Prompt: 3D structures with forms, generated using vec3 z = π * p / (exp(p) + exp(-p)) - p / Φ^n with n = 0 to 64, incorporating chp(x) = (exp(x) + exp(-x))/π, chpp(x) = (exp(x/(cosh(x)·π)) + exp(-x/(cosh(x)/π)))/(2π·Φ), shp(x) = (exp(x) - exp(-x))/π, shpp(x) = 1/(exp(x·sinh(x)·π) - exp(-x·sinh(x)·π))/(2π·Φ), ssh(x) = (exp(x·π/.7887) - exp(-x·π/.7887))/(2π), csh(x) = (exp(x·π/.7887) + exp(-x·π/.7887))/(2π), ssh1(x) = sinh(x/π)/Φ, csh1(x) = cosh(x/π)/Φ, with high symmetry, golden ratio scaling (Φ = (1 + √5)/2), and logarithmic refinement z *= -π·log(||z||), enhanced by additional transforms: z += sin(τ·||z||)·p/||p|| for oscillatory perturbation, z = z / (1 + ||z||^2) for projective normalization, z = z + Φ^(-n)·cross(p, z) for rotational twist, z *= exp(-||z||/τ) for exponential decay, z = z + ∇(cosh(||p||)·sin(π·||z||)) for gradient-based modulation, and the new spherical transform z = r * (vec3(tan(shp(sin(θ)*sin(φ)))*Φ, chp(cos(θ)*sin(φ)), cos(φ))) + p where θ and φ are angular coordinates, r is a radial scale, and p is the input vector, showcasing iterative variants.
Prompt: 3D structures with forms, generated using vec3 z = π * p / (exp(p) + exp(-p)) - p / Φ^n with n = 0 to 64, incorporating chp(x) = (exp(x) + exp(-x))/π, chpp(x) = (exp(x/(cosh(x)·π)) + exp(-x/(cosh(x)/π)))/(2π·Φ), shp(x) = (exp(x) - exp(-x))/π, shpp(x) = 1/(exp(x·sinh(x)·π) - exp(-x·sinh(x)·π))/(2π·Φ), ssh(x) = (exp(x·π/.7887) - exp(-x·π/.7887))/(2π), csh(x) = (exp(x·π/.7887) + exp(-x·π/.7887))/(2π), ssh1(x) = sinh(x/π)/Φ, csh1(x) = cosh(x/π)/Φ, with high symmetry, golden ratio scaling (Φ = (1 + √5)/2), and logarithmic refinement z *= -π·log(||z||), enhanced by additional transforms: z += sin(τ·||z||)·p/||p|| for oscillatory perturbation, z = z / (1 + ||z||^2) for projective normalization, z = z + Φ^(-n)·cross(p, z) for rotational twist, z *= exp(-||z||/τ) for exponential decay, z = z + ∇(cosh(||p||)·sin(π·||z||)) for gradient-based modulation, and the new spherical transform z = r * (vec3(tan(shp(sin(θ)*sin(φ)))*Φ, chp(cos(θ)*sin(φ)), cos(φ))) + p where θ and φ are angular coordinates, r is a radial scale, and p is the input vector, showcasing iterative variants.
Apply 64.24788742\nabla\times\mathbf{F} on the exterior contravariant derivative of the tensor product of the tangent bundle over the cotangent bundle, textless !
