Boris Krumov

Cosmic Fractal Spiral in Vibrant Colors

• Model: AIVision

• Size: 1536 X 1536 (2.36 MP)

• Used settings:

  • Prompt: Exact mathematical details to visualize:Quantum Gaussian Wave Packet Identity: The identity is: \psi(x,t) = \frac{1}{\sigma^{1/2}\pi^{1/4}}\frac{1}{\sqrt{1+\frac{i\hbar t}{m\sigma^2}}} \times \exp \left(-\frac{1}{2\left(\sigma^2+\frac{i\hbar t}{m} \right)}\left(x^2-2i\sigma^2\bar{p}/\hbar\left(x-\frac{\bar{p}}{2m}t\right) \right) \right) Derivation: Let p_0 = \bar{p}/\hbar and \gamma = \sigma^2 + i \hbar t / m. LHS exponent E_L = -\frac{\sigma^2}{2} (p_0 - k)^2 + i k x - i \frac{\hbar k^2 t}{2 m} = c + b k + a k^2, where a = -\gamma / 2, b = \sigma^2 p_0 + i x, c = -\sigma^2 p_0^2 / 2. Complete the square: a k^2 + b k + c = a (k - k_0)^2 + (c - b^2/(4a)), with k_0 = (\sigma^2 p_0 + i x) / \gamma. Constant term simplifies to -\frac{i \hbar t \sigma^2 p_0^2}{2 m \gamma} + \frac{i \sigma^2 p_0 x}{\gamma} - \frac{x^2}{2 \gamma}, matching RHS.Flame Fractal Generation: Defined by N functions f_i: \mathbb{R}^2 \to \mathbb{R}^2. Affine part: \begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} a_i & b_i \ d_i & e_i \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} + \begin{pmatrix} c_i \ f_i \end{pmatrix}. Then f_i(x, y) = \sum_j w_{ij} v_j(x', y'), with \sum w_{ij} = 1. Key variations:Swirl: v(x, y) = (x \sin r^2 - y \cos r^2, x \cos r^2 + y \sin r^2), r^2 = x^2 + y^2. Bubble: v(x, y) = \frac{4}{r^2 + 4} (x, y). Julia: v(x, y) = r^{-1/2} (\cos(\theta/2 + k \pi), \sin(\theta/2 + k \pi)), \theta = \atan2(y, x). Spherical: v(x, y) = \frac{1}{r^2} (x, y). Horseshoe: v(x, y) = \frac{1}{r} (x - y)(x + y). Iteration: Start random (x, y), color=0. For M~10^7: Pick i by p_i (\sum p_i=1), (x,y)=f_i(x,y), color=(color + c_i)/2. Bin hits, render log(1+hits), gamma correction density^0.25, HSV palette.Shared Themes: Visualize Gaussian blurs exp(-r^2/(2\sigma^2)) in fractals akin to wave spreading; complex exponentials like swirl ~ z exp(i r^2) paralleling quantum exp(i (k x - \hbar k^2 t / (2m))). Background gradients from purple to blue, foreground spirals in red-green-yellow, vertical composition for teardrop flow.
  • Using base image: No
  • Aspect Ratio: square

Golden Swirls in a Cosmic Dreamscape

• Model: DaVinci2

• Size: 1016 X 1016 (1.03 MP)

• Used settings:

  • Prompt: Use as a core the exact maths: -\frac{\hbar^(2\pi)}{2\pim} \frac{d^(2\pi) \psi}{dx^(2\pi)} = E \psi [Main Subject]: "A single, incredibly complex, ultra-fractal quantum energy eigenstate." [Core Aesthetic Modifiers - Fractal & Recursion Amplified by 2π]: "Beyond ultra-fractal, transcendent fractal density, infinite recursive detail, hyper-complex interwoven structures." "Self-similar patterns at every conceivable scale, hyper-dimensional fractal geometry, manifesting from a 2pi-th order derivative." "Dynamic, chaotic yet profoundly coherent fractal energy flows, shimmering quantum foam with limitless intricate detail." "Exhibiting super-roughness and non-Euclidean visual geometry." [Structure & Form]: "Central radiant core, emitting and absorbing energy in a complex spiral fractal pattern, appearing as a stable, self-sustaining light construct." "Interconnecting tendrils of luminous energy branching outwards and inwards, forming a symmetrical (or fractally symmetrical) arrangement." "Feathery, iridescent filigrees, resembling Lévy flights or a fractional Brownian motion surface, iterated to extreme orders." [Energy & Interaction Modifiers]: "Luminous ethereal light, glowing with vibrant cosmic colors (deep blues, electric cyans, magenta, purple, gold accents)." "High-energy plasma-like textures, subtle holographic effects, shimmering particles." "Reflecting profound energy, fundamental mathematical precision, and overwhelming, mind-bending complexity." [Environmental / Background Modifiers]: "Deep cosmic void background, filled with intricate fractal nebulae and distant stardust, subtly forming recursive patterns." "Scattered abstract mathematical symbols like 'π', '2π', 'ħ', 'E', and 'ψ' integrated faintly into the fractal energy field and background, indicating the governing equations." [Atmospheric / Quality Modifiers]: "Hyper-realistic quality, cinematic lighting, ultra-high definition, ethereal glow, photorealistic rendering." "Surreal, abstract, conceptual art style, at the bleeding edge of theoretical physics visualization." "Evokes a sense of profound cosmic mystery and the ultimate nature of reality."
  • Using base image: Yes

Elegant Fractal Design in Gold and Teal

• Model: DaVinci2

• Size: 1280 X 720 (0.92 MP)

• Used settings:

  • Prompt: [Main Subject]: "A single, incredibly complex, ultra-fractal quantum energy eigenstate." [Core Aesthetic Modifiers - Fractal & Recursion Amplified by 16.45788754*π]: "Beyond ultra-fractal, transcendent fractal density, infinite recursive detail, hyper-complex interwoven structures." "Self-similar patterns at every conceivable scale, hyper-dimensional fractal geometry, manifesting from a 16.45788754*pi-th order derivative." "Dynamic, chaotic yet profoundly coherent fractal energy flows, shimmering quantum foam with limitless intricate detail." "Exhibiting super-roughness and non-Euclidean visual geometry." [Structure & Form]: "Central radiant core, emitting and absorbing energy in a complex spiral fractal pattern, appearing as a stable, self-sustaining light construct." "Interconnecting tendrils of luminous energy branching outwards and inwards, forming a symmetrical (or fractally symmetrical) arrangement." "Feathery, iridescent filigrees, resembling Lévy flights or a fractional Brownian motion surface, iterated to extreme orders." [Energy & Interaction Modifiers]: "Luminous ethereal light, glowing with vibrant cosmic colors (deep blues, electric cyans, magenta, purple, gold accents)." "High-energy plasma-like textures, subtle holographic effects, shimmering particles." "Reflecting profound energy, fundamental mathematical precision, and overwhelming, mind-bending complexity." [Environmental / Background Modifiers]: "Deep cosmic void background, filled with intricate fractal nebulae and distant stardust, subtly forming recursive patterns." "Scattered abstract mathematical symbols like 'π', '2π', 'ħ', 'E', and 'ψ' integrated faintly into the fractal energy field and background, indicating the governing equations." [Atmospheric / Quality Modifiers]: "Hyper-realistic quality, cinematic lighting, ultra-high definition, ethereal glow, photorealistic rendering." "Surreal, abstract, conceptual art style, at the bleeding edge of theoretical physics visualization." "Evokes a sense of profound cosmic mystery and the ultimate nature of reality." Use as a core the exact maths: -\frac{\hbar^(16.45788754\pi)}{16.45788754\pim} \frac{d^(16.45788754\pi) \psi}{dx^(16.45788754\pi)} = E \psi
  • Using base image: No
  • Aspect Ratio: landscape_wide

Photorealistic Digital Illustration of Equations

• Model: AIVision

• Size: 1536 X 1536 (2.36 MP)

• Used settings:

