Prompt: Apply tensor product of the {1.6180339887498948482045868343656\pi\sqrt[\exp(1)]{2\cdot1.6180339887498948482045868343656}}-form of the tangent bundle of the orbifold over the {1.6180339887498948482045868343656\exp(1)\sqrt[\pi]{2\cdot1.6180339887498948482045868343656}}-form of the tangent bundle of the conifold !
Prompt: Create a highly detailed object, rendered as a glassy, translucent orb with central red spiked core transitioning to rippling blue outer surfaces, infinite self-similar details from power=11.24788742 iterations over exactly 3 loops, incorporating hyperbolic distortions via scaled sinh(x)/cosh(x) functions divided by π (chp(x)=(exp(x)+exp(-x))/π, shp(x)=(exp(x)-exp(-x))/π), advanced variants like chpp(x)=1/(exp(x/(cosh(x)/π))+exp(-x/(cosh(x)/π))) / (2π) * φ where φ=(√5+1)/2≈1.618, shpp(x)=(exp(x*sinh(x)*π)-exp(-x*sinh(x)*π))/(2π)*φ, ssh(x)=sinh(x*π/0.7887)/π, csh(x)=cosh(x*π/0.7887)/π, ssh1(x)=sinh(x/π)/φ, csh1(x)=cosh(x/π)/φ; initialization z = p / chpp(p) - p (component-wise), dr=1; per-iteration: r=||z||, if r>2 continue, θ=atan(z.y,z.x), ϕ=asin(z.z/r), dr=r^{power-1}*dr*power +1, r=r^power, θ*=power/φ, ϕ*=power/φ, direction vector (tan(shp(sin(θ)sin(ϕ)))*φ, chp(cos(θ)sin(ϕ)), cos(ϕ)), z=r*direction + p, p=reflect(p,z), r=||z||; distance estimator de=0.75 log(r) r / dr; full df(p)=shp(mandelBulb(p/2)*2); rendering with ray marching tolerance 0.00001, max marches 84, length 20, normals offset 0.000125, up to 8 bounces, Fresnel refraction index 1.05 (mat=vec3(0.8,0.5,1.05)), beer absorption -HSV(0.05,0.95,2.0), diffuse HSV(0.6,0.85,1.0), glow HSV(0.065,0.8,6.0), sky HSV(0.6,0.86,1.0) with planes at y=4/-6, light at (0,10,0), rotation matrix rot_x(1/((e*π)*chpp(1.221 t /π))/τ), camera at 0.6*(0,2,5) fov tan(τ/6), post-process ACES tonemap and sRGB gamma; blue gradient background, mandala-like rotational symmetry, crystalline waves, no animation, with "occlusion" achieved by: vec3 col = clamp(vec3(0.25/abs(reflect((reflect(rd*outerProduct(rd,ro),ro*g_rot)), chpp(ro*outerProduct(ro,rd)) ).z))*skyCol, 0.0, 1.618);
Prompt: Apply tensor product of the {1.6180339887498948482045868343656\sqrt[\exp(1.0)]{2\cdot1.6180339887498948482045868343656}}-form of the cotangent bundle of the orbifold over the {1.6180339887498948482045868343656\sqrt[\pi]{2\cdot1.6180339887498948482045868343656}}-form of the tangent bundle of the conifold !
Prompt: A highly detailed, symmetrical anthropomorphic visage emerging from precisely defined iterations by the following mathematical formulations and parameters: core distance estimator function mandelBulb(vec3 p) with power = 11.24788742, loops = 3, initial z = chp(p)*p - p where chp(x) = (exp(x) + exp(-x))/pi and pi = 3.1415926535897932384626433832795 / asinh(TIME), tau = 2*pi, then iterate r = length(z), theta = atan(z.x, z.y), phi = asin(z.z / r) + TIME*0.2 (animated), dr = pow(r, power - 1.0) * dr * power + 1.0, r = pow(r, power), theta = theta * power / PHI where PHI = (sqrt(5.0)*0.5 + 0.5), phi = phi * power / PHI, z = r * vec3(tan(shp(sin(theta)*sin(phi)))*PHI, chp(cos(theta)*sin(phi)), cos(phi)) + p where shp(x) = (exp(x) - exp(-x))/pi, with p = reflect(p, z) and final return 0.75 * log(r) * r / dr; overall distance field df(vec3 p) = shp(mandelBulb(p / 2.0) * 2.0) after applying global rotation g_rot = rot_x(((1.221*TIME + pi)/tau)); rendered via ray marching with max marches = 48, tolerance = 0.00001, normal offset = 0.00125, max length = 20.0, up to 8 bounces incorporating reflections, refractions with material vec3(0.8, 0.5, 1.05), Beer-Lambert absorption exp(-(st + 0.1)* -HSV2RGB(vec3(0.05, 0.95, 2.0))), Fresnel mixing, diffuse lighting from vec3(0.0, 10.0, 0.0), and sky color HSV2RGB(vec3(0.6, 0.86, 1.0)) with plane intersections; additional hyperbolic variants shpp(x) = (exp(x*(sinh(x)*pi)) - exp(-x*(sinh(x)*pi)))/tau*PHI, chpp(x) = (exp(x/(cosh(x)*pi)) + exp(-x/(cosh(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 applied in sky reflections and ray directions.
