#geometricabstract

Chrome Geometric Funnel Structure

• Model: Ideogram v4 (Pro)

• Size: 2496 X 3312 (8.27 MP)

• Used settings:

  • Prompt: To reverse-engineer the mathematics of this render, we have to dissect it on three distinct levels: **Algebraic Geometry (The Mirror Cavity)**, **Chaotic Scattering / Conformal Inversion (The Optics)**, and **Raymarching / Distance Estimation (The Rendering Pipeline)**.\n\n---\n\n### 1. The Cavity Geometry: Icosahedral Symmetry Group ($\\mathcal{I}_h$)\n\nThe bounding cage is a regular **pentagonal dodecahedron** with one or more faces removed to expose the interior.\n\n#### Mirror Planes\nA dodecahedron is governed by the full icosahedral group $\\mathcal{I}_h$ of order 120. Its 12 face planes are defined by normals $n_i$ aligned with the vertices of a dual regular icosahedron:\n\n$$n \\in \\left\\{ \\frac{1}{\\sqrt{\\phi^2 + 1}} (\\pm 1, \\pm \\phi, 0), \\; \\text{and cyclic permutations of coordinates} \\right\\}$$\n\nwhere $\\phi \u003d \\frac{1 + \\sqrt{5}}{2} \\approx 1.618$ is the golden ratio.\n\nThe inward-facing planar mirrors lie at distance $h$ from the origin:\n$$\\Pi_i : \\quad \\mathbf{x} \\cdot \\mathbf{n}_i - h \u003d 0$$\n\nPlanar reflections across these faces map any incident ray vector $\\mathbf{v}$ via the Householder reflection matrix:\n$$R_i(\\mathbf{v}) \u003d \\mathbf{v} - 2(\\mathbf{v} \\cdot \\mathbf{n}_i)\\mathbf{n}_i$$\n\n---\n\n### 2. The Spherical Core \u0026 Conformal Inversion\n\nAt the origin sits a reflective sphere of radius $R_s$:\n$$S : \\quad \\|\\mathbf{x}\\|^2 \u003d R_s^2$$\n\n#### The Geometric Inversion\nWhile the dodecahedron\u0027s walls induce affine reflections, the sphere introduces **spherical inversion** (a conformal mapping in $\\hat{\\mathbb{R}}^3$):\n$$\\mathcal{I}_S(\\mathbf{x}) \u003d \\frac{R_s^2}{\\|\\mathbf{x}\\|^2} \\mathbf{x}$$\n\nWhen an inward-directed ray hits the convex sphere at surface normal $\\mathbf{N} \u003d \\frac{\\mathbf{x}}{\\|\\mathbf{x}\\|}$, the reflected direction $\\mathbf{d}\u0027$ is:\n$$\\mathbf{d}\u0027 \u003d \\mathbf{d} - 2(\\mathbf{d} \\cdot \\mathbf{N})\\mathbf{N}$$\n\nBecause the sphere is **convex** ($\\kappa \u003e 0$), it acts as a dispersing optic. The interaction between:\n1. Flat planar foldings $\\{R_i\\}$\n2. Dispersing spherical reflections $\\{\\mathcal{I}_S\\}$\n\ncreates an **open Hamiltonian optical billiard**. \n\n#### Chaotic Scattering \u0026 Cantor Limit Sets\nLet $M$ be the bounce mapping $M : \\Sigma \\to \\Sigma$ on the Poincaré section of ray-surface intersections. Because the defocusing power of the sphere magnifies angular errors exponentially:\n$$\\delta \\theta_{k+1} \\approx \\left(1 + \\frac{2d}{R_s \\cos \\alpha_k}\\right) \\delta \\theta_k$$\n\nThe system has a strictly positive Lyapunov exponent $\\lambda \u003e 0$. The set of trapped rays that bounce indefinitely without escaping through the open face forms a **fractal repeller** (a Cantor set of measure zero) with Hausdorff dimension:\n$$0 \u003c d_H \u003c 3$$\nThe recursive rings and nested circular bands visible on the chrome surface are the visual projection of this fractal limit set (analogous to limit sets of Apollonian gaskets / Kleinian groups).\n\n---\n\n### 3. The SDF (Signed Distance Field) Formulation\n\nIn a shader (e.g., GLSL raymarching), this entire scene can be represented analytically as an implicit field $f(\\mathbf{p})$:\n\n$$\\begin{aligned}\nf_{\\text{sphere}}(\\mathbf{p}) \u0026\u003d \\|\\mathbf{p}\\| - R_s \\\\\nf_{\\text{dodec}}(\\mathbf{p}) \u0026\u003d \\max_{i\u003d1}^{12} (\\mathbf{p} \\cdot \\mathbf{n}_i - h) \\\\\nf_{\\text{edges}}(\\m
  • Using base image: No
  • Aspect Ratio: portrait