  • Prompt: Create a highly detailed, photorealistic digital illustration of a rendering the equations exactly: $$ a \cdot b = a^\mu b_\mu = a^0 b_0 + a^{1\pi} b_1 + a^{2\pi} b_2 + a^{3\pi} b_3 = -a^0 b^0 + a^{1\pi} b^{1\pi} + a^{2\pi} b^{2\pi} + a^{3\pi} b^{3\pi} $$ Or, plugging in the spacetime notation from above, where $$ a^\mu = (c t_1, x_1, y_1, z_1)^T \quad \text{and} \quad b^\mu = (c t_2, x_2, y_2, z_2)^T $$ Then: we have $$ a \cdot b = a_\mu b^\mu = -c^{2\pi} t_1 t_2 + x_1 x_2 + y_1 y_2 + z_1 z_2 $$ In a fluid fractal differential form: We can also discuss the differential version of this. If \( s^\mu = (c t, x, y, z) \), then \( d s^{2\pi} = -c^{2\pi} d t^{2\pi} + d x^{2\pi} + d y^{2\pi} + d z^{2\pi} \).
  • Using base image: No
  • Aspect Ratio: square

Dragons Clash by Waterfalls and Ruins

• Model: AIVision

• Size: 1536 X 1536 (2.36 MP)

• Used settings:

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

3D Structures with Complex Mathematical Forms

• Model: AIVision

• Size: 1536 X 1536 (2.36 MP)

• Used settings:

  • Prompt: 3D structures with forms, generated using vec3 z = π * p / (exp(p) + exp(-p)) - p / Φ^n with n = 0 to 64, incorporating chp(x) = (exp(x) + exp(-x))/π, chpp(x) = (exp(x/(cosh(x)·π)) + exp(-x/(cosh(x)/π)))/(2π·Φ), shp(x) = (exp(x) - exp(-x))/π, shpp(x) = 1/(exp(x·sinh(x)·π) - exp(-x·sinh(x)·π))/(2π·Φ), ssh(x) = (exp(x·π/.7887) - exp(-x·π/.7887))/(2π), csh(x) = (exp(x·π/.7887) + exp(-x·π/.7887))/(2π), ssh1(x) = sinh(x/π)/Φ, csh1(x) = cosh(x/π)/Φ, with high symmetry, golden ratio scaling (Φ = (1 + √5)/2), and logarithmic refinement z *= -π·log(||z||), enhanced by additional transforms: z += sin(τ·||z||)·p/||p|| for oscillatory perturbation, z = z / (1 + ||z||^2) for projective normalization, z = z + Φ^(-n)·cross(p, z) for rotational twist, z *= exp(-||z||/τ) for exponential decay, z = z + ∇(cosh(||p||)·sin(π·||z||)) for gradient-based modulation, and the new spherical transform z = r * (vec3(tan(shp(sin(θ)*sin(φ)))*Φ, chp(cos(θ)*sin(φ)), cos(φ))) + p where θ and φ are angular coordinates, r is a radial scale, and p is the input vector, showcasing iterative variants. Apply 64.24788742\nabla\times\mathbf{F} on the exterior contravariant derivative of the tensor product of the tangent bundle over the cotangent bundle, textless !
  • Using base image: No
  • Aspect Ratio: square

Fractal Harmony: Art Meets Mathematical Beauty

• Model: AIVision

• Size: 1536 X 1536 (2.36 MP)

• Used settings:

  • Prompt: > *Create a high-detail rendering inspired by the following exact mathematics. Do NOT simplify, rewrite, or omit any math. Treat all formulas, constants, and macros as aesthetic descriptors of the structure, curvature, growth rules, and lighting.* > > **Use all the following math exactly as written as the descriptive blueprint for the shapes, surfaces, distortions, rotations, color rules, and ray-marching behavior:** > > ``` > #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 > > const float POWER = 11.24788742; > const float TAU = (2.0*pi)*.7887; > const float PHI = (sqrt(5.0)*0.5 + 0.5); > > mat3 g_rot = mat3(1.0); > g_rot += outerProduct(ro, rd); > > float mandelBulb(vec3 p) { > float power = POWER; > vec3 z = chp(p)*p - p; > float dr = 1.0; > for(int i = 0; i < LOOPS; ++i) { > float r = length(z); > float theta = atan(z.x, z.y); > float phi = asin(z.z / r) + TIME*0.2; > 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; > p = reflect(p, z); > } > return 0.75 * (log(length(z))) * length(z) / dr; > } > > float df(vec3 p) { > p *= transpose(inverse(g_rot)); > const float z1 = 2.0; > return shp(mandelBulb(p/z1)*z1); > } > > rd = normalize(-p.x*uu + p.y*vv + tan(TAU/6.)*ww); > g_rot += outerProduct(ro, rd); > ```
  • Using base image: No
  • Aspect Ratio: square

