(r,power-1)*dr*power +1; r=pow(r,power); theta*=power/PHI; phi*=power/PHI; z = r * vec3(tan(shp(sin(theta)*sin(phi)))*PHI, chp(cos(theta)*sin(phi)), cos(phi)) + p; p=reflect(p,z); return 0.75*log(r)*r/dr.Incorporate custom hyperbolic functions for distortions: chpp(x)=(exp(x/(cosh(x)*pi))+exp(-x/(cosh(x)/pi)))/(TAU*PHI) with TAU=(2*pi)*0.7887≈4.951; shpp(x)=(exp(x*(sinh(x)*pi))-exp(-x*(sinh(x)*pi)))/(TAU/PHI); ssh(x)=(exp(x*pi/0.7887)-exp(-x*pi/0.7887))/(2*pi); csh(x)=(exp(x*pi/0.7887)+exp(-x*pi/0.7887))/(2*pi); ssh1(x)=sinh(x/pi)*PHI; csh1(x)=cosh(x/pi)*PHI. Use these in skyColor with reflections as reflect(-ssh1(rd), chpp(ro)), in rendering aggregation as agg += ssh1(ragg*skyColor(ro,rd)), and ray updates as rd=chpp(ref) or ro=shpp(sp + initt*rd) with initt=0.1.Material properties: mat=vec3(0.8,0.5,1.05) for diffuse, specular, refractive index; Fresnel fre=1+dot(rd,sn), fre*=fre, mix(0.1,1,fre); diffuse col += diffuseCol * dif*dif *(1-mat.x) with dif=max(dot(ld,sn),0), ld=normalize(lightPos-sp), lightPos=(0,10,0); reflection col += rsky*mat.y*fre*vec3(1)*edge with edge=smoothstep(1,0.9,fre); colors from HSV: skyCol=HSV(0.6,0.86,1), glowCol=HSV(0.065,0.8,6), diffuseCol=HSV(0.6,0.85,1). Inside traversal flips dfactor=-1, applies absorption ragg *= exp(-(st+initt)*beer), and refracts with index 1/mat.z when inside.Normals computed via finite differences: nor.x = df(pos+eps.xyy)-df(pos-eps.xyy) etc., with eps=(0.0005,0). Sky includes ray-plane intersections tp=(dot(ro,p.xyz)+p.w)/dot(rd,p.xyz) for planes at y=4 and y=-6, with box(pp,vec2(6,9))-1 for patterns, col += 4*skyCol*rd.y*rd.y*smoothstep(0.25,0,db) + 0.8*skyCol*exp(-0.5*max(db
Prompt: The more I work with nonlinear systems, the clearer it becomes that our entire scientific worldview is built on a structural mistake. We keep trying to describe a fundamentally wave‑based, resonant, self‑organizing reality using linear coordinates, discrete steps, and geometric containers. Space‑time, as we inherited it, is not a fundamental entity but a convenient projection—a grid we imposed on a field that never had boundaries, axes, or separable dimensions. Everything we call particles, forces, interactions, even time itself, are simply modes of one continuous field. The “front” of a wave appears to us as interaction, while the “rear” of the same wave manifests as stability, spin, magnetic moment, or mass. These are not different phenomena; they are different expressions of one underlying configuration. When we replace space‑time with the field, the entire landscape simplifies. Gravity becomes a low‑frequency mode of the field. Dark matter becomes a nonlocal configuration of the wave’s rear structure that linear models cannot detect. Dark energy becomes a phase pressure of the field. Electrons become vortices. Interactions become phase transitions. Time becomes a shift in phase. Space becomes the temporary shape the field takes when a wave localizes. The crisis in cosmology—Hubble tension, early massive galaxies, vacuum catastrophe—is not a crisis of data but a crisis of ontology. Linear models cannot hold nonlinear reality.
