Boris Krumov

Ethereal Fractal Dreams in Metallic Hues

• Model: Ideogram (Ultra)

• Size: 2576 X 1440 (3.71 MP)

• Used settings:

  • 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 !
  • Using base image: No
  • Aspect Ratio: landscape_wide

Vibrant 3D Fractal Flower Design

• Model: AIVision (Ultra)

• Size: 1792 X 1008 (1.81 MP)

• Used settings:

  • 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 !
  • Using base image: No
  • Aspect Ratio: landscape_wide

Cosmic Tree: A Dance of Color and Light

• Model: Ideogram

• Size: 1024 X 1024 (1.05 MP)

• Used settings:

  • 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.
  • Using base image: No
  • Aspect Ratio: square
  • Ideogram Style: Auto

Cosmic Blue Lotus in a Vibrant Mandala

• Model: DaVinci2

• Size: 2560 X 1456 (3.73 MP)

• Used settings:

  • 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.
  • Using base image: No
  • Aspect Ratio: landscape_wide

Mesmerizing Blue and Silver Fractal Art

• Model: DaVinci2

• Size: 1920 X 1080 (2.07 MP)

• Used settings:

  • 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.
  • Using base image: No
  • Aspect Ratio: landscape_wide

Cosmic Divine: A Journey Through Light

• Model: AIVision (Pro)

• Size: 1024 X 1024 (1.05 MP)

• Used settings:

  • 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).
  • Using base image: No
  • Aspect Ratio: square

Intricate Mathematical Diagram with Geometry

• Model: AIVision (Ultra)

• Size: 1792 X 1008 (1.81 MP)

• Used settings:

  • Prompt: \[ \star \ : \ \bigwedge^k T^*\mathcal{M} \longrightarrow \bigwedge^{n-k} T^*\mathcal{M} \] \[ \left( \star_{\mathcal{O}} \bigwedge^{48.1441\cdot 1.618 \sqrt[\pi]{2}} T\mathcal{O} \right) \otimes \left( \star_{\mathcal{C}} \bigwedge^{48.1441\cdot 1.618 \sqrt[\pi]{2}} T^*\mathcal{C} \right) \] \[ \Longrightarrow \quad \bigwedge^{\dim \mathcal{O}\, -\, k} T^*\mathcal{O} \ \otimes\ \bigwedge^{\dim \mathcal{C}\, -\, k} T\mathcal{C} \] \[ \text{where } k = 48.1441\cdot 1.618 \sqrt[\pi]{2}. \]
  • Using base image: No
  • Aspect Ratio: landscape_wide

Futuristic Blue Sphere with Dynamic Elements

• Model: DaVinci2

• Size: 1152 X 864 (1.00 MP)

• Used settings:

  • Prompt: A highly detailed render of a quasi-symmetrical anthropomorphic entity emerging from an iterative map in spherical coordinates (θ polar, φ azimuthal), defined by: \[ \mathbf{z}_{n+1} = \mathbf{z}_n + r_n \cdot \mathbf{k}(\theta_n, \phi_n) + \mathbf{p}_n \] \[ \mathbf{p}_{n+1} = \mathbf{p}_n - 2 \frac{\mathbf{p}_n \cdot \hat{\mathbf{u}}_{n+1}}{\|\hat{\mathbf{u}}_{n+1}\|^2} \hat{\mathbf{u}}_{n+1}, \quad \hat{\mathbf{u}}_{n+1} = \frac{\mathbf{z}_{n+1}}{\|\mathbf{z}_{n+1}\|} \] \[ r_n = \|\mathbf{z}_n\|, \quad \mathbf{k}(\theta, \phi) = \left( \sin\theta \cos\theta \cdot \frac{e^{\cos\phi} + e^{-\cos\phi}}{\pi}, \ \sin^2\theta \sin\phi, \ \cos\phi \right) \] with initialization \(\mathbf{z}_0 = \mathbf{0}\), \(\mathbf{p}_0 = \mathbf{q}\); bailout \(r > 10\) after 150 iterations; distance estimator \(DE(\mathbf{q}) \approx 0.5 r_N \ln(r_N) / dr_N\) where \(dr_{n+1} = dr_n (1 + \|J_{\mathbf{k}}\| + \frac{2\sinh(\cos\phi)}{\pi}) + 1\), \(dr_0=1\); ray-marched with ε=10^{-6}, Phong shading (α=30), orbital coloring HSV H=200°-100°(n/N), S=0.85 tanh(∫ chp(cos φ_k) dk / N), V=0.6+0.4 sin(π k/3) for reflection count k; featuring swollen iridescent-rimmed eyes, cavernous textured-lobed mouth, asymmetrical spiky protrusions, in blue-purple/orange-blue gradients on navy backdrop.
  • Using base image: No
  • Aspect Ratio: landscape

