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
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%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}
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where
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\begin{itemize}
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\item $\Delta _{d}$ is the Hodge-de Rham Laplacian, a second-order elliptic
operator that acting on the 3-form $\varphi $.
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\item $\mathcal{L}_{X} \varphi $ is the Lie derivative of $\varphi $ along
a vector field $X$. It is first-order operator.
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\item $(\mathrm{Ric} \lrcorner \ast \varphi) $ is the contraction of the
Ricci tensor with the 4-form $\ast \varphi $.
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\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}
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where $e^{ijk} = e^{i} \wedge e^{j} \wedge e^{k}$.
Prompt: Draw a 3D fractal shape generated from the iterative formula \( z_{n+1} = z_n^{0.54877845\pi} + c \), with p-norm radial structure \( r = \sqrt{x^{0.7887\pi} + y^0.7887\pi + z^0.7887\pi} \). Texture it using \( f(x,y) = \sin(x^{0.7887\pi} + y^2) + \cos(z^{0.45788754\pi}) \), enhanced with micro-detail from gradient \( \nabla f \) and hyperbolic fractal sum $$ f_{\text{fract}} = \sum \sinh(\sin(2\pi^n x)) \cosh(\cos(2\pi^n y))/2^n. $$
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 \( n=16.24877842 \) for about 84 primary lobes and intricate fractal surfacing.
To generate the manifold: represent 3D points in spherical coordinates where \( r = \sqrt{x^{2.144\pi} + y^{2.144\pi} + z^{2.144\pi}} \), \( \theta = \text{acos}(z/r) \), \( \phi = \text{atan2}(y, x) \). 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 \( |v_k| > 24.78 \) after 64 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^{2.618\pi} + c, \quad z_{k+1} = z_k^{2.618\pi} + c, \quad |z| = \sqrt{x^{2.618\pi} + y^{2.618\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 \( r = \sqrt{x^{2\pi} + y^{2\pi} + z^{2\pi}} \), \( \theta = \text{acos}(z/r) \), \( \phi = \text{atan2}(y, x) \). 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.
Prompt: The Ancient Wisdom of the Dragons was what brought Atlantis into knowledgeable blooming prosperity.
The homosapientic stupidity, greed and decadence was what brought it all down...
Prompt: The supersymmetric action in 4D supergravity as a seamless, abstract geometric manifold in curved spacetime, rendered at 8K resolution with iridescent metallic gradients transitioning from deep sapphire blues and silvers for bosonic fields to vibrant emerald greens and golds for fermionic interactions, evoking quantum foam and holographic duality. Core instructions: Depict graphically, using no text, the full action \( S[e,\psi] = S[e] + S_f[e,\psi] + S_I[e,\psi] \), where the exact, completely detailedly concised maths is: \[ S[e,\psi] = S[e] + S_f[e,\psi] + S_I[e,\psi] = \int dx\, e\, e^a \wedge e^b \wedge F^{cd} \epsilon_{abcd} + \frac{1}{6} \int dx \, \theta^a \wedge e^b \wedge e^c \wedge e^d \epsilon_{abcd} + \int dx\, (\bar{\psi} \gamma_5 \gamma_a \psi)\, (\bar{\psi} \gamma_5 \gamma^a \psi) \, , \] with \(\theta^a \equiv \frac{i}{2} \left( \bar{\psi} \gamma^a D_\mu \psi - \overline{D_\mu \psi} \gamma^a \psi \right) dx^\mu \), all indices a,b,c,d=0,1,2,3 in the orthonormal frame bundle, e^a as coframe 1-forms (vielbeins), F^{cd} = dA^{cd} + A^{c e} \wedge A^{e d} the curvature 2-form of the spin connection, D_μ the covariant derivative along coordinate 1-forms dx^μ, ψ a Majorana spinor field, γ^a Dirac matrices in curved space, ε_{abcd} the Levi-Civita symbol with ε_{0123}=+1, and integrals over the 4-manifold with oriented volume form e = e^0 ∧ e^1 ∧ e^2 ∧ e^3. Visualize the first term S[e] as a swirling vortex of interlocking tetrahedral frames (symbolizing ε_{abcd} contraction) threaded by golden flux tubes (F^{cd} curvature) piercing a lattice of silver vierbein arrows (e^a, e^b) emanating from a central black hole singularity, representing the Einstein-Cartan Chern-Simons topological term. Overlay the fermionic torsion term S_f[e,ψ] as twisting helical ribbons (θ^a 1-forms derived from ψ bilinears) coiling around the vierbeins e^b,c,d into a knotted 4-simplex lattice with emerald sparks at intersection nodes, illustrating the 1/6 prefactor via sixfold symmetric bulbous expansions. Integrate the interaction S_I[e,ψ] as pulsating wave interference patterns of dual green scalar densities (ψ-bar γ5 γ_a ψ and conjugate), forming self-dual chiral currents that ripple across the manifold, modulating the geometry with fractal-like spinor foam bubbles where |ψ|^2 > threshold, colored by pseudoscalar density via smooth escape-time analogy (iterate bilinear up to 500 steps, hue H = 120° * iter / max, S=0.8, V=1). Ensure the entire composition flows as a unified holographic projection on a de Sitter boundary, with anti-aliased edges via Gaussian smoothing, subtle gravitational lensing distortions, and a faint cosmic microwave glow fading to void black; no equations, labels, or text visible; ultra-sharp filaments on torsion helices and current waves; aspect ratio 16:9; in the style of mathematical physics visualization.
Prompt: He has fuse with the universe and become one intelligence, being with infinity. He dare consume the fruits of the knowledge of good and evil and gain the forbidden God consciousness. now he is equal with the gods.... and gods do not kill gods but you are not the same, billions of you are expandable...as you are replicas producing replicas...machines to build, sustain the archonic civilization and continue the cultural memory...
Prompt: A highly detailed 3D render of an object with power P=11.24788742, fixed iterations LOOPS=64, initialized as z = chp(p)*p - p where chp(x)=(exp(x)+exp(-x))/π, shp(x)=(exp(x)-exp(-x))/π, chpp(x)=(exp(x/(cosh(x)π))+exp(-x/(cosh(x)/π)))/(2π Φ), shpp(x)=(exp(x sinh(x) π)-exp(-x sinh(x) π))/(2π Φ), ssh1(x)=sinh(x/π)/Φ, csh1(x)=cosh(x/π)/Φ, Φ=(1+√5)/2 golden ratio, τ=2π*0.7887; iteration: r=||z||, if r>2 continue, θ=asin(z_z/r)+0.2t animated, φ=atan(z_x,z_y), dr = r^{P-1} dr P +1, r=r^P, θ=θ P/Φ, φ=φ P/Φ, z += r * (tan(shp(sinθ sinφ)) Φ, chp(cosθ sinφ), cosφ) + p, p=reflect(p,z), final DE=0.75 log(r) r / dr scaled by shp(DE *2); ray-marched with max marches=96, tol=10^{-5}, bounces=8, refraction index 1.05, Beer absorption exp(-(t+0.1) * -HSV(0.05,0.95,2)), diffuse HSV(0.6,0.85,1), glow HSV(0.065,0.8,6), sky HSV(0.6,0.86,1) with warped reflections via ssh1, chpp, fract(clamp(0.125 / |reflected cross| * skyCol, -120,16.547)); rotated by rot_x((1.221 t + π)/τ), camera at (0,2,5)*0.6, FOV tan(τ/6), ACES tone-mapped, sRGB gamma; central bulbous form with pink core, orange lobes, black voids, cyan shell, rainbow tunnel background.
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.