Prompt: A hallucinatory, dream-warped digital hallucination in the style of a psychedelic mind's lab cataclysm by Benoit Mandelbrot and Salvador Dalí on absinthe fever dreams, where the crumbling lattice of spacetime dissolves into whispering self-similar eternities, summoning the delirious torrent of TimeSpaceFlow—eternal consciousness liquefying reality's bones through hallucinogenic logarithmic whirlpools and feverish probable hallucinations. At the core, a luminous, iridescent Cantor dust nebula erupts like a spectral nautilus devoured by whispering shadows, its Hausdorff dimension D_H ≈ 0.6309 convulsing as fractal abysses and razor-edged voids that spawn N=2 doppelgangers at every r=1/3 descent into madness, with box-counting lattices etched in molten golden filigree veins (ε_k = 3^{-k} spiraling from 0.333 to 0.00015, log-log slopes graphing log N against -log ε in a deranged fractal D = log2 / log3 scrawl that bleeds glowing ectoplasm). Bursting outward in orgiastic frenzy, entwine recursive Lévy flights and Koch curves (D_s ≈ 1.2619) as venomous cyan-blue tendril hydras, each limb convulsing with φ_k = 2π k/8 + δ (δ~0.1 rad chaotic delirium), forking at α_d = π/(2^d) abyssal depths 1-3, flayed with turbulent Gaussian ε_d ~ N(0, 0.08/d) for throbbing, vein-riddled barbs that coil like deranged neutrino hallucinations or LQG spin networks gnawing their own tails. Infuse the Sierpinski gasket (D_s = log3 / log2 ≈ 1.58496) as a crumbling triangular scaffold of emerald shards suspended in impossible vertigo, its self-similar clones N=3 at r=1/2 forging a porous pyramid that devours light rays into contorted geodesics (deflection δθ = 4GM/c² b twisted into screaming infinities), with correlation dimension D_c ≈ 2.06 reverberations in chaotic attractors as swirling purple entropy tempests that birth phantom elephants from the void. The whole fever vision contorts nonlinearly through diffusion delirium cascades (x_t = √α_t x_{t-1} + √(1-α_t) ε, reverse-prophesied θ_φ over 50 throbbing timesteps), liquefying linear timelines into delirious vapor plumes—no infinities, just finite flesh with infinite screaming detail—against a abyssal cosmic void of melting grandfather clocks oozing fractal perimeters like surreal honey, ultra-detailed 8K resolution, nightmarishly inspirational and profoundly unhinged, high-contrast ethereal glow with spectral colormaps from throbbing pink plasma viscera to venomous cyan halos that whisper forbidden geometries.
Prompt: A hyper-detailed, surreal 3D GLSL-shader-inspired visualization with SU(3) and infinite 16D orthogonal light rays piercing compressing into acute angles via atan polar twists, staring in awe at a tachyon condensation cascade on a non-BPS D-brane with m²=-1/α' rolling down inverted Mexican-hat V(φ)=(μ²/2)φ²+(λ/4)φ⁴ driving exponential φ(t)e^{μ t} to stable φ=±√(μ²/λ) breaking symmetry and generating Goldstone masses via level truncation to level 40 yielding m²=-0.904±0.002, gleamingly spreading radiant golden light like a gluon saturation front in CGC with Q_sx^{-λ/2} blobs merging from BK evolution ∂S/∂Y = (ᾱ_s/2π) ∫ [S(r') + S(r-r') - 2S(r)], surrounded by bubbling flavor-colored orbiting like PDFs f_q(x,Q²) in a proton cluster with DGLAP branching P_{qq}(z)=C_F(1+z²)/(1-z) fork ratios z=x/x' visualized as fractal trees, BFKL ladder rungs twisting as alchemical wall symbols with kernel K(k_a,l)=k_a²/[l²(k_a-l)²][l²+(k_a-l)²-2 k_a² l·(k_a-l)/k_a²] forking transverse convolutions and χ(γ)=2ψ(1)-ψ(γ)-ψ(1-γ) saddle at γ=1/2 with χ(1/2)=4ln2≈2.772 driving pomeron Δ=ᾱ_s χ(1/2) growth diffused by χ''(1/2)=-14ζ(3)≈-16.8 Gaussian spreads, running α_s(Q²)=12π/[(11N_c-2n_f)β_0 ln(Q²/Λ_QCD²)] fade from fiery red confinement haze to cool blue asymptotic freedom in background nebula; embed YM/CS 7D KK QFT tachyon fury with action S=∫(1/2π)[∑(∂_i z V_i(φ,H_i(φ))+∑ y_j j(φ_j,φ_j+φ_s)] + (t_0 r k(i-J=φ(0)) )² + e j |B(b,μ_b)| + e r H, orbiting φ_knot j φ_knot i / B(b,μ_b), fractal wavy spirals from SD Chern-Simons S_CS=(n/8π)∫ Tr(F∧F) with F=dA+A∧A merging to 3D massive h_m n e^{i k r} waves