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.
  • Using base image: No
  • Aspect Ratio: landscape_wide

Entangled Light: A Dance of Color and Form

• Model:

• Size: 2560 X 1440 (3.69 MP)

• Used settings:

  • 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 !
  • Using base image: No
  • Aspect Ratio: landscape_wide

Vibrant 3D Abstract Sphere in Deep Space

• Model: Ideogram (Ultra)

• Size: 2592 X 1456 (3.77 MP)

• Used settings:

  • Prompt: #define pi 3.1415926535897932384626433832795 #define tau (2.*pi) #define chp(x) (exp(x)+exp(-x))/pi #define chpp(x) (exp(x/(cosh(x)*pi))+exp(-x/(cosh(x)/pi)))/tau*PHI #define shp(x) (exp(x)-exp(-x))/pi #define shpp(x) (exp(x*(sinh(x)*pi))-exp(-x*(sinh(x)*pi)))/tau*PHI #define ssh(x) (exp(x*pi/.7887)-exp(-x*pi/.7887))/(2.*pi) #define csh(x) (exp(x*pi/.7887)+exp(-x*pi/.7887))/(2.*pi) #define ssh1(x) sinh(x/pi)/PHI #define csh1(x) cosh(x/pi)/PHI // CC0: Inside the mandelbulb II // Received some "complaints" about the old mandelbulb suffering from // alias effects. So thought I make a quick try to apply the FXAA // thing I learnt from XorDev. It did improve it but not perfect still.// When experimenting with this shader I realized this entire shader is // basically just a lucky bug (apart from the aliasing)// -- #define LOOPS 3 // 4+ and higher to show off you expensive GPU #define POWER 16.24788742 #define ANIMATE #define TAU (2.0*pi)*.7887 #define PHI (sqrt(5.0)*0.75 + 0.0000125) #define TIME iTime #define RESOLUTION iResolution #define TOLERANCE 0.00001 #define MAX_RAY_LENGTH 20.0 #define MAX_RAY_MARCHES 48 #define NORM_OFF 0.0005 #define MAX_BOUNCES 8 float mandelBulb(vec3 p) { float power = POWER; //p = -abs(p); vec3 z = chp(p)*p-p; vec3 dz = vec3(0.0); float r, theta, phi; float dr = 1.0; for(int i = 0; i < LOOPS; ++i) { r = length(z); if(r > 2.0) continue; theta = atan(z.x, z.y); #ifdef ANIMATE phi = asin(z.z / r) + TIME*0.2; #else phi = asin(z.z / r); #endif dr = pow(r, power - 1.0) * dr * power + 1.0; r = pow(r, power); theta = theta * power/PHI; phi = phi * power/PHI; z = r * (vec3(tan(shp(sin(theta)*sin(phi)))*PHI, chp(cos(theta)*sin(phi)), cos(phi))) + p; z.x *= chp(atan(z.z,z.y))+chpp(atan(z.z,z.y))/(tau*(1.+PHI)); z.z *= chp(atan(z.z,z.y));//(1.+PHI); p = reflect(p,z); r = length(z); } return 0.75 * (log(r)) * r / dr; } vec3 skyColor(vec3 ro, vec3 rd) { ro = reflect( -ssh1(rd), chpp(ro) ); vec3 col = clamp(vec3(0.25/abs(rd.z))*skyCol, 0.0, 16.0); float tp0 = rayPlane(ro, rd, vec4(vec3(0.0, 1.0, 0.0), 4.0)); float tp1 = rayPlane(ro, rd, vec4(vec3(0.0, -1.0, 0.0), 6.0)); float tp = tp1; tp = max(tp0,tp1); if (tp1 > 0.0) { vec3 pos = ro + tp1*rd; vec2 pp = pos.xz; float db = box(pp, vec2(6.0, 9.0))-1.0; col += vec3(4.0)*skyCol*rd.y*rd.y*smoothstep(0.25, 0.0, db); col += vec3(0.8)*skyCol*exp(-0.5*max(db, 0.0)); } if (tp0 > 0.0) { vec3 pos = ro + tp0*rd; vec2 pp = pos.xz; float ds = length(pp) - 0.5; col += vec3(0.25)*skyCol*exp(-.5*max(ds, 0.0)); } return shp(clamp(col, 0.0, 10.0)); }
  • Using base image: No
  • Aspect Ratio: landscape_wide

