Prompt: <lora:Intricacy Vibe:1.0>
A lotus in a cosmic background, representing a transcendentally-warped TimeSpaceFlow with the exact metric ds^{12.78544587\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^{\pi\phi} + \sinh^{\sqrt[\pi]{3}\pi} x , d\Omega^{12.78544587\pi}, \phi = (1 + \sqrt{5})/2; central glowing golden core as singularity with amber-orange light rays, nonsymmetrical translucent cyan-blue lotus petals with intricate golden vein fractals exhibiting non-integer oscillations and mirror symmetry spirals, recursive self-similar golden-ratio helicoidal curls along petal edges, ethereal volumetric glow and caustics, all followint the exact, precise, concise and full mathematics provided.
Apply tensor product of the cotangent bundle of the orbifold over the tangent bundle of the conifold; then TimeSpaceFlow wave mirror symmetralize them !
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: "Let \( \mathcal{O} \) be a smooth Calabi–Yau orbifold and \( \mathcal{C} \) the resolved small conifold (smooth Calabi–Yau 3-fold). Fix compatible Ricci-flat metrics on both.
Apply the scalar multiple
\( 24.24788742 \, (\nabla \times \mathbf{F}) \), where \( \mathbf{F} \in \Gamma(\mathbb{R}^3) \) is a compactly supported vector field on a local Euclidean chart, to the tensor product
\( d_\nabla \, \omega \otimes \mathcal{L}_\xi \, \alpha \), where \( d_\nabla : \Gamma(\Lambda^\bullet T^*\mathcal{O} \otimes T\mathcal{O}) \to \Gamma(\Lambda^{\bullet+1} T^*\mathcal{O} \otimes T\mathcal{O}) \) is the exterior covariant derivative induced by the Levi-Civita connection on the tangent orbibundle of \( \mathcal{O} \), - \( \omega \) is a smooth section of \( \Lambda^1 T^*\mathcal{O} \otimes T\mathcal{O} \),
\( \mathcal{L}_\xi \) denotes the Lie derivative along a Killing vector field \( \xi \) on the resolved conifold \( \mathcal{C} \),
\( \alpha \in \Omega^2(\mathcal{C}) \) is a Kähler (1,1)-form, evaluated at the unique stratum-preserving orbifold-conifold correspondence point in the moduli space where the stringy Kähler moduli align at the conifold locus under mirror symmetry after analytic continuation through the 24.24788742-th branch of the Picard–Fuchs equations.(We further demand that the entire expression be Wick-rotated, smeared over a Gaussian regulator of width \( e^{-24.24788742} \), and uplifted to eleven dimensions just for the vibes.)"
Prompt: 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,0)), and similar for circular ds=length(pp)-0.5, clamped and shaped with shp(clamp(col,0,10)).
Prompt: Over the hybrid space \( O \cup_C \) generate the rank-\( (\dim O + 6) \) vector bundle \( E = TO \boxtimes T^*C \) (external tensor product of the tangent orbibundle of \( O \) and cotangent bundle of each resolved conifold \( C \)).
Create a smooth section \( Z \in \Gamma(E) \cong \Gamma(TO) \times \Gamma(T^*C) \).
Generate the contravariant exterior covariant derivative \( \nabla^\sharp : \Gamma(E) \rightarrow \Gamma(E \otimes E) \) as (2,0)-tensor field lines.
Create a compactly supported vector field \( F \) on \( O \), its curl \( \Phi = \kappa (\nabla \times F) \) with \( \kappa \) is floating number, defined below, equal to 12.144578875441.
Final central object: the vector field \( \Psi = \iota_\Phi (\nabla^\sharp Z) \in \Gamma(E) \) living directly on the tensor-product bundle \( E \), depicted as flow lines emerging from the conifold resolutions and wrapping the entire structure.
Gaussian regulator \( e^{-\kappa} \) as soft halo, Wick rotation suggested by subtle time-like streaks.