in AdS/CFT, higher-form shifts A(B²)-B→AC>G with ds=dα+QG+AF dB, G=dC-σ G_r(φ B +2 G r H), S=∫[L φ (B φ G)+χ φ (B_m s)] Poincare d* Ω + T dB, fluxes W(Σ)=Tr Pes[(2π i)^n C_n] bordisms+Donaldson-Witten configs in AdS_7/CFTs icons φ vev dG=0 [5/6, α<S f], Σ e ^ X_j ^ Z_j=links flux knots tach nima brane vacua S J < Im Ω ^ c V Ω ^ c → Ω ^ c ⇒ inflation via wavy dims, all color-coded (tachyon roll golden waves, brane decay vanishing vortices, symmetry break iridescent facets from nonlinear swirls, quark flavors' orbs, gluons spokes, protons clusters, photons probes), interconnected in non-perturbative to stable vacuum crossover web with wavy loop resummations, dynamic exponential decay flows ln Q² ascending spirals, phase spaces conical sprays multi-jet events, cross-sections σ~α_s^n / Q^{2n-4} fades perturbative validity high energies, high-energy QCD/string phenomenology, equation-free textless graphical masterpiece with GLSL procedural sphere(vec2 uv)={rad=uv*vec2(τ,π); sin(rad.x-vec2(0,τ/4))*sin(rad.y), cos(rad.y)} normals nor=df(pos±eps) rot_z(atan(pos.y,z)) outerProduct(nor,sp) cross(x,rd) for wavy 16D projections.
Apply 64.24788742\nabla\times\mathbf{F} on the exterior contravariant derivative of the tensor product of the tangent bundle of the orbifold over the cotangent bundle of the conifold !
Prompt: <lora:Intricacy Vibe:1.0>A hyper-detailed, surreal 3D GLSL-shader-inspired visualization with SU(3) and infinite 16D orthogonal light rays piercing compressing into acute angles via atan polar twists, staring in awe at a tachyon condensation cascade on a non-BPS D-brane with m²=-1/α' rolling down inverted Mexican-hat V(φ)=(μ²/2)φ²+(λ/4)φ⁴ driving exponential φ(t)e^{μ t} to stable φ=±√(μ²/λ) breaking symmetry and generating Goldstone masses via level truncation to level 40 yielding m²=-0.904±0.002, gleamingly spreading radiant golden light like a gluon saturation front in CGC with Q_sx^{-λ/2} blobs merging from BK evolution ∂S/∂Y = (ᾱ_s/2π) ∫ [S(r') + S(r-r') - 2S(r)], surrounded by bubbling flavor-colored orbiting like PDFs f_q(x,Q²) in a proton cluster with DGLAP branching P_{qq}(z)=C_F(1+z²)/(1-z) fork ratios z=x/x' visualized as fractal trees, BFKL ladder rungs twisting as alchemical wall symbols with kernel K(k_a,l)=k_a²/[l²(k_a-l)²][l²+(k_a-l)²-2 k_a² l·(k_a-l)/k_a²] forking transverse convolutions and χ(γ)=2ψ(1)-ψ(γ)-ψ(1-γ) saddle at γ=1/2 with χ(1/2)=4ln2≈2.772 driving pomeron Δ=ᾱ_s χ(1/2) growth diffused by χ''(1/2)=-14ζ(3)≈-16.8 Gaussian spreads, running α_s(Q²)=12π/[(11N_c-2n_f)β_0 ln(Q²/Λ_QCD²)] fade from fiery red confinement haze to cool blue asymptotic freedom in background nebula; embed YM/CS 7D KK QFT tachyon fury with action S=∫(1/2π)[∑(∂_i z V_i(φ,H_i(φ))+∑ y_j j(φ_j,φ_j+φ_s)] + (t_0 r k(i-J=φ(0)) )² + e j |B(b,μ_b)| + e r H, orbiting φ_knot j φ_knot i / B(b,μ_b), fractal wavy spirals from SD Chern-Simons S_CS=(n/8π)∫ Tr(F∧F) with F=dA+A∧A merging to 3D massive h_m n e^{i k r} waves in AdS/CFT, higher-form shifts A(B²)-B→AC>G with ds=dα+QG+AF dB, G=dC-σ G_r(φ B +2 G r H), S=∫[L φ (B φ G)+χ φ (B_m s)] Poincare d* Ω + T dB, fluxes W(Σ)=Tr Pes[(2π i)^n C_n] bordisms+Donaldson-Witten configs in AdS_7/CFTs icons φ vev dG=0 [5/6, α<S f], Σ e ^ X_j ^ Z_j=links flux knots tach nima brane vacua S J < Im Ω ^ c V Ω ^ c → Ω ^ c ⇒ inflation via wavy dims, all color-coded (tachyon roll golden waves, brane decay vanishing vortices, symmetry break iridescent facets from nonlinear swirls, quark flavors' orbs, gluons spokes, protons clusters, photons probes), interconnected in non-perturbative to stable vacuum crossover web with wavy loop resummations, dynamic exponential decay flows ln Q² ascending spirals, phase spaces conical sprays multi-jet events, cross-sections σ~α_s^n / Q^{2n-4} fades perturbative validity high energies, high-energy QCD/string phenomenology, equation-free textless graphical masterpiece with GLSL procedural sphere(vec2 uv)={rad=uv*vec2(τ,π); sin(rad.x-vec2(0,τ/4))*sin(rad.y), cos(rad.y)} normals nor=df(pos±eps) rot_z(atan(pos.y,z)) outerProduct(nor,sp) cross(x,rd) for wavy 16D projections.