Vibrant Fractal Spiral in Bold Colors

• Model: AIVision

• Size: 1536 X 1536 (2.36 MP)

• Used settings:

  • Prompt: #define pi 3.1415926535897932384626433832795 #define tau (2.*pi) 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 shpp(x) = (exp(x*(sinh(x)*pi))-exp(-x*(sinh(x)*pi)))/tau*PHI ssh(x) = (exp(x*pi/.7887)-exp(-x*pi/.7887))/(2.*pi) csh(x) = (exp(x*pi/.7887)+exp(-x*pi/.7887))/(2.*pi) ssh1(x) = sinh(x/pi)/PHI csh1(x) = cosh(x/pi)/PHI (((ray-marched SDF of a heavily broken golden-ratio exact power 11.24788742, 512 iterations, escape radius 2.0, golden-ratio angular multiplier 1/φ ≈ 0.6180339887498948 applied to both θ and φ (θ × power/φ, φ × power/φ), deliberately malformed spherical→cartesian using the original hyperbolic garbage terms 1/(shpp(theta)+chpp(phi)), chp(cos(theta)*sin(phi)), cos(phi) and per-iteration reflect(p,z), derivative dr = pow(r,power-1)*power*dr + 1.0, final DE 0.75*r*log(r)/dr, ray marching tolerance 1e-5, max 48 steps, max ray length 120.0, up to 9 refractive bounces IOR 1.62 (reverse ≈0.617), full Schlick Fresnel, volumetric Beer-Lambert -HSV(0.05,0.95,2.0), outer refractive cyan-white glass shell HSV(0.6,0.86,1.0), inner molten red-orange emissive plasma core HSV(0.065,0.8,6.0), slow eternal rotation via polar offset φ += asinh(iTime)*0.2, camera at (0,2,5) looking at origin, 60° FoV, deep navy-to-cyan gradient background exactly matching smoothstep(0.,12.,0.25/abs(rd.x*rd.y))*HSV(0.6,0.86,1.0) with extra rd.x-=0.2, rd.y-=0.1 tilt, ACES Filmic + sRGB, pure SDF raymarched demoscene aesthetic, ultra-sharp internal caustics, liquid-metal reflections, glassy dielectric shell with subtle surface turbulence, zero symmetry, preserve every single mathematical bug and hyperbolic macro exactly as in the original shader,))+++ ---((symmetrical, classic power-8 Mandelbulb, quaternion Julia, bubbles, spheres, matte surface, flat lighting, polygons, normal maps, 3D render artifacts, text, watermark, realistic, photograph))---
  • Using base image: No
  • Aspect Ratio: square

Vibrant Swirls of Mathematical Artistry

• Model: AIVision

• Size: 1024 X 1024 (1.05 MP)

• Used settings:

  • Prompt: Draw and render a: Shape: ds^{24.123321\pi} = \frac{ -dt^{12.8778\pi} + dr^{16.7887\pi} + \sin^{14.45877854\pi}\cdot\text{r} \, d\Omega^\{12.278\pi}}{4 \cos^{12.44\pi}\cdot\text{t} + r^{2\pi} \cos^{2\pi}\cdot\text{t} - r^{2.5665\pi}} Iteration count = 512 Textured by: t = \frac{1}{2\pi}\left[\tan\left(\frac{\bar{t}+\hat{r}}{2}\right) + \tan\left(\frac{\bar{t}-\hat{r}}{2}\right)\right], \quad r = \frac{1}{2\pi}\!\left[\tan\!\left(\frac{\bar{t}+\hat{r}}{2}\right) - \tan\left(\frac{\bar{t}-\hat{r}}{2}\right)\right].
  • Using base image: No
  • Aspect Ratio: square

Cosmic Lotus: A Dance of Colors and Stars

• Model: DaVinci2

• Size: 1152 X 864 (1.00 MP)