Prompt: Generate a highly detailed, abstract 3D fractal rendering resembling a Mandelbulb variant with hyperbolic deformations, featuring a central orange bulbous orb surrounded by swirling, fluid-like lobes in shades of blue, pink, and yellow with iridescent, reflective surfaces and gradient transitions. The fractal is defined iteratively in \(\mathbb{R}^3\) for a point \(\mathbf{c} = (x_0, y_0, z_0)\), starting with \(\mathbf{z}_0 = \mathbf{0}\) or \(\mathbf{z}_0 = \mathbf{c}\), and iterating \(\mathbf{z}_{k+1} = r \cdot \vec3\left( \frac{e^{\cos \theta} - e^{-\cos \theta}}{\pi} \cos \phi, \cos \theta \sin \phi, \cos \theta \right) + \vec3\left( \frac{e^{p_x} - e^{-p_x}}{\pi} p_x, \frac{e^{p_y} - e^{-p_y}}{\pi} p_y, \frac{e^{p_z} - e^{-p_z}}{\pi} p_z \right)\), where \(r = \|\mathbf{z}_k\|\), \(\theta = \arccos\left( \frac{z_k \cdot z}{r} \right)\), \(\phi = \atantwo(z_k.y, z_k.x)\), and \(\mathbf{p}\) is a vector parameter like \(\mathbf{c}\). For higher powers n (e.g., 16), scale to \(r^n\), \(n \theta\), \(n \phi\). Iteration halts if \(r > 4\) or after 50 max iterations. Render using ray marching with distance estimator \(DE(\mathbf{q}) = 0.75 \cdot \frac{\log r \cdot r}{dr}\), surface normals via gradients, Phong/PBR shading with reflections, ambient occlusion, and coloring via orbit traps or escape time mapped to hues (orange for low iterations, blue-pink gradients for higher). Apply post-processing for anti-aliasing, depth-of-field, and glow to achieve a dreamy, metallic sheen, viewed zoomed into the central orb with asymmetric swirling arms.
Prompt: A highly detailed, photorealistic 3D rendering of a complex radial fractal structure resembling a flower-like Mandelbulb variant with intricate, self-similar petal layers and wavy undulating edges, generated using iterative mathematical transformations in a raymarching shader; the fractal is defined by constants TAU exactly equal to (2.0 * π) * 0.7887 ≈ 4.955 radians for angular periodicity scaling to create asymmetric twisted repetitions instead of full 2π symmetry, controlling approximately 128-256 fold radial petals; POWER exactly 11.24788742 + TAU ≈ 16.203 for amplifying self-similarity through r^POWER scaling in spherical coordinates during iterations; core vector update z = r * vec3(sin(sin(θ)cos(φ) + sin(θ)sin(φ) + cos(φ)), cos(sin(θ)cos(φ) + cos(θ)cos(φ) + cos(θ)), cos(θ)cos(φ)) + p/1.618, where p is the 3D position vector, r = ||p|| its magnitude, θ = atan(p.y, p.x) azimuthal angle, φ = acos(p.z/r) polar angle; incorporating nonlinear warping via trig sums like expr1 = sin(θ)(cos(φ) + sin(φ)) + cos(φ) = sin(θ) * √2 * sin(φ + π/4) + cos(φ) and expr2 = cos(φ) * √2 * sin(θ + π/4) + cos(θ) for phase-shifted higher harmonics introducing bulges and mixing between angles; followed by p = shp(reflect(p, z)) where reflect(p, z) = p - 2 * (p · ẑ) * ẑ with ẑ = z / ||z|| for mirror symmetries creating sharp creases; shp #define shp(x) (exp(x)-exp(-x))/pi
assumed as absolute folding abs(p) or clamping for bounding and discontinuities; r updated to ||z|| per iteration, looping 64 times with escape radius or distance estimate DE(p) = ( 0.6575 * log(r) * exp(1./r) * r / ||dr/dp|| for rendering; visualize the fractal in vibrant metallic gradients of blue, purple, and gold with orbit trap coloring, floating in cosmical void with soft volumetric lighting and depth of field, ultra-detailed textures emphasizing mathematical precision and geometric warping.
