Prompt: A highly detailed volumetric fractal rendering inspired by derived hyperbolic Fibonacci-like functions: incorporate the simplified geometry formula 2 * sinh(π * x * sinh(x)) * φ / π for symmetric, explosively growing bulbous structures with even parity and golden ratio scaling; nuance with the asymmetric shading expression φ * (exp(x / (π * cosh(x))) + exp(-π * x / cosh(x))) for uneven glow decay, creating fiery orange internal emissions that fade to translucent icy blue exteriors; emphasize infinite self-similarity, wavy refractive boundaries, and organic alien forms on a deep blue cosmic background, in ultra-high resolution with ray-traced volumetrics and subtle particle effects.
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.
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.
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 tangent bundle of the conifold; then TimeSpaceFlow wave mirror symmetralize them !
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 !
Illuminated Vacuum Chamber with Particle Visualization
Model:
Nano Banana Pro
(Pro)
Size:
2560 X 1440
(3.69 MP)
Used settings:
Prompt: A Bose-Einstein Condensate (BEC) is a unique state of matter where atoms, cooled to near absolute zero, lose individual identities and behave as one single quantum entity, a macroscopic wave, showing quantum effects on a large scale, like a superfluid or "atom laser". Predicted by Satyendra Nath Bose and Albert Einstein in the 1920s, it was first created in 1995, revealing bizarre quantum behaviors that challenge classical physics.
Prompt: A G$_{2}$-structure on a 7-dimensional manifold is characterized by a 3-form
$\varphi $, which reduces the structure group to the exceptional Lie group
G$_{2}$. When $\varphi $ is both closed and co-closed, the structure is
torsion-free, and the associated metric is Ricci-flat. The G$_{2}$-Ricci flow is defined by the following equation
%
%e3 #&#
\begin{equation}
\frac{\partial \varphi}{\partial t} = \Delta _{d} \varphi +
\mathcal{L}_{X} \varphi + \mathrm{Ric} \lrcorner \ast \varphi + T(\varphi ),
\label{eq3}
\end{equation}
%
where
%
\begin{itemize}
%
\item $\Delta _{d}$ is the Hodge-de Rham Laplacian, a second-order elliptic
operator that acting on the 3-form $\varphi $.
%
\item $\mathcal{L}_{X} \varphi $ is the Lie derivative of $\varphi $ along
a vector field $X$. It is first-order operator.
%
\item $(\mathrm{Ric} \lrcorner \ast \varphi) $ is the contraction of the
Ricci tensor with the 4-form $\ast \varphi $.
%
\item $T(\varphi )$ represents the torsion of the G$_{2}$-structure, which measures the deviations from the torsion-free condition.
\begin{equation}
\varphi = e^{123} + e^{145} + e^{167} + e^{246} - e^{257} - e^{347} - e^{356},
\label{eq1}
\end{equation}
%
where $e^{ijk} = e^{i} \wedge e^{j} \wedge e^{k}$.
Prompt: A highly detailed 3D rendering of the quintic Calabi-Yau 3-fold hypersurface in ℂℙ⁴ defined by ∑_{i=0}^4 z_i^5 = 0, a compact complex manifold of complex dimension 3 with trivial canonical bundle K_X ≅