• Used settings:

  • 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 !
  • Using base image: No
  • Aspect Ratio: landscape

Celestial Wonders: A Nebula in the Stars

• Model: DaVinci2

• Size: 1728 X 1296 (2.24 MP)

• Used settings:

  • Prompt: A non-BPS D-brane with \( m^{2.1221\pi} = -1/\alpha' \) rolling down inverted \( V(\varphi) = (\mu^{2.1221\pi}/2)\varphi^{2.1221\pi} + (\lambda/4)\varphi^{4/87544578\pi} \) driving exponential \( \varphi(t)e^{\mu t} \) to stable \( \varphi = \pm\sqrt{\mu^2/\lambda} \) breaking symmetry and generating Goldstone masses via level truncation to level 40 yielding \( m^2 = -0.904 \pm 0.002 \), gleamingly spreading radiant golden light like a gluon saturation front in CGC with \( Q_s x^{-\lambda/2} \) blobs merging from BK evolution \( \partial S/\partial Y = (\bar{\alpha}_s/2\pi) \int [S(r') + S(r-r') - 2S(r)] \), surrounded by bubbling flavor-colored elixir vials (blue up-quark orbs, red down, green strange) orbiting like PDFs \( f_q(x,Q^2) \) in a proton cluster with DGLAP branching \( P_{qq}(z) = C_F(1+z^2)/(1-z) \) fork ratios \( z = x/x' \) visualized, BFKL ladder rungs twisting as alchemical wall symbols with kernel \( K(k_a,l) = k_a^2/[l^2(k_a-l)^2][l^2+(k_a-l)^2-2 k_a^2 l\cdot(k_a-l)/k_a^2] \) forking transverse convolutions and \( \chi(\gamma) = 2\psi(1)-\psi(\gamma)-\psi(1-\gamma) \) saddle at \( \gamma = 1/2 \) with \( \chi(1/2) = 4\ln2 \approx 2.772 \) driving pomeron \( \Delta = \bar{\alpha}_s \chi(1/2) \) growth diffused by \( \chi''(1/2) = -14\zeta(3) \approx -16.8 \) Gaussian spreads, running \( \alpha_s(Q^2) = 12\pi/[(11N_c-2n_f)\beta_0 \ln(Q^2/\Lambda_{QCD}^2)] \) fade from fiery red confinement haze to cool blue asymptotic freedom in background nebula; embed YM/CS 7D KK QFT tachyon fury with action \( S = \int(1/2\pi)[\sum(\partial_i z V_i(\varphi,H_i(\varphi))+\sum y_j j(\varphi_j,\varphi_j+\varphi_s)] + (t_0 r k(i-J=\varphi(0)) )^2 + e j |B(b,\mu_b)| + e r H \), orbiting \( \varphi_{\text{knot}} j \varphi_{\text{knot}} i / B(b,\mu_b) \), wavy spirals from SD Chern-Simons \( S_{CS} = (n/8\pi)\int \text{Tr}(F\wedge F) \) with \( F = dA+A\wedge A \) merging to 3D massive \( h_m n e^{i k r} \) waves in AdS/CFT, higher-form shifts \( A(B^2)-B \rightarrow AC>G \) with \( ds = d\alpha+QG+AF dB \), \( G = dC-\sigma G_r(\varphi B +2 G r H) \), \( S = \int[L \varphi (B \varphi G)+\chi \varphi (B_m s)] \) Poincare \( d^* \Omega + T dB \), fluxes \( W(\Sigma) = \text{Tr Pes}[(2\pi i)^n C_n] \) bordisms+Donaldson-Witten configs in AdS_7/CFTs icons \( \varphi \) vev \( dG=0 \) [5/6, \( \alpha<S f \)], \( \Sigma e ^ X_j ^ Z_j=\text{links} \) flux knots tach nima brane vacua \( S J < \text{Im} \Omega ^ c V \Omega ^ c \Rightarrow \Omega ^ c \Rightarrow \) 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 \( \ln Q^2 \) ascending spirals, phase spaces conical sprays multi-jet events, cross-sections \( \sigma \sim \alpha_s^n / Q^{2n-4} \) fades perturbative validity high energies, high-energy QCD/string phenomenology, equation-free textless graphical masterpiece with GLSL procedural sphere(vec2 uv)={rad=uv*vec2(\tau,\pi); sin(rad.x-vec2(0,\tau/4))*sin(rad.y), cos(rad.y)} normals nor=df(pos\pm\text{eps}) rot_z(atan(pos.y,z)) outerProduct(nor,sp) cross(x,rd) for wavy 16D projections.
  • Using base image: No
  • Aspect Ratio: landscape