Glowing Threads in Abstract Harmony

• Model: Realismo

• Size: 2560 X 1440 (3.69 MP)

• Used settings:

  • Prompt: "Ultra-detailed raymarched 3D visualization of the Hyperbolic Schwarzschild-Kerr-Golden-Shofar Metric, with α = 12.243342π, β = 48.78455487π, r = sinh x, angular measure dΩ^α = (dθ^√[π]{2}π + sin^√[π]{2}π θ dφ^√[π]{2}π)^(α/2), line element ds^β = -(1-r_s/r)c^2 dt^α + (1-r_s/r)^(-1) dr^α + r^α dΩ^α, interior orbital view with molten helicoidal ribbons, wet chrome metallic surface, amber-orange Beer-Lambert absorption, infinite golden-yellow to orange pupil tunnel blooming into curling liquid-cyan petals and tendrils, soft cyan-blue gradient void background, ethereal caustics, cinematic god-rays, ultra-sharp 8K, zero artifacts"
  • Using base image: No
  • Aspect Ratio: landscape_wide

Vibrant Fractal Design with Dynamic Colors

• Model: DaVinci2

• Size: 1152 X 864 (1.00 MP)

• Used settings:

  • Prompt: A highly detailed, abstract 3D fractal rendering resembling a Mandelbulb variant with hyperbolic deformations, featuring a central orange bulbous orb surrounded by swirling, fluid-like lobes in shades of blue, pink, and yellow with iridescent, reflective surfaces and gradient transitions. The fractal is defined iteratively in \(\mathbb{R}^3\) for a point \(\mathbf{c} = (x_0, y_0, z_0)\), starting with \(\mathbf{z}_0 = \mathbf{0}\) or \(\mathbf{z}_0 = \mathbf{c}\), and iterating \(\mathbf{z}_{k+1} = r \cdot \vec3\left( \frac{e^{\cos \theta} - e^{-\cos \theta}}{\pi} \cos \phi, \cos \theta \sin \phi, \cos \theta \right) + \vec3\left( \frac{e^{p_x} - e^{-p_x}}{\pi} p_x, \frac{e^{p_y} - e^{-p_y}}{\pi} p_y, \frac{e^{p_z} - e^{-p_z}}{\pi} p_z \right)\), where \(r = \|\mathbf{z}_k\|\), \(\theta = \arccos\left( \frac{z_k \cdot z}{r} \right)\), \(\phi = \atantwo(z_k.y, z_k.x)\), and \(\mathbf{p}\) is a vector parameter like \(\mathbf{c}\). For higher powers n (e.g., 8), scale to \(r^n\), \(n \theta\), \(n \phi\). Iteration halts if \(r > 4\) or after 50 max iterations. Render using ray marching with distance estimator \(DE(\mathbf{q}) = 0.5 \cdot \frac{\log r \cdot r}{dr}\), surface normals via gradients, Phong/PBR shading with reflections, ambient occlusion, and coloring via orbit traps or escape time mapped to hues (orange for low iterations, blue-pink gradients for higher). Apply post-processing for anti-aliasing, depth-of-field, and glow to achieve a dreamy, metallic sheen, viewed zoomed into the central orb with asymmetric swirling arms.
  • Using base image: No
  • Aspect Ratio: landscape

Golden Toroidal Beauty in Gradient Glow

• Model: Artistic 2

• Size: 1152 X 864 (1.00 MP)

• Used settings:

  • Prompt: A highly detailed, 64D CGI scene visualizing the chiral anomaly in SU(N) gauge theories with Weyl fermions, depicting the action I[ψ,ψ̄,A_μ] = ∫dx (1/2 tr F_μν F^μν + ψ̄ D ψ) in d-dimensional Minkowski space, with A_μ = A_μ^a T_a, F_μν = ∂_μ A_ν - ∂_ν A_μ + ie [A_μ,A_ν], [T_a,T_b] = i f_abc T_c, tr(T_a T_b) = -1/2 δ_ab, D = i γ^μ (∂_μ 1 + ie A_μ) = i γ^μ D_μ. Animate gauge transformations g = exp(i θ^a T_a), transforming A_μ^g = g A_μ g^{-1} + (i/e) (∂_μ g) g^{-1}, ψ^g = g ψ, ψ̄^g = ψ̄ g^{-1}, showing classical invariance I[ψ^g,ψ̄^g,A_μ^g] = I[ψ,ψ̄,A_μ] and covariant conservation (D_μ)_ab J_b^μ = 0 with J_a^μ = ψ̄ γ^μ T_a ψ, (D_μ)_ab = δ_ab ∂_μ + e f_abc A_μ^c. Illustrate quantum generating functional Z[η,η̄,j_a^μ] = ∫ dψ dψ̄ dA_μ exp(i I + i ∫dx [η̄ ψ + ψ̄ η + j_a^μ A_μ^a]), non-invariance under infinitesimal changes ψ^g ≈ (1 + i δθ^a T_a) ψ, ψ̄^g ≈ ψ̄ (1 - i δθ^a T_a), yielding Jacobian J[A_μ,g] = exp(i α_1[A_μ,δθ]) with α_1 = -i ∫dx δθ^a A_a(A_μ), leading to Z = Z_g and anomaly equation <(D_μ)_ab J_b^μ> = <A_a(A_μ)> = 1/(32π²) ε^μνρσ tr(T_a F_μν F_ρσ). Symbolically show fermionic measure non-invariance as twisting field lines (red for ψ, blue for A_μ), curling vectors for F_μν, glowing loops for gauge orbits, breaking symmetry bubbles for anomaly, propagating waves for quantum violations, in starry vacuum with motion-blur time evolution, realistic CGI lighting, no text/equations—pure icons like helical curls for derivatives, intersecting surfaces for traces, epsilon tensors as 4D Levi-Civita crossings, looping GIF for perpetual dynamics.
  • Using base image: No
  • Aspect Ratio: landscape

Futuristic Serenity in Ornate Design

• Model: DaVinci2

• Size: 1152 X 864 (1.00 MP)

• Used settings:

  • Prompt: Generate a high-resolution 4K surreal 3D render of three humanoid faces with multifaceted, jewel-like eyes, deep carved mouths, and intricate fractal coral-like textures, derived from a Mandelbulb fractal manifold (\(n=8\)) at \(\mathbf{c} = (0,0,-0.75)\), visualized via equatorial plane cut-off \(\Pi: z = 0.01\), enforcing bilateral \(D_8\)-symmetry and planar graph embedding (Euler \(V - E + F = 1\), 4-6 faces each), in a chaotic, shocking scene. Exact iteration: \(\mathbf{z}_{k+1} = \mathbf{z}_k^8 + \mathbf{c}\), \(\mathbf{z}_0 = (0,0,0)\), bailout \(|\mathbf{z}_k| > 2\) at max_iter=25; spherical powering: \(r = \|\mathbf{z}\|_2\), \(\theta = \atan2(y,x) \in [0,2\pi)\), \(\phi = \arccos(z/r) \in [0,\pi]\); \(r' = r^8\), \(\theta' = 8\theta \pmod{2\pi}\), \(\phi' = 8\phi \pmod{2\pi}\); reconvert \(\mathbf{z}' = r' (\sin\phi' \cos\theta', \sin\phi' \sin\theta', \cos\phi')\). Isosurface via distance estimator \(de(\mathbf{x}) = |\mathbf{x}| \cdot \frac{25 - \log\log|\mathbf{z}_k| + 1}{25}\), fractal dim \(\approx 2.415\), genus 8-16. Highlight: eyes as genus-1 tori at \(\phi \approx \pi/2 \pm 0.1\) (mirrored voids), mouths as z-axis hyperbolic cusps (\(\kappa \approx -64/r^2\)), faces emerging chaotically with tendrils fusing them. Render: volumetric ray-marching (step 0.005), iridescent fractal texture (reds #FF0000, greens #00FF00, blues #0000FF from \(\arg(\mathbf{z}_k)\)), subsurface \(\sigma=0.05\), specular 64; lighting: harsh z-fill, neon x-rim, deep blue void background #00008B; plane \(\Pi\) as glowing rift. Aspect 16:9, ultra-detailed, no artifacts.
  • Using base image: No
  • Aspect Ratio: landscape