Prompt: A surreal psychedelic 3D raymarched landscape of infinite triply periodic hyperbolic gyroid minimal surfaces with mean curvature H=0, defined by implicit level set f(p)≈0 with f=dot(cos p, sin(p.zxy)) +1.1618= cos x sin z + cos y sin x + cos z sin y +1.1618, offset-duplicated as min(gyroid(p), gyroid(p-vec3(0,0,π≈3.1416))); warped by mat2 rotations M=inv\begin{pmatrix} c & -s \ s & c \end{pmatrix} with c=sinh(cos a)∈[-1.175,1.175], s=cosh(sin a)∈[1,1.543], det M=(cosh(2 cos a)+cosh(2 sin a))/2>1 for non-orthogonal shear/scaling; look-at mat3 with up=(0,1.618 tan(sinh(cosh(1/iTime))),0), rt=normalize(tan(reflect(reflect(sinh(cosh(dir/cosh(iTime))), up), up))) composing tan-boosted reflections/hyperbolics; featuring mirrored canyon-like splits from interpenetrating networks, twisting hyperbolic tunnels from exponential warps, vibrant pink-yellow-orange-green gradients via golden-ratio albedos alb_m1=(0.618,0.618,0.81)max(1.618,smoothstep(0,12.5,freck)), alb_m2=(0.618,0.83,0.0618)same with freck=∑ cosh(23 p_i)=(e^{23p}+e^{-23p})/2 per coord for high-contrast exponential spots; soft quadratic fog 1-exp(-0.008 d²) attenuating throughput=1-fog; 2 reflective bounces with rd=reflect(rd,sn), offset ro=p+sn0.01, throughput*=0.9*fres^1, fres=1-max(0,-rd·sn); noisy normals sn=normalize(∇map + 0.1 pow(|cos(64 p)|,16)) via tetrahedral finite diff ∇map≈(1/2ε)∑ e_k map(p+e_k) with ε=(0.618,-0.618)*12.21 / sinh(cosh(1/iTime)) vec2, then sn=sinh(cosh(sn)); lighting with ld=normalize(lp-p), lp=(10,-10,-10+ro.z), diff=max(0,0.5+2 sn·ld), diff2=(||sin(2 sn)0.5+0.5||)^2, diff3=max(0,0.5+0.5 sn·(0,1,0)), spec=max(0,reflect(-ld,sn)·-rd), col+=(0.3,0.25,0.25) spec^4 8 + (0.4,0.6,0.9)diff + (0.5,0.1,0.1)diff2 + (0.9,0.1,0.4)diff3, col=albgetAO; AO=clamp(1-occ,0,1.618) with occ=∑(t-map(reflect(p,sn)+sn t)) for t=0.04 i, i=0..7; camera ro=(π/2,0,-0.5 t), rd=normalize(vec3(-sin uv, -0.3425)) with sin(uv) fisheye, mouse rd.zy=rot(mo.y π) sinh(rd.zy), rd.xz=rot(-mo.x π) rd.xz, auto-rot sin(0.2 t)/cos(0.4 t); vignette smoothstep(0,1,1.2-||0.9 uv||), gamma col^{0.4545}.
Prompt: Apply tensor product of the {12.2481441842*1.618\sqrt[\pi]{2}}-form of the cotangent bundle of the orbifold over the {12.2481441842*1.618\sqrt[\pi]{2}}-form of the tangent bundle of the conifold !
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: Apply tensor product of the {1.6180339887498948482045868343656\sqrt[\exp(1.0)]{1.6180339887498948482045868343656\pi}}-form of the cotangent bundle of the orbifold over the {1.6180339887498948482045868343656\sqrt[\pi]{1.6180339887498948482045868343656\exp(1.0)}}-form of the tangent bundle of the conifold !
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: 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 !
Illuminated Vacuum Chamber with Particle Visualization
Model:
Nano Banana 2
(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: (Surreal dreamscape:1.5) featuring a (cosmic portal entity:1.4) formed from (frozen starlight and liquid nebulae:1.3). A (radiant peach soul:1.4) hides within a (cavernous violet void:1.2). Surrounding structure of (fractal ice shards:1.3) and (ethereal mists:1.2). (Electric blue aura:1.2), (deep shadows:1.3), (tactile hallucinations:1.1), (mystical atmosphere:1.4), (complex geometry:1.2), (isolated in infinity:1.2), (abstract expressionism:1.3).
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: Pure TOPOLOGICAL ANTHROPOMORPHISM, emergent primate-alien visage isosurface born solely from rigorous custom GLSL fractal topology. A symmetrical 3D biomechanical face crystallizing out of volatile mathematical equations, no pareidolia but absolute geometric necessity. Formulated with high-iteration transcendental metrics: power = 11.24788742, loops = 3, tau = (2.0 * pi) * 0.7887, and golden ratio PHI. Hyperbolic space warping driven by custom functions chp(x) = (exp(x)+exp(-x))/pi and shp(x) = (exp(x)-exp(-x))/pi*PHI. Extreme fine micro-detail resolved via pseudo-reflection formula p -= pow(PHI, pi/PHI) * dot(p,z) * z inside the loop. Intricate concentric spirals in the glowing, high-contrast hyper-detailed eyes, sharp metallic nose, and carved jawline blending seamlessly into explosive, symmetrical frozen liquid splashes. Photorealistic raymarching material with complex internal refraction, iridescent rose quartz skin tones, and bright copper-orange bioluminescent organic electrical filaments. Stark contrast against a deep, dark royal blue volumetric studio background. 8k resolution, flawless ray-traced reflections, highly polished glassy and fleshy textures, mathematically perfect balance.
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 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 $e^{ijk} = e^{i} \wedge e^{j} \wedge e^{k}$.
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.