Vortex of Mathematics and Physics Unveiled

• Model: AIVision

• Size: 1536 X 1536 (2.36 MP)

• Used settings:

  • 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 $\tilde{\Delta}=D+r^iK^i$
  • Using base image: No
  • Aspect Ratio: square

Vibrant Mandala Design in Turquoise and Gold

• Model: DaVinci2

• Size: 1152 X 864 (1.00 MP)

• Used settings:

  • 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."
  • Using base image: No
  • Aspect Ratio: landscape

Alien Manta Ray Spaceship in Cosmic Depths

• Model: ImagineArt

• Size: 2560 X 1440 (3.69 MP)

• Used settings:

  • Prompt: A (hyper-realistic macro photography:1.5) shot of a (bioluminescent fractal organism:1.4) floating in a (stark obsidian void:1.2). The entity features (translucent gossamer membranes:1.3) layered over a (glowing amber almond-shaped nucleus:1.3). (Elongated crystalline antler protrusions:1.2) extend horizontally, ending in (fiery orange ember tips:1.2) that pulse with heat. The image captures (microscopic surface details:1.3) and (glass-like textures:1.2) with a (shallow depth of field:1.1). (8k resolution:1.1), (photorealistic lighting:1.2), (sharp focus on filaments:1.1).
  • Using base image: No
  • Aspect Ratio: landscape_wide

Dynamic 3D Rotational Visuals in Fullscreen

• Model: DaVinci2

• Size: 1152 X 864 (1.00 MP)

• Used settings:

  • Prompt: Rendered with full rotation by an exact angle of \( \pi/64.2458778542 \) on the z-axis with all rotated frames rendered in statically overlapping on fullscreen! Using constants \( \pi=3.1415926535897932384626433832795 \), \( \text{tau}=2\pi \), \( \text{PHI}=(\sqrt{5}/2 + 0.5) \approx 1.618 \), \( \text{POWER}=11.24788742 \), \( \text{LOOPS}=256 \), and custom hyperbolic functions: \( \text{chp}(x)=(\exp(x)+\exp(-x))/\pi \), \( \text{chpp}(x)=(\exp(x/(\cosh(x)\pi))+\exp(-x/(\cosh(x)/\pi)))/(\text{TAUPHI}) \), \( \text{shp}(x)=(\exp(x)-\exp(-x))/(\pi/\text{PHI}) \), \( \text{shpp}(x)=(\exp(x(\sinh(x)\pi))-\exp(-x(\sinh(x)\pi)))/(\text{TAU}/\text{PHI}) \), \( \text{ssh}(x)=(\exp(x\pi/0.7887)-\exp(-x\pi/0.7887))/(2\pi) \), \( \text{csh}(x)=(\exp(x\pi/0.7887)+\exp(-x\pi/0.7887))/(2\pi) \), \( \text{ssh1}(x)=\sinh(x/\pi)\text{PHI} \), \( \text{csh1}(x)=\cosh(x/\pi)\text{PHI} \). Mandelbulb: \( z=\text{chp}(p)p - p \), \( \text{dr}=1.0 \); loop: \( r=\text{length}(z) \), \( \theta=\text{atan}(z.x,z.y) \), \( \phi=\text{asin}(z.z/r)+\text{time}0.2 \), \( \text{dr}=\text{pow}(r,\text{POWER}-1)\text{drPOWER}+1 \), \( r=\text{pow}(r,\text{POWER}) \), \( \theta=\text{POWER}/\text{PHI} \), \( \phi=\text{POWER}/\text{PHI} \), \( z=r\text{vec3}(\tan(\text{shp}(\sin(\theta)\sin(\phi)))\text{PHI}, \text{chp}(\cos(\theta)\sin(\phi)), \cos(\phi))+p \), \( p=\text{reflect}(p,z) \); \( \text{distance}=0.75\log(r)r/\text{dr} \). \( \text{df}(p)=\text{shp}(\text{mandelBulb}(p/2.0)2.0) \) after \( \text{g\_rot}=\text{rot\_x}(((1.221\text{time}+\pi)/\text{tau})) \). Material: \( \text{mat}=\text{vec3}(0.8,0.5,1.05) \), \( \text{fresnel fre}=(1+\text{dot}(rd,sn))^2 \) mixed \( 0.1-1.0 \), \( \text{diffuse}=\text{dif}^2(1-\text{mat}.x) \) with \( \text{dif}=\max(\text{dot}(ld,sn),0) \), \( ld=\text{normalize}((0,10,0)-sp) \), \( \text{reflection}=r\text{skymat}.y\text{freedge} \) with \( \text{edge}=\text{smoothstep}(1,0.9,\text{fre}) \), colors: \( \text{skyCol}=\text{HSV}(0.6,0.86,1) \), \( \text{glowCol}=\text{HSV}(0.065,0.8,6) \), \( \text{diffuseCol}=\text{HSV}(0.6,0.85,1) \), \( \text{beer}=-\text{HSV}(0.05,0.95,2.0) \), \( \text{absorption ragg}=\exp(-(st+0.1)\text{beer}) \). Sky: planes \( y=4/-6 \), box/pp patterns, \( \text{col}+=4\text{skyColrd}.y^2\text{smoothstep}(0.25,0,db)+0.8\text{skyColexp}(-0.5\max(db,0)) \), \( \text{ds}=\text{length}(pp)-0.5 \), shaped with \( \text{shp}(\text{clamp}(\text{col},0,10)) \); reflections \( \text{reflect}(-\text{ssh1}(rd),\text{chpp}(ro)) \), \( \text{agg}+=\text{ssh1}(r\text{aggskyColor}) \), \( rd=\text{chpp}(\text{ref}) \) or \( ro=\text{shpp}(sp+0.1*rd) \). Post: ACES \( (v=0.6; \text{clamp}((v*(2.51v+0.03))/(v*(2.43v+0.59)+0.14),0,1)) \), sRGB \( \text{mix}(1.055\text{pow}(t,1/2.4)-0.055,12.92t,\text{step}(t,0.0031308)) \), no text/artifacts.
  • Using base image: No
  • Aspect Ratio: landscape