Prompt: "Create a hyper-detailed, surreal digital artwork in the style of a quantum field theory mandala fused with topological knot diagrams, holographic projections, and Kaluza-Klein compactifications, rendered in glowing neon blues, purples, electric golds, and shimmering tachyon reds on a cosmic black void background evoking infinite energy heat-up in a 7D collider singularity. At the center, a radiant 7D holographic orb pulses with core equations: E = f φ μ B + η H (scalar Zeeman energy) orbiting the variational action S = ∫ [∑i (1/2)<ψ_i|Ĥ_i|ψ_i> + ∑{i,j} (1/3) w_{ij} <ψ_i|ψ_j * ψ_j> + ∑_{i,j} λ_{ij} (f_i/f_j - φ)^2 + ∑i κ_i |B_i · μ_i| + η H + ∑{i,j} γ_{ij} w_{knot,ij}] dτ (many-body overlaps, constraints, magnetic dots, knot weights). Radiating in fractal spirals: Left arc, 3D Chern-Simons TQFT S_CS = (k/4π) ∫ Tr(A ∧ dA + (2/3) A ∧ A ∧ A) (U(1) flat F=0, integer k invariance), Wilson loops W_R(γ) = Tr[P exp(i ∮γ A)] braiding Jones knots as w{knot,ij} linking for anyons in quantum Hall. Right arc, 4D Yang-Mills S_YM = -1/(4g²) ∫ Tr(F ∧ *F) with F = dA + A ∧ A (gluon propagation), boundary-merging to massive 3D YM. Upper cascade, form shifts: 1-form A (3D loops) → 2-form B ∈ Ω²(M) (5D surfaces, H = dB or Ω₂ = dB + A▹B in crossed module G→H▹ with Ω₁ = dA + [A,A]/2 - α(B); action ∫ (1/2) H ∧ H + (k/24π²) B ∧ H ∧ H + 2CS ⟨A,Ω₂⟩ + ⟨Ω₁,B⟩, EOM dH + (k/12π²) H ∧ H = J_{(1)} for 1-branes, topological m from Stueckelberg) → 3-form C ∈ Ω³(M) (7D volumes, G = dC or Ω₃ = dC + [A,C] + [B,B] in 2-crossed module G→H→K▹δ with Ω₁=0, Ω₂=0, Peiffer δΩ₁=[Ω₁,B]; merged action ∫ (1/2) G ∧ *G + (k/(2π)^3 · 3!) CS_7(C) = Tr(C ∧ dC ∧ (dC)^2 + (3/2) C ∧ C ∧ dC ∧ dC + (3/5) C³ ∧ dC + (1/7) C⁴) + 3CS ⟨A,Ω₃⟩ + ⟨B,Ω₂⟩ + ⟨C,Ω₁⟩ + (1/2) Tr(Ω₃ ∧ *Ω₃) + m² Tr(C ∧ C), EOM dΩ₃ + [A,*Ω₃] + (k/4π) Ω₂ = J_{(2)} for 2-branes). Lower vortex, applications: Tachyon condensation V(T) = -(μ²/2)T² + (λ/4)T⁴ rolling unstable vacua to <T>~√(μ²/λ) breaking Spin(7)→G₂, stabilizing C-flux on T³/CY₃ KK compactification (ds⁷² = ds⁴² + g_{mn} dy^m dy^n, C_{μmn} dx^μ ∧ dy^m ∧ dy^n modes, θ-term axion from ∫_T³ C, chiral matter from wrapped M5s), bordism invariants W(Σ³)=Tr P exp(∫_Σ³ C) linking 3-manifolds, Donaldson polys post-reduction, AdS₇ CFT duals, cosmic strings as codim-3 defects in GUT scales. Interweave icons: Higgs vev φ, Bianchi dG=0, Peiffer terms, Gauss-volume linking for Σ_i³ × Σ_j³, early-universe flux knots, tachyon minima curving to brane-stabilized vacua. Text overlays in elegant LaTeX script: 'From 1-Form Loops to 3-Form Volumes: Merged YM/CS in 7D KK Knotty QFT with Tachyon Fury'. Ultra-high resolution, intricate linework like exploded Feynman diagrams in Escher-KK topology, vibrant clashing distortions for aesthetic conceptual heat."
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: <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: 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: A smooth, highly reflective bulbous geometric form whose 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 = 16.4877884, 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 faint rotational echoes.
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).
Depict this mathematical object as a large glossy hyperbolic fractal sphere with smooth curvature, concentric internal rings, deep warm core transitioning to cool blue rim, intense grazing-angle highlights, and a soft blue background, evoking nonphysical hyperbolic refraction and warped exponential geometry.
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 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.
Prompt: <lora:Intricacy Vibe:1.0>Using the action:
\[
S[e,\psi] = S[e] + S_f[e,\psi] + S_I[e,\psi] = \int dx\, e\, e^a \wedge e^b \wedge F^{cd} \epsilon_{abcd} + \frac{1}{6} \int dx \, \theta^a \wedge e^b \wedge e^c \wedge e^d \epsilon_{abcd} + \int dx\, (\bar{\psi} \gamma_5 \gamma_a \psi)\, (\bar{\psi} \gamma_5 \gamma^a \psi) \, ,
\] 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;
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.