Profile in Mosaic

• Model: DaVinci2

• Size: 1264 X 944 (1.19 MP)

• Used settings:

  • Prompt: 3d titanium abstract portrait with a smooth, elegant profile. She has a distinctive hairstyle formed from swirling black and white stripes that blend seamlessly into the background. The background is composed of similar vertical stripes, creating a sense of depth and movement. The woman’s expression is serene and contemplative, with closed eyes and soft lips. The overall atmosphere is a blend of modern art and surrealism, emphasizing the harmony between the figure and the abstract design. Capture a high-contrast look with dramatic lighting that enhances the curves and lines of the hair and face.
  • Using base image: Yes

columns

• Model: DaVinci2

• Size: 1992 X 3024 (6.02 MP)

• Used settings:

  • Prompt: portrait, stylization 1.4, 1girl, liminal surreal effect, prudish, brightness 1.3, plasticity 1.3, mip-mapping style of Scott Wade and Wayne Barlow, 1girl. line-art motif by Saul Leiter & Yevgeny Zamyatin, Vibrant ink art in motion, collage art reminiscence, female, 1face, solo, embossed details, juminocore. Vibrant hatchings, intricate textures, linework of squiggly lines, pastel watercolors, dynamic scratching, splashing, blotting with a quill pen, busy, bustling, fidgety, lively, teeming irregular strokes and texturing. American Gothic-inspired woodblock print blending elements of Post-Avantgarde and pointillism, layered stylization, watery and wavery elements and shapes, abstracted composition, tesselated, tetradic coloring scheme light-pink dark-blue starfleet-blue sky-blue, foampunk details, (2/3 rule composition:0.5). FS, NixPort, novuschroma, iridescent, lacquer, prismatic light, 69yottea_style_illu, mΨh0r_a97Q, refraction, contrast side lighting, displacement mapping 1.3, 3d effects, coloring by Mike Zeck & Barclay Shaw, polished orphism, serene dimmed lighting, zentangle (watanabe tomari:1.1), (watari \(nijimukiokuiro\):0.5), soft atmospheric gradients, <lora:USNR STYLE_XL_loha:0.65>, (chiaroscuro,high contrast:0.1), (sunbeam), WAI-illustrious-SDXL, rigid texture mapping 1.3, illuminated by glimmering light blending with textures, coloring by Mike Zeck & Barclay Shaw, orphism, intricate textures, linework of squiggly lines, wet pastel watercolors, dynamic scratching, foampunk details, (2/3 rule composition:0.5). FS, NixPort, novuschroma, iridescent, permissiveness, expressed liberalness, licentiousness, slight dissoluteness, high coloring for stark contrasts, stylization 1.4, complex sketchy hatchings with watercolors, vvisible textures, american gothic 1.3, hues, tints, style of Félix Ziem, raytraced shadows 1.3, intuition, boleromix,metallic polished surfaces, coloring by Ed Binkley, glossy, shiny, joyful, phong shading 1.3, gradient, brightened complex sketchy hatchings with watercolors, heroine, Line-art, rough surfaces, hues and tints, vivid cubism, orphism, tesselated panels, refracting scales and shapes,displacement mapping, shadow mapping1.8, mythp0rt, evocative color palette, dirty aquarelle tones, splatter effects overspray of colors,mythp0rt, g0thicPXL in style of adepta sororitas, liminal, highly detailed, linework, coloring by Ed Binkley, light-brown and greyish gradients, MechaP0s, <lora:aidmaGLOW-FLUX-
  • Using base image: Yes