Apply 64.24788742\nabla\times\mathbf{F} on the exterior contravariant derivative of the tensor product of the tangent bundle of the orbifold over the cotangent bundle of the conifold !
Prompt: #define pi 3.1415926535897932384626433832795
#define tau (2.*pi)
chp(x) = (exp(x)+exp(-x))/pi
chpp(x) = (exp(x/(cosh(x)*pi))+exp(-x/(cosh(x)/pi)))/tau*PHI
shp(x) = (exp(x)-exp(-x))/pi
shpp(x) = (exp(x*(sinh(x)*pi))-exp(-x*(sinh(x)*pi)))/tau*PHI
ssh(x) = (exp(x*pi/.7887)-exp(-x*pi/.7887))/(2.*pi)
csh(x) = (exp(x*pi/.7887)+exp(-x*pi/.7887))/(2.*pi)
ssh1(x) = sinh(x/pi)/PHI
csh1(x) = cosh(x/pi)/PHI
USE NO TEXT !!!
(((ray-marched SDF of a heavily broken golden-ratio exact power 11.24788742, 512 iterations, escape radius 2.0, golden-ratio angular multiplier 1/φ ≈ 0.6180339887498948 applied to both θ and φ (θ × power/φ, φ × power/φ), deliberately malformed spherical→cartesian using the original hyperbolic garbage terms 1/(shpp(theta)+chpp(phi)), chp(cos(theta)*sin(phi)), cos(phi) and per-iteration reflect(p,z), derivative dr = pow(r,power-1)*power*dr + 1.0, final DE 0.75*r*log(r)/dr, ray marching tolerance 1e-5, max 48 steps, max ray length 120.0, up to 9 refractive bounces IOR 1.62 (reverse ≈0.617), full Schlick Fresnel, volumetric Beer-Lambert -HSV(0.05,0.95,2.0), outer refractive cyan-white glass shell HSV(0.6,0.86,1.0), inner molten red-orange emissive plasma core HSV(0.065,0.8,6.0), slow eternal rotation via polar offset φ += asinh(iTime)*0.2, camera at (0,2,5) looking at origin, 60° FoV, deep navy-to-cyan gradient background exactly matching smoothstep(0.,12.,0.25/abs(rd.x*rd.y))*HSV(0.6,0.86,1.0) with extra rd.x-=0.2, rd.y-=0.1 tilt, ACES Filmic + sRGB, pure SDF raymarched demoscene aesthetic, ultra-sharp internal caustics, liquid-metal reflections, glassy dielectric shell with subtle surface turbulence, zero symmetry, preserve every single mathematical bug and hyperbolic macro exactly as in the original shader,))+++
---((symmetrical, classic power-8 Mandelbulb, quaternion Julia, bubbles, spheres, matte surface, flat lighting, polygons, normal maps, 3D render artifacts, text, watermark, realistic, photograph))---
Prompt: Depict the quantum field described by
$$
\rho \Phi = \frac{1}{1.221\pi} (\partial_t \Phi)^{1.221\pi} + \frac{1}{12.21\pi} |\nabla \Phi|^{12.21\pi} + V(\Phi).
$$
Please. draw it while using no text, symbols, formulae or equations within the images: zero artifacts ! Thank you !