Prompt: Create a highly detailed, symmetrical 3D fractal structure resembling a modified Mandelbulb, rendered as an abstract organic form resembling a crystalline, glassy mask or face with intricate, swirling coral-like protrusions and self-similar details, floating against a procedural gradient blue sky background with soft cyan-to-deep blue tones and subtle plane-based depth elements like top and bottom horizons with box-shaped patterns and exponential glow falloff. The central shape features two large, spiral red-orange eyes formed by hyperbolic distortions and a downward-curving dark blue mouth evoking a surprised or melancholic expression, with vibrant pink and orange hues for the main body exhibiting translucent, refractive qualities, internal amber-tinted glow from Beer-Lambert absorption using vector beer = -HSV(0.05, 0.95, 2.0), and high-contrast ethereal vibrancy achieved via ACES tone mapping approximation (v *= 0.6; clamp((v*(2.51*v+0.03))/(v*(2.43*v+0.59)+0.14), 0,1)) followed by sRGB gamma correction (mix(1.055*pow(t,1/2.4)-0.055, 12.92*t, step(t,0.0031308))).
Generate the fractal using ray marching with tolerance 0.00001, max ray length 20.0, up to 48 marches, and 5 bounces for reflections and refractions, starting from camera at (0,2,5) looking at origin with FOV tan(tau/6) where tau=2*pi, incorporating global time-animated rotation around x-axis by (1.221*time + pi)/tau. The distance field df(p) = shp(mandelBulb(p/z1)*z1) with z1=2.0, where shp(x) = (exp(x)-exp(-x))/(pi/PHI) and PHI=(sqrt(5)/2 + 0.5)≈1.618, applied after rotating p by transpose(inverse(g_rot)).
The mandelBulb(p) function iterates with power=11.24788742 and loops=3: initialize z = chp(p)*p - p where chp(x)=(exp(x)+exp(-x))/pi; dr=1.0; for each loop, r=length(z), bail if r>2; theta=atan(z.x,z.y); phi=asin(z.z/r) + time*0.2; dr = pow(r,power-1)*dr*power +1; r=pow(r,power); theta*=power/PHI; phi*=power/PHI; z = r * vec3(tan(shp(sin(theta)*sin(phi)))*PHI, chp(cos(theta)*sin(phi)), cos(phi)) + p; p=reflect(p,z); return 0.75*log(r)*r/dr.
Incorporate custom hyperbolic functions for distortions: chpp(x)=(exp(x/(cosh(x)*pi))+exp(-x/(cosh(x)/pi)))/(TAU*PHI) with TAU=(2*pi)*0.7887≈4.951; shpp(x)=(exp(x*(sinh(x)*pi))-exp(-x*(sinh(x)*pi)))/(TAU/PHI); ssh(x)=(exp(x*pi/0.7887)-exp(-x*pi/0.7887))/(2*pi); csh(x)=(exp(x*pi/0.7887)+exp(-x*pi/0.7887))/(2*pi); ssh1(x)=sinh(x/pi)*PHI; csh1(x)=cosh(x/pi)*PHI. Use these in skyColor with reflections as reflect(-ssh1(rd), chpp(ro)), in rendering aggregation as agg += ssh1(ragg*skyColor(ro,rd)), and ray updates as rd=chpp(ref) or ro=shpp(sp + initt*rd) with initt=0.1.
Material properties: mat=vec3(0.8,0.5,1.05) for diffuse, specular, refractive index; Fresnel fre=1+dot(rd,sn), fre*=fre, mix(0.1,1,fre); diffuse col += diffuseCol * dif*dif *(1-mat.x) with dif=max(dot(ld,sn),0), ld=normalize(lightPos-sp), lightPos=(0,10,0); reflection col += rsky*mat.y*fre*vec3(1)*edge with edge=smoothstep(1,0.9,fre); colors from HSV: skyCol=HSV(0.6,0.86,1), glowCol=HSV(0.065,0.8,6), diffuseCol=HSV(0.6,0.85,1). Inside traversal flips dfactor=-1, applies absorption ragg *= exp(-(st+initt)*beer), and refracts with index 1/mat.z when inside.