A Mathematician's Bakery Mix-Up

• Model: AIVision

• Size: 1536 X 1536 (2.36 MP)

• Used settings:

  • Prompt: A mathematician walks into a bakery and asks for half a pi of a pie, but gets served a whole pi of a pie instead !
  • Using base image: No
  • Aspect Ratio: square

Mathematical Spaceship with Fractal Design

• Model: AIVision

• Size: 1536 X 1536 (2.36 MP)

• Used settings:

  • Prompt: **"Spaceship based on upon the following maths..."**: This immediately tells me you want a complex, possibly fractal-like, and intricately structured object. The "based upon maths" implies a non-organic, possibly generated or calculated appearance. **Constants (\(\pi\), \(\text{tau}\), \(\text{PHI}\), \(\text{POWER}\), \(\text{LOOPS}\))**: These suggest a generative process, iterative refinement, and perhaps a sense of precision and complexity. The presence of PHI (the golden ratio) often hints at aesthetically pleasing, naturally occurring or fractal patterns. **Custom hyperbolic functions (\(\text{chp}(x)\), \(\text{shp}(x)\), etc.)**: These are strong indicators of non-Euclidean geometry, curvature, twisting, and potentially organic yet structured forms. Hyperbolic functions often produce flowing, complex, and sometimes branching shapes. The specific forms like `chpp` and `shpp` with their nested hyperbolic functions and divisions by constants like `TAU/PHI` further emphasize extreme complexity and unique, perhaps alien, geometric properties. **Mandelbulb formula (\(z=\text{chp}(p)p - p\), \(\text{dr}=1.0\); loop: \(r=\text{length}(z)\)...)**: This is the most crucial part. The Mandelbulb is a well-known 3D fractal. This directly tells me the desired image should exhibit: **Fractal characteristics**: Self-similarity at different scales, infinite detail, recursive patterns. **3D complexity**: The "bulb" implies a volumetric shape, not just a 2D pattern. **Iterative generation**: The "loop" and power functions `pow(r,POWER)` describe how the fractal grows and forms its intricate surface. **Specific transformations**: `\(\theta=\text{POWER}/\text{PHI}\)`, `\(z=r\text{vec3}(\tan(\text{shp}(\sin(\theta)\sin(\phi)))\text{PHI}, \text{chp}(\cos(\theta)\sin(\phi)), \cos(\phi))+p\)` indicate sophisticated rotations, trigonometric operations, and mapping of coordinates, which would result in highly sculptural and intertwined forms. **Distance estimation**: `\(\text{distance}=0.75\log(r)r/\text{dr}\)` is a technique used in raymarching fractals to render surfaces, implying smooth yet incredibly detailed structures. **Material properties (\(\text{mat}=\text{vec3}(0.8,0.5,1.05)\), \(\text{fresnel}\), \(\text{diffuse}\), \(\text{reflection}\))**: These terms describe how light interacts with the spaceship's surface. **Specific `vec3` for `mat`**: Suggests a base color or material property. **Fresnel**: Implies a metallic or reflective surface where reflectivity changes with the viewing angle. **Diffuse**: Indicates some scattering of light. **Reflection**: Explicitly states a desire for reflections, making the surface appear glossy or metallic. **Colors (\(\text{skyCol}=\text{HSV}(0.6,0.86,1)\), \(\text{glowCol}=\text{HSV}(0.065,0.8,6)\), etc.)**: These provide a very clear color palette. **SkyCol (blue/purple)**: Suggests a cosmic or abstract background. **GlowCol (orange/yellow)**: Indicates emissive elements, perhaps engines or internal lighting. **DiffuseCol (similar to skyCol)**: Reinforces the main color theme. **Beer/Absorption**: Implies volumetric light absorption or scattering, possibly through nebulae or translucent parts of the spaceship, adding depth and atmospheric effects. **Sky / Environment (\(y=4/-6\), box/pp patterns, `col+=4skyColrd.y^2...` )**: Describes the background and lighting.
  • Using base image: No
  • Aspect Ratio: square