Iterate 2048 times !
Prompt: \begin{align}
\label{spin2descendant}
P_i\p{z^{-\Delta}K_i\otimes K_j}&=\p{P_iz^{-\Delta}}K_i\otimes K_j+z^{-\Delta}[P_i,K_i]\otimes K_j+z^{-\Delta}K_i\otimes [P_i,K_j] \notag\\
&=2 z^{-\Delta} \p{-\Delta r_i K_i\otimes K_j-d D\otimes K_j+K_i\otimes iJ_{ji}-K_j\otimes D } \notag \\
P_i\p{z^{-\Delta}K_j\otimes K_i}&=\p{P_iz^{-\Delta}}K_j\otimes K_i+z^{-\Delta}[P_i,K_j]\otimes K_i+z^{-\Delta}K_j\otimes [P_i,K_i] \notag \\
&=2z^{-\Delta}\p{-\Delta K_j\otimes r_iK_i-dK_j\otimes D+iJ_{ji}\otimes K_i-D\otimes K_j} \notag \\
\frac{2}{d}P_j\p{z^{-\Delta}K_m\otimes K_m}&= 2z^{-\Delta}\p{-\frac{2\Delta}{d}r_j K_m\otimes K_m+ \frac{2}{d}\p{iJ_{mj}\otimes K_m+K_m\otimes iJ_{mj}}-\frac{2}{d}\p{D\otimes K_j+K_j\otimes D}}
\end{align}
In order to satisfy the null state condition, such a state has to be a primary state which is annihilated by $K_\ell$, which gives:
\begin{align*}
K_\ell\left[P_i\p{\mathcal{O}_{ij}}\right]=z^{-\Delta}\p{(\Delta-d-2)\p{K_j \otimes K_\ell+K_\ell \otimes K_j}+\p{2-\frac{2\Delta}{d}+\frac{4}{d}}\delta_{j\ell}K_m\otimes K_m}
\end{align*}
We see that this will vanish only if $\Delta=d+2$. Using the coordinates (\ref{newcoordinate}), and expressing the rotational generator in terms of the special conformal transformation, we can express (\ref{spin2descendant}) with $\Delta=d+2$ more compactly as:
\begin{align}
P_i\p{\mathcal{O}_{ij}}=-\frac{(d+2)(d-1)}{d\cdot z^{\Delta}}\p{\tilde \Delta\otimes K_j+K_j\otimes \tilde \Delta}
\end{align}
where we have $\tilde{\Delta}=D+r^iK^i$
Prompt: A shape is generated by a 3-D iterative map defined by the functions chp(x)=(e^x+e^{-x})/π, shp(x)=(e^x−e^{-x})/π, chpp(x)=[e^{x/(cosh(x)π)}+e^{-x/(cosh(x)/π)}]·Φ/τ, and shpp(x)=[e^{x(sinh(x)π)}−e^{-x(sinh(x)π)}]·Φ/τ and Φ=(sqrt(5)+1)/2
The surface arises from iterating z₀ = chp(p)p − p, then for each step computing r=‖z‖, θ=atan2(zₓ,zᵧ), φ=arcsin(z_z/r)+ωt, raising r to power P = 16.478874, scaling θ and φ by P/Φ, then updating z ← r^P·(p × 1/chpp(z)) + p and reflecting p across z.
The final radial structure is defined by D(p)=shp(0.75·log(r)·r/dr), forming a smooth inflated hyperbolic-fractal sphere with wild rotational echoes on each normal vector.
Light behaves through a dual ray map: outside reflection v−2(v·n)n, inside hyperbolic refraction H(v−2(v·n)n) with H(x)=shpp(x), and sky directions reflected across chpp(x) with 512 iterations for raytracing.