Pure abstract geometry, cosmic bright background, precise mathematical beauty, no text using:
$$
\kappa = 12.144578875441
$$
$$
E = TO \boxtimes T^*C
$$
\( \nabla^\sharp Z \)
$$
\Phi = \kappa (\nabla \times F)
$$
$$
\Psi = \iota_\Phi (\nabla^\sharp Z) \in \Gamma(E)
$$
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: 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: Create a highly detailed, vibrant digital artwork of a 3D manifold structure, rendered in glowing shades of purple, cyan, and blue, resembling a futuristic crystalline flower or starburst emerging from a cosmic starry night sky background with a deep blue-purple gradient. The central fractal object should be highly symmetric with pointed, spiky lobes radiating outward in a self-similar pattern, evoking infinite complexity and detail, specifically using the Mandelbulb formula with power parameter \( n=8 \) for about 7-8 primary lobes and intricate fractal surfacing.
To generate the manifold: represent 3D points in spherical coordinates where \( r = \sqrt{x^2 + y^2 + z^2} \), \( \theta = \text{acos}(z/r) \), \( \phi = \text{atan2}(y, x) \). The power operation \( v^n = r^n \cdot [\sin(n\theta) \cos(n\phi), \sin(n\theta) \sin(n\phi), \cos(n\theta)] \). Iteration: \( v_{k+1} = v_k^n + c \), starting from \( v_0 = (0,0,0) \), with escape if \( |v_k| > 2 \) after many iterations. Use ray marching with distance estimator \( DE(p) \approx (1/2) \cdot (r - R) / |dr/dv| \) for rendering, applying escape-time coloring, orbit traps, and Phong shading for neon glow effects.
Use also:
\sum_{n=0}^\infty \left(\frac{1}{2^n}\right), \quad \int_{-\infty}^\infty e^{-x^2} \, dx = \sqrt{\pi}, \quad f(x) = x^2 + c, \quad z_{k+1} = z_k^2 + c, \quad |z| = \sqrt{x^2 + y^2}, \quad z = r e^{i\theta}, \quad z^2 = r^2 e^{i2\theta}, \quad x' = r^2 \cos(2\theta), \quad y' = r^2 \sin(2\theta)
Ensure the composition is centered on the fractal with soft glows, high resolution, surreal and mathematical aesthetic, similar to AI-generated fractal art in a cosmic math universe.
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: Create a highly detailed, vibrant digital artwork of a 3D manifold structure, rendered in glowing shades of purple, cyan, and blue, resembling a futuristic crystalline flower or starburst emerging from a cosmic starry night sky background with a deep blue-purple gradient. The central fractal object should be highly symmetric with pointed, spiky lobes radiating outward in a self-similar pattern, evoking infinite complexity and detail, specifically using the Mandelbulb formula with power parameter \( n=8 \) for about 7-8 primary lobes and intricate fractal surfacing.
To generate the manifold: represent 3D points in spherical coordinates where \( r = \sqrt{x^{2\pi} + y^{2\pi} + z^{2\pi}} \), \( \theta = \text{acos}(z/r) \), \( \phi = \text{atan2}(y, x) \). The power operation \( v^n = r^n \cdot [\sin(n\theta) \cos(n\phi), \sin(n\theta) \sin(n\phi), \cos(n\theta)] \). Iteration: \( v_{k+1} = v_k^n + c \), starting from \( v_0 = (0,0,0) \), with escape if \( |v_k| > 2 \) after many iterations. Use ray marching with distance estimator \( DE(p) \approx (1/2) \cdot (r - R) / |dr/dv| \) for rendering, applying escape-time coloring, orbit traps, and Phong shading for neon glow effects.
Use also:
\sum_{n=0}^\infty \left(\frac{1}{2^n}\right), \quad \int_{-\infty}^\infty e^{-x^{2\pi}} \, dx = \sqrt{\pi}, \quad f(x) = x^{2\pi} + c, \quad z_{k+1} = z_k^{2\pi} + c, \quad |z| = \sqrt{x^{2\pi} + y^{2\pi}}, \quad z = r e^{i\theta}, \quad z^2 = r^2 e^{i2\theta}, \quad x' = r^2 \cos(2\theta), \quad y' = r^2 \sin(2\theta)
Ensure the composition is centered on the fractal with soft glows, high resolution, surreal and mathematical aesthetic, similar to AI-generated fractal art in a cosmic math universe.
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: 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: "Let \( \mathcal{O} \) be a smooth Calabi–Yau orbifold and \( \mathcal{C} \) the resolved small conifold (smooth Calabi–Yau 3-fold). Fix compatible Ricci-flat metrics on both.
