#mathematical

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

Abstract Geometry Meets Advanced Science

• Model: AIVision

• 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

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

Art Meets Mathematics

• Model: DaVinci2

• Size: 1016 X 1016 (1.03 MP)

• Used settings:

  • Prompt: Abstract painting with yellow, red, gold, blue, black and white, Basquiat style, washer gypsy woman Irish washerwoman, surrounded by mathematics and mathematical formulas and geometric drawings, carvaggio lighting, Edvard Munch design with locs and black and white illusion lines and you can see she loves math and math equations Jean-Michel Basquiat Basquiat colourful Zdzisław Beksiński Picasso Van Gogh naive art high contrast Henri Matisse Alex Grey SALVADOR DALI Ernst Ludwig Kirchner Tim Burton graffiti zentangle Jackson Pollock Francis Bacon Egon Schiele M.C. Escher Joan Miró Louis Wain Victo Ngai Enki Bilal art brut Remedios Varo milo manara primitivism Jean-Michel Basquiat Raoul Dufy abstract expressionism Ray Caesar Roberto Matta Niko Pirosmani Chaim Soutine Adam Martinakis Arshile Gorky rauschenberg Tanya Shatseva Francesco Clemente Bridget Bate Tichenor Psychosomatic surrealism Gayane Khachaturian faszinierend
  • Using base image: Yes

Was it realy liberation day on April 2nd?

• Model: AIVision

• Size: 1024 X 1024 (1.05 MP)

• Used settings:

  • Prompt: The middle aged gnome with his pointed hat takes issue with Trump’s tariff formula Δτₖ = (xₖ - mₖ) / (ε • φ • mₖ)
  • Using base image: No
  • Aspect Ratio: square

Albert Einstein in Quiet Reflection

• Model: FluX

• Size: 864 X 1152 (1.00 MP)

• Used settings:

  • Prompt: Albert Einstein in Quiet Reflection: Albert Einstein is seated at a simple wooden desk, his iconic wild white hair textured with bold impasto strokes. His thoughtful eyes, painted with clean minimalist lines, reflect his deep contemplation. The background is a gradient of soft blues and whites, emphasizing his serene yet dynamic presence. A chalkboard with faint mathematical formulas sits in the background. Primary Artist: Blissful Minimalism for clean facial details Enhancement Artist: Impasto for hair and texture-rich clothing
  • Using base image: No
  • Aspect Ratio: portrait

Colorful Pencil Sketch of Pi's Gate

• Model: AIVision

• Size: 1024 X 1024 (1.05 MP)

• Used settings:

  • Prompt: Pi is a gate on a mathematical road
  • Using base image: No
  • Aspect Ratio: square

Life, the Universe and Everything

• Model: AIVision

• Size: 4096 X 4096 (16.78 MP)

• Used settings:

  • Prompt: dripping cracked paint and ink. dark ethereal surreal cloaked hooded woman, double exposure, portrait, extremely detailed universe within, stars, galaxy, nebula, visible mathematical equations. holding a DNA strain in her hand. no-eyes, empty eye sockets, no-face. dark surrealism, hyperdetailed, 8K, High Dynamic Range (HDR), 8K resolution, 3D, Octane Render, Cinematic Post-processing, black+crimson red+purple+yellow+orange, epic award winning masterpiece, crisp quality
  • Using base image: No
  • Aspect Ratio: square

Equation of Nature: Butterfly Harmony

• Model: AIVision

• Size: 1792 X 1024 (1.84 MP)

• Used settings:

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

Golden Teal Spiral: Cosmic Infinity

• Model: Artistic

• Size: 896 X 704 (0.63 MP)

• Used settings:

  • Prompt: Life's forms emerge, bound by math - fractals and Fibonacci, nature's coded dance.
  • Using base image: No
  • Aspect Ratio: landscape

Celestial Fusion: Art Meets Science

• Model: AIVision

• Size: 1024 X 1024 (1.05 MP)

• Used settings:

  • Prompt: Math Art, beautiful woman visualizing complex math formulas, with hidden mathematical symbols in the background, dress made of mathematical graphs, Generative Art, Surreal Photography, style mix of Linda Allison, Alberto Vargas, Manfred Mohr, Andreas Lie, complex scene, intricate details, intricate filigree details, fashion by Balmain, inspired by 'A Beautiful Mind'
  • Using base image: No
  • Aspect Ratio: square

Mathematics Is Us

• Model: AIVision

• Size: 1792 X 1024 (1.84 MP)

• Used settings:

  • Prompt: Intricate geometric figure man, composed of mathematical calculations and formulas, abstract shapes, lines, triangles, circles, squares, intricate patterns, black and white contrast, sharp lines, detailed shading, monochromatic palette, digital art, high resolution, precise angles, complex design, minimalist background, precise symmetry, Escher-esque illusion, dynamic lighting to enhance depth and dimension.
  • Using base image: No
  • Aspect Ratio: landscape