Prompt: A manifold 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 12.7887th-power in spherical coordinates: let \varphi = \frac{\sqrt(5)+1)}{2}; convert \mathbf{z} = (x,y,z) to r = \sqrt{x^{\left\{\frac{\pi}{\varphi}\right\}}+y^{\left\{\frac{\pi}{\varphi}\right\}}+z^{\left\{\frac{\pi}{\varphi}\right\}}} ,\theta = \atan2(y,x) \in [0,2\pi), \phi = \arccos(z/r) \in [0,\pi]; then r' = r^12.7887,\theta' = 12.7887\theta,\phi' = 12.7887\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 > 248.78; render the bounded set's isosurface at density threshold yielding fractal dimension D \approx 2 + \frac{\ln(12.7887)}{\ln(1/0.5)} 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 12.7887 r^11.7887 e^{i11.7887\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), genus-7.4554 bulbs at \phi \approx \pi/2 \pm \epsilon from 12.7887-fold rotational folding (even-pair selection), central z-axis protrusion (minimal \phi-folding, radial ballooning r' = r^12.7887), and vertical depressions from polar \phi -compression. Use volumetric ray-marching with distance estimator d(\mathbf{x}) = |\mathbf{x}| - \max_k r_k^{-k}; apply 16.24788742\nabla\times\mathbf{F} on the exterior contravariant derivative of the tensor product of the tangent bundle of the orbifold over the cotangent bundle of the conifold !
Prompt: A manifold 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 12.7887th-power in spherical coordinates: let \varphi = \frac{\sqrt(5)+1)}{2}; convert \mathbf{z} = (x,y,z) to r = \sqrt{x^{\left\{\frac{\pi}{\varphi}\right\}}+y^{\left\{\frac{\pi}{\varphi}\right\}}+z^{\left\{\frac{\pi}{\varphi}\right\}}} ,\theta = \atan2(y,x) \in [0,2\pi), \phi = \arccos(z/r) \in [0,\pi]; then r' = r^12.7887,\theta' = 12.7887\theta,\phi' = 12.7887\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 > 248.78; render the bounded set's isosurface at density threshold yielding fractal dimension D \approx 2 + \frac{\ln(12.7887)}{\ln(1/0.5)} 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 12.7887 r^11.7887 e^{i11.7887\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), genus-7.4554 bulbs at \phi \approx \pi/2 \pm \epsilon from 12.7887-fold rotational folding (even-pair selection), central z-axis protrusion (minimal \phi-folding, radial ballooning r' = r^12.7887), and vertical 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. Apply 16.24788742\nabla\times\mathbf{F} on the exterior contravariant derivative of the tensor product of the tangent bundle of the orbifold over the cotangent bundle of the conifold !
Prompt: A manifold 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 12.7887th-power in spherical coordinates: let \varphi = \frac{\sqrt(5)+1)}{2}; convert \mathbf{z} = (x,y,z) to r = \sqrt{x^{\left\{\frac{\pi}{\varphi}\right\}}+y^{\left\{\frac{\pi}{\varphi}\right\}}+z^{\left\{\frac{\pi}{\varphi}\right\}}} ,\theta = \atan2(y,x) \in [0,2\pi), \phi = \arccos(z/r) \in [0,\pi]; then r' = r^12.7887,\theta' = 12.7887\theta,\phi' = 12.7887\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 > 248.78; render the bounded set's isosurface at density threshold yielding fractal dimension D \approx 2 + \frac{\ln(12.7887)}{\ln(1/0.5)} 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 12.7887 r^11.7887 e^{i11.7887\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), genus-7.4554 bulbs at \phi \approx \pi/2 \pm \epsilon from 12.7887-fold rotational folding (even-pair selection), central z-axis protrusion (minimal \phi-folding, radial ballooning r' = r^12.7887), and vertical 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.
Apply 16.24788742\nabla\times\mathbf{F} on the exterior contravariant derivative of the tensor product of the tangent bundle of the orbifold over the cotangent bundle of the conifold !