Vibrant Rainbow Vortex Mosaic Design

• Model: DaVinci2

• Size: 1152 X 864 (1.00 MP)

• Used settings:

  • Prompt: A colourful oil painting of a never before seen colour. Quillow a bit like yellow. Mixed with reds, blues, greens, purple, black and orange. A psychedelic groovy explosion of colour.
  • Using base image: No
  • Aspect Ratio: landscape

Vibrant Floral Mosaic in Watercolor Harmony

• Model: DaVinci2

• Size: 888 X 1336 (1.19 MP)

• Used settings:

  • Prompt: Translucent flower Hold me close Translucent flower Don't let go Quiet beauty On your face Quiet beauty Melted lace Paint a dreamy hazy breezy wet on wet watercolour artwork, using minimal paints and maximum loose water effects, EmmAI melancholy emotive colour palette, melancholy composition, minimalist, impressionist, with no text, no artifacts.
  • Using base image: Yes

Futuristic Elegance in Vibrant Colors

• Model: DaVinci2

• Size: 1088 X 1088 (1.18 MP)

• Used settings:

  • Prompt: In the style of Leonora Carrington Modigliani;Pollock,Vasarely robotic female with long a neck , red,green tones, insanely detailed and intricate, elegant, ornate, hyper realistic, super detailed, oil on cracked canvas, highly textured. splash of colorful liquid
  • Using base image: Yes

Vibrant Geometric Vortex of Science Art

• Model: (Mode 20)

• Size: 1792 X 1008 (1.81 MP)

• Used settings:

  • Prompt: {\frac {1}{\sqrt {2(1+c)}}}{\Big (}(1+c)\cos(\sech\theta ),a\sin(\csch\theta )-b\cos(\sech\theta ),a\cos(\sech\theta )+b\sin(\csch\theta ),(1+c)\sin(\csch\theta ){\Big )}
  • Using base image: No
  • Aspect Ratio: landscape_wide

Vibrant Abstract Cat Illustration in Colorful Design

• Model: DaVinci2

• Size: 1280 X 720 (0.92 MP)

• Used settings:

  • Prompt: Beautiful cat in style of Rosina Wachtmeister
  • Using base image: No
  • Aspect Ratio: landscape_wide

Vibrant Mosaic of a Stylish Woman in Hat

• Model: DaVinci2

• Size: 760 X 1352 (1.03 MP)

• Used settings:

  • Prompt: Mosaic, cubism. Post-impressionist ink wash illustration in the style of Edward Vuillard, Andrei Protsouk, Bernard Buffet, Robert McGinnis.
  • Using base image: Yes

Cosmic Harmony: A Tapestry of Unity

• Model:

• Size: 1024 X 1024 (1.05 MP)

• Used settings:

  • Prompt: a realidade , dividida em 5 aspectos da realidade, matéria, mente, emoção, energia e espírito, Arte digital, pintura digital detalhada com foco em iluminação atmosférica
  • Using base image: No
  • Aspect Ratio: square

Fag- Man

• Model:

• Size: 2981 X 1677 (5.00 MP)

• Used settings:

  • Prompt: An abstracted, colourful yet dominated by the colour red portrait, yet Art Deco influenced portrait of a Big Yellow Pac-man Hippy with big laughing face facing forward but slightly to the right with head with cigarette in the mouth, raised, constructed entirely out of 3d Lego cubes bricks facing forward with mouth with cigarette featuring multi-layered hair interwoven with mystical symbols, and sacred geometry in three-dimensional layers over a background of 3d fractal jungle plants.
  • Using base image: No
  • Aspect Ratio: landscape_wide

Bees in a Honeycomb Landscape

• Model:

• Size: 1677 X 2981 (5.00 MP)

• Used settings:

  • Prompt: An abstracted, colourful, yet Art Deco influenced portrait of a bunch of TYPICAL BEES facing forward with well defined skin raised constructed entirely out of 3d cubes facing forward with big eyes and STERN features featuring multi-layered feathers interwoven with mystical symbols, and sacred geometry in three-dimensional layers over a background of 3d fractal jungle plants.
  • Using base image: No
  • Aspect Ratio: portrait_tall