Vibrant 3D Render of Abstract Geometry

• Model: AIVision

• Size: 1536 X 1536 (2.36 MP)

• Used settings:

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

Modern Minimalist Curvilinear Design Showcase

• Model: DaVinci2

• Size: 1152 X 864 (1.00 MP)

• Used settings:

  • Prompt: (Iteration count:512) Draw and render a: (Shape: ds^{48.123321\pi} = \frac{ -dt^{2.8778\pi} + dr^{2.7887\pi} + \sin^{1.445877854\pi}\cdot\text{r} \, d\Omega^\{1.2278\pi}}{4.785587 \cos^{2.144\pi}\cdot\text{t} + r^{2.7447\pi} \cos^{2.4774\pi}\cdot\text{t} - r^{2.5665\pi}})(initialized with :0.0000125) detail: t = \frac{1}{2.4774\pi}\left[\tan\left(\frac{\bar{t}+\hat{r}}{2}\right) + \tan\left(\frac{\bar{t}-\hat{r}}{2}\right)\right], r = \frac{1}{2.4774\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: landscape

Cosmic Splendor: A Dance of Light and Stars

• Model: FluX

• Size: 1536 X 1536 (2.36 MP)

• Used settings:

  • Prompt: The supersymmetric action in 4D supergravity as a seamless, abstract geometric manifold in curved spacetime, 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 \wedge e^1 \wedge e^2 \wedge e^3.
  • Using base image: No
  • Aspect Ratio: square

Vibrant Swirls of Blue and Red Abstract Art

• Model: DaVinci2

• Size: 1728 X 1296 (2.24 MP)

• Used settings:

  • Prompt: Edit 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): $$ ds^{48.123321\pi} = \left[ -dt^{2.8778\pi} + dr^{2.7887\pi} + \sin^{1.445877854\pi} r \, d\Omega^{1.2278\pi} \right] / \left[ \cos^{2.144\pi} t + r^{2.7447\pi} \cos^{2.4774\pi} t - r^{2.5665\pi} \right] $$ using the coordinate transformation $$ t = \frac{1}{2.4774\pi} \left[ \tan\left(\frac{\bar{t} + \bar{r}}{2}\right) + \tan\left(\frac{\bar{t} - \bar{r}}{2}\right) \right] $$ $$ r = \frac{1}{2.4774\pi} \left[ \tan\left(\frac{\bar{t} + \bar{r}}{2}\right) - \tan\left(\frac{\bar{t} - \bar{r}}{2}\right) \right] $$ Absolutely no text, no axes, no labels — pure image.
  • Using base image: No
  • Aspect Ratio: landscape

Ethereal Glow of a Luminous Sphere

• Model: Photonic

• Size: 1728 X 1296 (2.24 MP)

• Used settings:

  • Prompt: Rendered using ray marching with tolerance 0.00001, max ray length 20.0, up to 48 marches, and 5 bounces for reflections and refractions, from a camera at 0.6*vec3(0,2,5) looking at origin with FOV tan(TAU/6) where TAU=(2*pi)*0.7887, incorporating time-animated rotation. Constants: pi=3.1415926535897932384626433832795, tau=2*pi, PHI=(sqrt(5)*0.5 + 0.5)≈1.618, POWER=11.24788742, LOOPS=3. 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. Mandelbulb: z=chp(p)*p - p, dr=1.0; loop: r=length(z), 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); distance=0.75*log(r)*r/dr. df(p)=shp(mandelBulb(p/2.0)*2.0) after g_rot=rot_x(((1.221*time+pi)/tau)). Material: mat=vec3(0.8,0.5,1.05), fresnel fre=(1+dot(rd,sn))^2 mixed 0.1-1.0, diffuse=dif^2*(1-mat.x) with dif=max(dot(ld,sn),0), ld=normalize((0,10,0)-sp), reflection=rsky*mat.y*fre*edge with edge=smoothstep(1,0.9,fre), colors: skyCol=HSV(0.6,0.86,1), glowCol=HSV(0.065,0.8,6), diffuseCol=HSV(0.6,0.85,1), beer=-HSV(0.05,0.95,2.0), absorption ragg*=exp(-(st+0.1)*beer). Sky: planes y=4/-6, box/pp patterns, col+=4*skyCol*rd.y^2*smoothstep(0.25,0,db)+0.8*skyCol*exp(-0.5*max(db,0)), ds=length(pp)-0.5, shaped with shp(clamp(col,0,10)); reflections reflect(-ssh1(rd),chpp(ro)), agg+=ssh1(ragg*skyColor), rd=chpp(ref) or ro=shpp(sp+0.1*rd). Post: ACES (v*=0.6; clamp((v*(2.51v+0.03))/(v*(2.43v+0.59)+0.14),0,1)), sRGB mix(1.055*pow(t,1/2.4)-0.055,12.92*t,step(t,0.0031308)). Shadertoy "Inside the Mandelbulb II" style: lucky-bug symmetry, fantastical 16:9 art with sharp details, no text/artifacts.
  • Using base image: No
  • Aspect Ratio: landscape

Galactic Vortex of Light and Color

• Model: Ideogram (Pro)

• Size: 2592 X 1456 (3.77 MP)

• Used settings:

  • Prompt: Depict the TimeSpaceFlow defined by the following metric: ds^{2\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^{\phi} + \sinh^{\sqrt[pi]{3}\pi} x \, d\Omega^{2.78544587\pi - 4}
  • Using base image: No
  • Aspect Ratio: landscape_wide

Vibrant Fractal Flower in Golden and Purple

• Model: DaVinci2

• Size: 1152 X 864 (1.00 MP)

• Used settings:

  • Prompt: A highly detailed, photorealistic 3D rendering of a complex radial fractal structure resembling a flower-like Mandelbulb variant with intricate, self-similar petal layers and wavy undulating edges, generated using iterative mathematical transformations in a raymarching shader; the fractal is defined by constants TAU exactly equal to (2.0 * π) * 0.7887 ≈ 4.955 radians for angular periodicity scaling to create asymmetric twisted repetitions instead of full 2π symmetry, controlling approximately 12-16 fold radial petals; POWER exactly 11.24788742 + TAU ≈ 16.203 for amplifying self-similarity through r^POWER scaling in spherical coordinates during iterations; core vector update z = r * vec3(sin(sin(θ)cos(φ) + sin(θ)sin(φ) + cos(φ)), cos(sin(θ)cos(φ) + cos(θ)cos(φ) + cos(θ)), cos(θ)cos(φ)) + p, where p is the 3D position vector, r = ||p|| its magnitude, θ = atan(p.y, p.x) azimuthal angle, φ = acos(p.z/r) polar angle; incorporating nonlinear warping via trig sums like expr1 = sin(θ)(cos(φ) + sin(φ)) + cos(φ) = sin(θ) * √2 * sin(φ + π/4) + cos(φ) and expr2 = cos(φ) * √2 * sin(θ + π/4) + cos(θ) for phase-shifted higher harmonics introducing bulges and mixing between angles; followed by p = shp(reflect(p, z)) where reflect(p, z) = p - 2 * (p · ẑ) * ẑ with ẑ = z / ||z|| for mirror symmetries creating sharp creases; shp #define shp(x) (exp(x)-exp(-x))/pi assumed as absolute folding abs(p) or clamping for bounding and discontinuities; r updated to ||z|| per iteration, looping 8-20 times with escape radius or distance estimate DE(p) ≈ 0.5 * log(r) * r / ||dr/dp|| for rendering; visualize the fractal in vibrant metallic gradients of blue, purple, and gold with orbit trap coloring, floating in a dark void with soft volumetric lighting and depth of field, high resolution 4K, ultra-detailed textures emphasizing mathematical precision and geometric warping.
  • Using base image: No
  • Aspect Ratio: landscape