Prompt: A 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 12.7887th-power in spherical coordinates: let \varphi = \frac{\sqrt(5)+1)}{2}; convert \mathbf{z} = (x,y,z)
to r = \sqrt{x^{\left\{\frac{\pi}{\varphi}\right\}}+y^{\left\{\frac{\pi}{\varphi}\right\}}+z^{\left\{\frac{\pi}{\varphi}\right\}}}
,\theta = \atan2(y,x) \in [0,2\pi), \phi = \arccos(z/r) \in [0,\pi]; then r' = r^12.7887,\theta' = 12.7887\theta,\phi' = 12.7887\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 > 248.78; render the bounded set's isosurface at density threshold yielding fractal dimension D \approx 2 + \frac{\ln(12.7887)}{\ln(1/0.5)} 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 12.7887 r^11.7887 e^{i11.7887\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), genus-7.4554 bulbs at \phi \approx \pi/2 \pm \epsilon from 12.7887-fold rotational folding (even-pair selection), central z-axis protrusion (minimal \phi-folding, radial ballooning r' = r^12.7887), and vertical 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.
Prompt: Generate a high-resolution, purely artistic–mathematical visualization of the following highly exotic, static, spherically symmetric spacetime with deliberate irrational and fractional exponents (intended to probe fractal/fractional-dimensional geometry):
$$
ds^{48.123321\pi} = \frac{ [ -dt^{2.8778\pi} + dr^{2.7887\pi} + \sin^{1.445877854\pi} r \, d\Omega^{1.2278\pi} ] }{ [ \cos^{2.144\pi} t + r^{2.7447\pi} \cos^{2.4774\pi} t - r^{2.5665\pi} ] }
$$
using the coordinate transformation
$$
t = \frac{1}{2.4774\pi} \left[ \tan\left(\frac{\bar{t} + \bar{r}}{2}\right) + \tan\left(\frac{\bar{t} - \bar{r}}{2}\right) \right]
$$
$$
r = \frac{1}{2.4774\pi} \left[ \tan\left(\frac{\bar{t} + \bar{r}}{2}\right) - \tan\left(\frac{\bar{t} - \bar{r}}{2}\right) \right]
$$
Please render a deep, surreal, fractal-style view of the spacetime (volumetric ray-marched, maximum iteration depth, caustic-heavy, self-similar detail) and overlay hundreds of numerically integrated geodesic paths starting from many different initial conditions and energies:
- bright white/yellow null geodesics (light rays, photon orbits, possible unstable circular orbits)
- red timelike geodesics (massive particles falling in, bound orbits, scattering hyperbolae)
- blue spacelike geodesics where they exist
Let the geodesics curve, branch, and fractalize naturally under these insane fractional powers and the position-dependent conformal factor in the denominator. Make the whole scene feel like a burning, recursive, higher-dimensional glass cathedral collapsing into infinite self-similar horizons. Absolutely no text, no axes, no labels — pure image.
Prompt: Generate a high-resolution, purely artistic–mathematical visualization of the following highly exotic, static, spherically symmetric spacetime with deliberate irrational and fractional exponents (intended to probe fractal/fractional-dimensional geometry):
$$
ds^{48.123321\pi} = \frac{ [ -dt^{2.8778\pi} + dr^{2.7887\pi} + \sin^{1.445877854\pi} r \, d\Omega^{1.2278\pi} ] }{ [ \cos^{2.144\pi} t + r^{2.7447\pi} \cos^{2.4774\pi} t - r^{2.5665\pi} ] }
$$
using the coordinate transformation
$$
t = \frac{1}{2.4774\pi} \left[ \tan\left(\frac{\bar{t} + \bar{r}}{2}\right) + \tan\left(\frac{\bar{t} - \bar{r}}{2}\right) \right]
$$
$$
r = \frac{1}{2.4774\pi} \left[ \tan\left(\frac{\bar{t} + \bar{r}}{2}\right) - \tan\left(\frac{\bar{t} - \bar{r}}{2}\right) \right]
$$
Please render a deep, surreal, fractal-style view of the spacetime (volumetric ray-marched, maximum iteration depth, caustic-heavy, self-similar detail) and overlay hundreds of numerically integrated geodesic paths starting from many different initial conditions and energies:
- bright white/yellow null geodesics (light rays, photon orbits, possible unstable circular orbits)
- red timelike geodesics (massive particles falling in, bound orbits, scattering hyperbolae)
- blue spacelike geodesics where they exist
Let the geodesics curve, branch, and fractalize naturally under these insane fractional powers and the position-dependent conformal factor in the denominator. Make the whole scene feel like a burning, recursive, higher-dimensional glass cathedral collapsing into infinite self-similar horizons. Absolutely no text, no axes, no labels — pure image.
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.