Normals computed via finite differences: nor.x = df(pos+eps.xyy)-df(pos-eps.xyy) etc., with eps=(0.0005,0). Sky includes ray-plane intersections tp=(dot(ro,p.xyz)+p.w)/dot(rd,p.xyz) for planes at y=4 and y=-6, with box(pp,vec2(6,9))-1 for patterns, col += 4*skyCol*rd.y*rd.y*smoothstep(0.25,0,db) + 0.8*skyCol*exp(-0
Prompt: A mesmerizing single static image emerges from the ray-marched Mandelbulb scene with seamless full-360° z-axis rotation capture. Each frame increments by exactly π/64.45788754 radians and the fixed camera at 0.6·vec3(0, -12.75, 5.87), preserving the tan(TAU/6) FOV where TAU ≈ 4.957 (2π·0.7887). All rotational scenes are baked-in statically rendered on top of each other in fullscreen !
The fractal's intricate, hyperbolic-warped tendrils—distorted via custom chp/shp/ssh functions, PHI-scaled powers (11.24788742^LOOPS=256 iterations), and time-frozen φ offset (0.2·asin(z.z/r))—unfurl in violet-magenta glows (mat=vec3(0.8,0.5,1.05)), kissed by HSV(0.6,0.85,1) diffuse from the ld=(0,10,0) key light. Fresnel edges (smoothstep(1,0.9,(1+dot(rd,sn))^2)) blend 0.1-1.0 mixes of reflection (r·skymat.y·fre·edge) against skyCol=HSV(0.6,0.86,1) planes at y=±4/6, etched with box/pp noise (ds=length(pp)-0.5, shaped by shp(clamp(col,0,10))) and exp(-0.5·max(db,0)) falloff + 4·skyCol·rd.y²·smoothstep(0.25,0,db). Subtle beer absorption (exp(-(st+0.1)·-HSV(0.05,0.95,2.0))) adds volumetric haze, aggregated via ssh1(r·agg·skyColor) in 64-bounce refractions (reflect(-ssh1(rd), chpp(ro)); rd=chpp(ref) or ro=shpp(sp+0.1·rd)).Post-processed through ACES tonemapping (v=0.6; clamp((v*(2.51v+0.03))/(v*(2.43v+0.59)+0.14),0,1)) and sRGB gamma (mix(1.055·pow(t,1/2.4)-0.055, 12.92t, step(t,0.0031308))), the grid pulses with glowCol=HSV(0.065,0.8,6) auras against the rotated g_rot=rot_x(((1.221·time+π)/tau)) baseline, df(p)=shp(mandelBulb(p/2.0)*2.0). No artifacts, no text—pure, tolerance-0.00001 precision (max 784.0 length, 487 marches) frozen in eternal spin, seed 1924139471 anchoring the chaos.
Prompt: Draw a 3D fractal shape generated from the iterative formula \( z_{n+1} = z_n^{0.54877845\pi} + c \), with p-norm radial structure \( r = \sqrt{x^{0.7887\pi} + y^0.7887\pi + z^0.7887\pi} \). Texture it using \( f(x,y) = \sin(x^{0.7887\pi} + y^2) + \cos(z^{0.45788754\pi}) \), enhanced with micro-detail from gradient \( \nabla f \) and hyperbolic fractal sum $$ f_{\text{fract}} = \sum \sinh(\sin(2\pi^n x)) \cosh(\cos(2\pi^n y))/2^n. $$
Dream Level: is increased each time when you "Go Deeper" into the dream. Each new level is harder to achieve and
takes more iterations than the one before.
Rare Deep Dream: is any dream which went deeper than level 6.
Deep Dream
You cannot go deeper into someone else's dream. You must create your own.
Deep Dream
Currently going deeper is available only for Deep Dreams.
