#torus

Geometric Torus with Mathematical Elegance

• Model: AIVision

• Size: 1536 X 1536 (2.36 MP)

• Used settings:

  • Prompt: (initialized with:0.00125): t = \frac{1}{2.4774\pi}\left[\tan\left(\frac{\tilde{t}+\tilde{r}}{2}\right) + \tan\left(\frac{\tilde{t}-\tilde{r}}{2}\right)\right], r = \frac{1}{2.4774\pi}\left[\tan\left(\frac{\tilde{t}+\tilde{r}}{2}\right) - \tan\left(\frac{\tilde{t}-\tilde{r}}{2}\right)\right] (Iteration count:2048) 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}};
  • Upscale & Enhance: 1
  • Aspect Ratio: square

Sleek Black and Purple Textured Torus Design

• Model: DaVinci2

• Size: 1152 X 864 (1.00 MP)

• Used settings:

  • Prompt: A render of an object with power \( P = 11.24788742\pi \), fixed iterations \( \text{LOOPS} = 256 \), initialized as \( z = \text{chp}(p) * p - p \) where \( \text{chp}(x) = \frac{(\exp(x) + \exp(-x))}{\pi} \), \( \text{shp}(x) = \frac{(\exp(x) - \exp(-x))}{\pi} \), \( \text{chpp}(x) = \frac{(\exp(x / (\cosh(x) \pi)) + \exp(-x / (\cosh(x) / \pi)))}{2 \pi \Phi} \), \( \text{shpp}(x) = \frac{(\exp(x \sinh(x) \pi) - \exp(-x \sinh(x) \pi))}{2 \pi \Phi} \), \( \text{ssh1}(x) = \frac{\sinh(x / \pi)}{\Phi} \), \( \text{csh1}(x) = \frac{\cosh(x / \pi)}{\Phi} \), \( \Phi = \frac{(1 + \sqrt{5})}{2} \) golden ratio, \( \tau = 2 \pi * 0.7887 \); iteration: \( r = ||z|| \), if \( r > 2 \) continue, \( \theta = \text{asin}(z_z / r) + 0.2t \) animated, \( \varphi = \text{atan}(z_x, z_y) \), \( dr = r^{P - 1} dr P + 1 \), \( r = r^P \), \( \theta = \theta P / \Phi \), \( \varphi = \varphi P / \Phi \), \( z += r * (\tan(\text{shp}(\sin\theta \sin\varphi)) \Phi, \text{chp}(\cos\theta \sin\varphi), \cos\varphi) + p \), \( p = \text{reflect}(p, z) \), final \( \text{DE} = 0.75 \log(r) r / dr \) scaled by \( \text{shp}(\text{DE} * 2) \); ray-marched with max marches = 96, tol = 10^{-5}, bounces = 8, refraction index 1.01275, Beer absorption \( \exp(-(t + 0.1) * -\text{HSV}(0.05, 0.95, 2)) \), diffuse \( \text{HSV}(0.6, 0.85, 1) \), glow \( \text{HSV}(0.065, 0.8, 6) \), sky \( \text{HSV}(0.6, 0.86, 1) \) with warped reflections via ssh1, chpp, \( \text{fract}(\text{clamp}(0.125 / |\text{reflected cross}| * \text{skyCol}, -120, 16.547)) \); rotated by \( \text{rot\_x}((1.221 t + \pi) / \tau) \), camera at \( (0, 2, 5) * 0.6 \), FOV \( \tan(\tau / 6) \), ACES tone-mapped, sRGB gamma; central bulbous form with pink core, orange lobes, black voids, cyan shell, rainbow tunnel background. Apply: TE \otimes_{TB} TF \xrightarrow{h_E \otimes h_F} TE \otimes_{TB} TF \xrightarrow{\ \ \ \ \tau\ \ \ \ } T(E \otimes F). vec3 col = fract(clamp(vec3(0.125/abs(reflect(cross(rd,ro),reflect(shpp(ro),chpp(rd))).z))*skyCol, -120.0, 16.54788745));
  • Upscale & Enhance: 0
  • Aspect Ratio: landscape

Surreal Geometry in a Dreamlike Landscape

• Model: AIVision

• Size: 2236 X 2236 (5.00 MP)

• Used settings:

  • Prompt: Show a real-world example of Euler's identity. Include a Klein bottle, a mobius strip, or a pseudosphere as part of the image.
  • Upscale & Enhance: 5
  • Aspect Ratio: square

Colorful Torus on Black Surface

• Model: ArtFusion

• Size: 1607 X 1244 (2.00 MP)

• Used settings:

  • Prompt: A perfect depiction of a Klein bottle, drawn in the style of Theodore Roczak, but with oil pastels
  • Upscale & Enhance: 2
  • Aspect Ratio: landscape

Glow

• Model: AIVision

• Size: 1792 X 1024 (1.84 MP)

• Used settings:

  • Prompt: Purple and Pink Crystal Ball filled with a black hole surrounded by mini suns of different sizes and luminance .
  • Upscale & Enhance: 0
  • Aspect Ratio: landscape

Ethereal Galactic Torus

• Model: Artistic

• Size: 896 X 704 (0.63 MP)

• Used settings:

  • Prompt: Doxa fuego bioluminiscente
  • Upscale & Enhance: 0
  • Aspect Ratio: landscape

isogeneous

• Model: Artistic

• Size: 896 X 704 (0.63 MP)

• Used settings:

  • Prompt: isogeneous
  • Upscale & Enhance: 0
  • Aspect Ratio: landscape

torus 6.1.9

• Model: Artistic

• Size: 1449 X 1449 (2.10 MP)

• Used settings:

  • Prompt: change nothing
  • Using base image: Yes (Effect strength - 30%)
  • Base image resolution: 1MP
  • Upscale & Enhance: 2

torusausage

• Model: Artistic

• Size: 896 X 704 (0.63 MP)

• Used settings:

  • Prompt: torusausage
  • Upscale & Enhance: 0
  • Aspect Ratio: landscape

toroidal sausages

• Model: Artistic

• Size: 896 X 704 (0.63 MP)

• Used settings:

  • Prompt: toroidal sausages
  • Upscale & Enhance: 0
  • Aspect Ratio: landscape

An armchair in the shape of a torus

• Model: Artistic

• Size: 896 X 704 (0.63 MP)

• Used settings:

  • Prompt: An armchair in the shape of a torus
  • Upscale & Enhance: 0
  • Aspect Ratio: landscape

A combination of a torus and a donut

• Model: Artistic

• Size: 896 X 704 (0.63 MP)

• Used settings:

  • Prompt: A combination of a torus and a donut
  • Upscale & Enhance: 0
  • Aspect Ratio: landscape