Apply the scalar multiple
\( 24.24788742 \, (\nabla \times \mathbf{F}) \), where \( \mathbf{F} \in \Gamma(\mathbb{R}^3) \) is a compactly supported vector field on a local Euclidean chart, to the tensor product
\( d_\nabla \, \omega \otimes \mathcal{L}_\xi \, \alpha \), where \( d_\nabla : \Gamma(\Lambda^\bullet T^*\mathcal{O} \otimes T\mathcal{O}) \to \Gamma(\Lambda^{\bullet+1} T^*\mathcal{O} \otimes T\mathcal{O}) \) is the exterior covariant derivative induced by the Levi-Civita connection on the tangent orbibundle of \( \mathcal{O} \), - \( \omega \) is a smooth section of \( \Lambda^1 T^*\mathcal{O} \otimes T\mathcal{O} \),
\( \mathcal{L}_\xi \) denotes the Lie derivative along a Killing vector field \( \xi \) on the resolved conifold \( \mathcal{C} \),
\( \alpha \in \Omega^2(\mathcal{C}) \) is a Kähler (1,1)-form, evaluated at the unique stratum-preserving orbifold-conifold correspondence point in the moduli space where the stringy Kähler moduli align at the conifold locus under mirror symmetry after analytic continuation through the 24.24788742-th branch of the Picard–Fuchs equations.(We further demand that the entire expression be Wick-rotated, smeared over a Gaussian regulator of width \( e^{-24.24788742} \), and uplifted to eleven dimensions just for the vibes.)"
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 digital rendering of an abstract, symmetrical fractal structure resembling a surreal, organic face floating against a gradient blue sky background, generated using a modified Mandelbulb fractal algorithm viewed from the inside with ray marching. Incorporate precise mathematical details: Define constants pi = 3.1415926535897932384626433832795, tau = 2*pi, TAU = (2*pi)*0.7887, PHI = (sqrt(5)*0.5 + 0.5) ≈1.618 golden ratio, POWER = 11.24788742 for exponentiation, LOOPS = 3 iterations, TOLERANCE = 0.00001, MAX_RAY_LENGTH = 20.0, MAX_RAY_MARCHES = 48, NORM_OFF = 0.0005, MAX_BOUNCES = 5. Custom hyperbolic functions: 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/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. The Mandelbulb distance estimator mandelBulb(p): Initialize z = chp(p)*p - p, dr=1.0; for i=0 to LOOPS-1, r=length(z), theta=atan(z.x,z.y), phi=asin(z.z/r) + optional time*0.2 for animation; dr = r^(POWER-1) * dr * POWER + 1; r = 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 distance 0.75 * log(r) * r / dr. Overall distance function df(p) = shp(mandelBulb(p/2.0)*2.0) after applying rotation matrix g_rot = rot_x(((1.221*time + pi)/tau)). Render with ray marching from camera at 0.6*vec3(0,2,5) looking at origin, FOV tan(TAU/6), incorporating bounces for reflection (reflect(rd,sn)), refraction (refract(rd,sn,1.0/mat.z or inverse)), fresnel fre=1+dot(rd,sn) squared and mixed 0.1-1.0, diffuse dif=max(dot(ld,sn),0)^2 * (1-mat.x) with ld to light at (0,10,0), material mat=(0.8,0.5,1.05), beer absorption exp(-(st+0.1)* -HSV(0.05,0.95,2.0)). Sky background: Procedural with planes at y=4 and y=-6, box bounds, exponential falloff, colored HSV(0.6,0.86,1.0). Colors: Glow HSV(0.065,0.8,6.0), diffuse HSV(0.6,0.85,1.0), post-processed with ACES tonemapping aces_approx(v) = clamp((v*(2.51v+0.03))/(v*(2.43v+0.59)+0.14),0,1) after *0.6, and sRGB gamma mix(1.055*t^(1/2.4)-0.055,12.92*t,step(t,0.0031308)). The structure features two large spiral-eyed voids as eyes, a curved dark blue mouth-like opening at the bottom, elaborate branching tendrils and crystalline edges with subtle particle specks dissipating at sides, ethereal pinkish-orange glow, edge fresnel effects, hyper-realistic yet fantastical Shadertoy-inspired 3D art in 16:9 aspect ratio with sharp details and no text or artifacts.
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.