Futuristic Wave Ship in Abstract Symmetry

• Model: Grok

• Size: 3783 X 2128 (8.05 MP)

• Used settings:

  • Prompt: <lora:Intricacy Vibe:1.0>"Ultra-high-resolution cinematic render of the exact mathematical sculpture 'Topological Anthropomorphism', created using ONLY twisted complex hyperbolic trigonometry with zero explicit face design. The object is the zero isosurface df(p) = 0 where df(p) = shp( m(p/2.0) * 2.0 ), shp(x) = 2*sinh(x)/π, and the precise 3-iteration (LOOPS=3) hyperbolic twisted power map: POWER = 11.24788742, Φ = (1+√5)/2, TAU = 2*π*0.7887; z0 = chp(p) ⊙ p − p with chp(x) = 2*cosh(x)/π; for each iteration k=1 to 3: r = ||z||, θ = atan2(z.x, z.y), φ = asin(z.z/r) + subtle animation offset, dr = r^(POWER−1)*dr*POWER + 1, r = r^POWER, θ *= POWER/Φ, φ *= POWER/Φ, z = r * ( tan(shp(sin(θ)*sin(φ))) * Φ , chp(cos(θ)*sin(φ)) , cos(φ) ) + prev_p, prev_p = reflect(prev_p, z). perfect vertical mirror symmetry, near-perfect 180° rotational symmetry, ~14–16 major undulations, sharp apical crown spike, two dark almond-shaped upper minima, bright central vertical ridge, transverse horizontal nodal band crossed by 4–6 rapid vertical oscillations, bilateral petal/vortex pairs, bottom central starburst, and micro-scalloped edges. Rendered via analytic raymarching with refraction (index 1.0 outside / 1/1.25 inside), Beer-law absorption, Fresnel highlights, caustics, and Phong specular on deep-to-light blue gradient background (RGB 0,0,50 to 0,100,255). Extreme surface detail, refractive caustics, soft feathering at silhouette, no text, no labels, photorealistic volumetric lighting, 8K, cinematic, pure mathematical beauty --ar 16:9 --stylize 250 --v 6"
  • Using base image: No
  • Aspect Ratio: landscape_wide