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: 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: 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: 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: 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: 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: 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.
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.
